$\displaystyle=\sigma_{k}\partial^{\eta}_{k}\sigma_{j}P_{(j,\omega)}|_{U}$
$\displaystyle=\sigma_{j}\sigma_{k-1}\partial^{\eta}_{k-1}P_{(j,\omega)}|_{U}$
$\displaystyle=\sigma_{j}\sigma_{k-1}\partial^{\omega}_{j}P_{(k,\eta)}|_{U}$
$\displaystyle=\sigma_{j}\partial^{\omega}_{j}\sigma_{k}P_{(k,\eta)}|_{U}=\sigma_{k}P_{(k,\eta)}|_{U}$

shows the claim. Here, we used the block form of $P_{(j,\omega)}$ in the first
line, cubical identities in the second line, and the assumed compatibility in
the third line. ∎

###### Proof of Proposition 4.4.11.

Let
$\partial\Box[n]_{\bullet}\rightarrow\widetilde{\Psi\mathrm{Dir}}_{\bullet}(M)$
be given by the family of block Dirac operators

$\\{P_{(j,\omega)}\in\widetilde{\Psi\mathrm{Dir}}_{n-1}(M)\,:\,\partial^{\omega}_{j}P_{(k,\eta)}=\partial^{k-1}_{\eta}P_{(j,\omega)}\text{
for }j<k\\},$

where $(j,\omega)\in\\{1,\dots,n\\}\times\mathord{\mathbb{Z}}_{2}$. Since we
are given only finitely many block Dirac operators, we can pick a sufficiently
large $\rho>0$ such that the core of each $P_{(j,\omega)}$ is a subset of
$M\times\rho I^{n-1}$. All $\sigma_{j}P_{(j,\omega)}$ are again block Dirac
operators whose cores are contained in $M\times\rho I^{n}$, so they restrict
to $\\{\omega x_{j}>\rho\\}$ and all of their intersections. The compatibility
assumption yields that the operators agree on the intersections by Lemma
4.4.12.

Thus, these block Dirac operators can be patched together with a suitable
choice of the partition of unity from Definition 4.2.5 to an operator $P$ on
$\mathfrak{S}_{g}\rightarrow M\times\mathord{\mathbb{R}}^{n}\setminus\rho
I^{n}$, where $g$ is the union of all (compatible) underlying Riemannian
metrics $g_{P_{(j,\omega)}}$. Extend the metric $g$ to a block metric on
$M\times\mathord{\mathbb{R}}^{n}$ as in the proof of Proposition 3.1.8 and
denote the extension again with $g$.

Let $\chi\colon M\times\mathord{\mathbb{R}}^{n}\rightarrow[0,1]$ be a smooth
function that is identically $1$ on an open neighbourhood of $M\times\rho
I^{n}$ and identically zero on
$M\times\mathord{\mathbb{R}}^{n}\setminus(\rho+2)I^{n}$. Let $\psi\colon
M\times\mathord{\mathbb{R}}^{n}\rightarrow[0,1]$ be a function that is
identically $1$ on $\mathrm{supp}\,(1-\chi)$ and supported within the
complement $M\times\rho I^{n}$. The operator
$\mathtt{P}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\psi P(1-\chi)+{\not{\mathfrak{D}}}_{g}\chi$ is a block operator on
$M\times R^{n}$ that satisfies
$\partial^{\omega}_{j}\mathtt{P}=P_{(j,\omega)}$ because it agrees with $P$
away from $M\times(\rho+2)I^{n}$. Symbol calculus and the choice of $\chi$ and
$\psi$ imply that the principal symbol of $\mathtt{P}$ agrees with the
principal symbol of ${\not{\mathfrak{D}}}_{g}$. From
$\mathtt{P}-{\not{\mathfrak{D}}}_{g}=\psi(P-{\not{\mathfrak{D}}}_{g})(1-\chi)$
follows that $P$ differs from ${\not{\mathfrak{D}}}_{g}$ by a bounded operator
$H^{0}$. If $\mathtt{P}$ were symmetric, then it would be a block Dirac
operator and a filler. Although $\mathtt{P}$ might not be symmetric, its
formal adjoint exists and is given by

$\mathtt{P}^{\ast}=\chi{\not{\mathfrak{D}}}_{g}+(1-\chi)P\psi.$

It is again a block operator of Dirac type. Thus a filler is given by
$P_{\mathrm{fill}}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=(\mathtt{P}^{\ast}+\mathtt{P})/2$, so
$\widetilde{\Psi\mathrm{Dir}}_{\bullet}(M)$ is combinatorially contractible. ∎

We now describe the comparison map
$\Psi\mathrm{Dir}_{\bullet}(M)\rightarrow\widetilde{\Psi\mathrm{Dir}}_{\bullet}(M)$.
To this end, recall that for a block map
$P\colon\mathord{\mathbb{R}}^{n}\rightarrow\Psi\mathrm{Dir}(M)$ with
underlying block map of Riemannian metrics
$h\colon\mathord{\mathbb{R}}^{n}\rightarrow\mathrm{Riem}(M)$, the spinor
bundle decomposes $\mathfrak{S}_{\mathrm{susp}\,h}\cong\mathfrak{S}_{h}\otimes
Cl_{n,0}$ and that we have defined the _operator suspension_
$\mathrm{susp}(P)$ as the operator that corresponds under the isomorphism
$\Gamma_{c}(\mathfrak{S}_{\mathrm{susp}\,h})\cong\Gamma_{c}(\mathfrak{S}_{h}\otimes
Cl_{n,0})$ to

$\mathrm{susp}(P)=P\otimes\mathrm{id}+\sum_{k=1}^{n}\partial_{t_{k}}\cdot\nabla_{\partial_{t_{k}}}^{\mathfrak{S}_{\mathrm{susp}\,h}}.$

######  Lemma 4.4.13.

The suspensions are well defined, affine maps and assemble to a cubical map
$\Psi\mathrm{Dir}_{\bullet}(M)\rightarrow\widetilde{\Psi\mathrm{Dir}}_{\bullet}(M)$

###### Proof.

We already know from Lemma 4.2.12 that $\mathrm{susp}(P)$ is a block operator
of Dirac type. In remains to show that $\mathrm{susp}(P)$ is symmetric and
differs from ${\not{\mathfrak{D}}}_{\mathrm{susp}\,h}$ by a bounded operator.
In the following, we will abbreviate $\mathrm{susp}\,h$ with $g$.

To show that $\mathrm{susp}(P)$ is symmetric, we will show that the summands
$P^{\mathrm{ex}}=P\otimes\mathrm{id}$ and
$D\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\sum\partial_{k}\cdot\nabla_{k}^{\mathfrak{S}_{g}}$ are symmetric
operators. It is enough to consider compactly supported sections of the form
$\sigma(m,t)\otimes e_{I}$. Using Fubinis theorem we derive symmetry of
$P^{\mathrm{ex}}$ by the following lines:

$\displaystyle\left(P^{\mathrm{ex}}(\sigma_{1}\otimes e_{I}),\sigma_{2}\otimes
e_{J}\right)$
$\displaystyle=\int_{\mathord{\mathbb{R}}^{n}}\left(P_{t}\sigma_{1}(\cdot,t)\otimes
e_{I},\sigma_{2}(\cdot,t)\otimes
e_{J}\right)_{H^{0}(M,\mathfrak{S}_{h(t)})\otimes Cl_{n,0}}\mathrm{d}t$
$\displaystyle=\int_{\mathord{\mathbb{R}}^{n}}\left(P_{t}\sigma_{1}(\cdot,t)e_{I},\sigma_{2}(\cdot,t)\right)_{H^{0}(M,\mathfrak{S}_{h(t)})}\langle
e_{I},e_{J}\rangle_{Cl_{n,0}}\mathrm{d}t$
$\displaystyle=\int_{\mathord{\mathbb{R}}^{n}}\left(\sigma_{1}(\cdot,t),P_{t}\sigma_{2}(\cdot,t)\right)_{H^{0}(M,\mathfrak{S}_{h(t)})}\langle
e_{I},e_{J}\rangle_{Cl_{n,0}}\mathrm{d}t$
$\displaystyle=\dots=\left(\sigma_{1}\otimes
e_{I},P^{\mathrm{ex}}(\sigma_{2}\otimes e_{J})\right).$

The operator $D$ is symmetric as it is a $\mathfrak{S}_{g}$-twisted Dirac
operator on $Cl_{n,0}\rightarrow M\times\mathord{\mathbb{R}}^{n}$ in the sense
of [LawsonMichelsonSpin].

Symbol calculus and Theorem B.0.30 show that

$\mathrm{susp}(P)-{\not{\mathfrak{D}}}_{g}=(P-{\not{\mathfrak{D}}}_{h})\otimes\mathrm{id}\in
Op^{0}(H^{0}(\mathfrak{S}_{g})))$

because $P-{\not{\mathfrak{D}}}_{h}$ is a block map with values in pseudo
differential operators of order zero. The operator
$\mathrm{susp}(P)-{\not{\mathfrak{D}}}_{g}$ thus decomposes in the same manner
as a block operator666The only thing that prevents the operator to be a block
operator is that it only lives in $Op^{0}$ instead of
$\overline{\Psi\mathrm{DO}}^{0}$.. The proof of Lemma 4.3.1 does not require
that the operator can be approximated by actual pseudo differential operators
of order zero, so $\mathrm{susp}(P)-{\not{\mathfrak{D}}}_{g}$ induces a
bounded endomorphism on $H^{0}(\mathfrak{S}_{g})$.

We have seen in the proof of Lemma 4.2.12 that $P$ decomposes on
$U_{r}({\bm{\varepsilon}})$, provided $r$ is sufficiently large, as follows

$\displaystyle(\Phi_{\bm{\varepsilon}})_{\ast}\mathrm{susp}(P)|_{U({\bm{\varepsilon}})}$
$\displaystyle=\mathrm{susp}(P({\bm{\varepsilon}}))\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes\sum_{k=1}^{|{\mathrm{dom}\,}{\bm{\varepsilon}}|}\left((\Phi_{\bm{\varepsilon}})_{\ast}\nabla^{\mathfrak{S}_{g}}\right)_{\partial_{t_{k}}}$
$\displaystyle=\mathrm{susp}(P({\bm{\varepsilon}}))\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes\sum_{k=1}^{|{\mathrm{dom}\,}{\bm{\varepsilon}}|}\partial_{t_{k}}\cdot\frac{\partial}{\partial
t_{k}},$

which implies
$\partial^{\varepsilon}_{i}\mathrm{susp}(P)=\mathrm{susp}(\partial^{\varepsilon}_{i}P)$.

Using the isomorphism
$\mathfrak{S}_{\sigma_{n+1}g}\cong\mathfrak{S}_{g}\boxtimes Cl_{1,0}$ we can
identify $\mathrm{susp}(\sigma_{n+1}P)$ with
$\mathrm{susp}(P)\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}=\sigma_{n+1}\mathrm{susp}(P)$,
which shows that $\mathrm{susp}$ commutes with $\sigma_{n+1}$. It is a bit
cumbersome but not hard to prove that

$\mathrm{Cycl}(i,n)_{\ast}\mathrm{susp}(P)=\mathrm{susp}(\mathrm{cycl}(i,n)_{\ast}P)=\mathrm{susp}(P\circ\mathrm{cycl}(i,n)^{-1})$

for every $n\geq 1$, which yields
$\sigma_{i}\mathrm{susp}(P)=\mathrm{susp}(\sigma_{i}P)$. ∎

As in the case of psc metrics, it is not true that the operator suspension
sends block maps of invertible operators to an invertible block operator. In
order to correct this defect, we will construct a smaller model for
$\Psi\mathrm{Dir}^{\times}_{\bullet}(M)$ in the next section.

###  4.5 The Comparison Map

The operator suspension of a smooth, invertible pseudo Dirac operator valued
block map does not need to be invertible. To repair this defect, we will
construct a weakly equivalent cubical subset $B_{\bullet}$ of
$\Psi\mathrm{Dir}^{\times}_{\bullet}(M)$, which is Kan and such that the image
of the operator suspension map lies in
$\widetilde{\mathcal{R}^{+}_{\bullet}}(M)$ and
$\widetilde{\Psi\mathrm{Dir}}^{\times}_{\bullet}(M)$. Essentially,
${B_{\bullet}}$ is the subset of smooth block maps that are sufficiently
“slowly” parameterised.

Before we start, recall that an inner product $h$ on a finite dimensional
vector space $V$ induces an isomorphism $h^{\flat}\colon V\rightarrow
V^{\vee}$ via $v\mapsto h(v,\text{--})$ whose inverse is denoted by
$h_{\sharp}$. If $b$ is another bilinear form on $V$, we denote its induced
endomorphism $h_{\sharp}\circ b$ with $b^{\mathrm{op}}$.

For each block map of Riemannian metrics
$h\colon\mathord{\mathbb{R}}^{n}\rightarrow\mathrm{Riem}(M)$, the spinor
bundle decomposes into
$\mathfrak{S}_{\mathrm{susp}\,h}\cong\mathfrak{S}_{h}\otimes Cl_{n,0}$, where
$\mathfrak{S}_{h}\rightarrow M\times\mathord{\mathbb{R}}^{n}$. We need to
properly investigate the spinor connection of a suspension metric
$\mathrm{susp}\,h$. We denote the standard basis on $\mathord{\mathbb{R}}^{n}$
with $\partial_{j}=\partial_{t_{j}}$. inline]Rewrite the beginning. See Thomas
remarks!

######  Lemma 4.5.1.

Let $g_{0}\in\mathrm{Riem}(M)$ be a Riemannian metric. Denote the constant
block map with values $g_{0}$ and its suspension with the same letter. For
each block map of Riemannian metrics
$h\colon\mathord{\mathbb{R}}^{n}\rightarrow\mathrm{Riem}(M)$, the spinor
connection on $\mathfrak{S}_{\mathrm{susp}\,h}$ satisfies

$p\mathcal{G}(\mathrm{susp}\,h,g_{0})_{\ast}(\nabla^{\mathfrak{S}_{\mathrm{susp}\,h}})_{\partial_{i}}=\partial_{i}$

for all $1\leq i\leq n$.

Informally speaking, the Christoffel endomorphism
$\Gamma_{i}=\partial_{i}-\nabla^{\mathfrak{S}_{\mathrm{susp}\,h}}_{\partial_{i}}$
vanishes for all $1\leq i\leq n$.

###### Proof.

Write $g$ for $\mathrm{susp}\,h$, abuse notation with
$g_{0}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\mathrm{susp}\,g_{0}$ and let
$p\mathcal{G}(g,g_{0})\colon\mathfrak{S}_{g}\rightarrow\mathfrak{S}_{g_{0}}$
be the pre-gauge map. The push-forward
$p\mathcal{G}(g,g_{0})_{\ast}(\nabla^{\mathfrak{S}_{g}})$ is a connection
$\mathfrak{S}_{g_{0}}$. Since
$g_{0}=g_{0}\oplus\langle\cdot,\cdot\rangle_{\mathord{\mathbb{R}}^{n}}$ is a
product metric, the space of section
$\Gamma(M\times\mathord{\mathbb{R}}^{n},\mathfrak{S}_{g_{0}})$ can be
identified with
$\mathcal{C}^{\infty}\bigl{(}\mathord{\mathbb{R}}^{n},\Gamma(M,\mathfrak{S}_{g_{0}}\otimes
Cl_{n,0})\bigr{)}$, so that all $\partial_{i}$ are well defined differential
operators. Thus, the difference

$\Gamma_{i}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\partial_{i}-p\mathcal{G}(\mathrm{susp}\,h,g_{0})_{\ast}(\nabla^{\mathfrak{S}_{\mathrm{susp}}{\,h}})_{\partial_{i}}$

is a bundle endomorphism of $\mathfrak{S}_{g_{0}}$, the Christoffel
endomorphism for
$p\mathcal{G}(g,g_{0})_{\ast}(\nabla^{\mathfrak{S}_{g}})_{\partial_{i}}$. This
means, that if we pick an orthogonal frame of $TM$ complementing the standard
basis of $\mathord{\mathbb{R}}^{n}$, then the matrix representation of that
endomorphism with respect to the chosen frame yield the Christoffel symbols of
$p\mathcal{G}(g,g_{0})_{\ast}(\nabla^{\mathfrak{S}_{g}})$. (Of course, the
identity $\Gamma_{i}=0$ is independent of the chosen frame on $TM$.)

The Christoffel symbols for
$p\mathcal{G}(g,g_{0})_{\ast}(\nabla^{\mathfrak{S}_{g}})_{\partial_{i}}$ with
respect to a frame induced by $(e_{1-d},\dots,e_{0}$,
$\partial_{1}\dots,\partial_{n})$ agree with the Christoffel symbols for
$\nabla^{\mathfrak{S}_{g}}_{\partial_{i}}$ with respect to the frame induced
by $p\mathcal{G}(g_{0},g)(e_{(1-d)}),\dots,p\mathcal{G}(g_{0},g)(e_{0})$,
$\partial_{1},\dots,\partial_{n}$.

The Christoffel symbols of the spinor connection with respect to an arbitrary
orthonormal frame $(e_{i})$ on the manifold and the induced orthonormal frame
$(\sigma_{\alpha})$ on the spinor bundle is given by the formula
[LawsonMichelsonSpin]*Theorem II.4.14

$\nabla_{e_{i}}^{\mathfrak{S}_{g}}\sigma_{\alpha}=\frac{1}{2}\left(\sum_{j<k}\Gamma_{ij}^{k}e_{j}e_{k}\right)\cdot\sigma_{\alpha},$

where $\Gamma_{ij}^{k}$ are the Christoffel symbols of the Levi-Cevita
connection with respect to the orthonormal frame $(e_{i})$.

To determine $\Gamma_{i}=\Gamma^{L.C.,g}_{i}$ at $(m_{0},t_{0})$, the
Christoffel endomorphism of the Levi-Cevita connection on
$M_{t_{0}}=M\times\\{t_{0}\\}$, we will choose a local orthonormal frame that
is particularly adapted to this point. Let $(y_{1-d},\dots,y_{0})\colon
U\subset M\rightarrow V\subseteq\mathord{\mathbb{R}}^{d}$ be geodesic
coordinates for $h(t_{0})$ centered at $m_{0}$. These coordinates map $m_{0}$
to $0$. Let $\partial_{y_{(1-d)}},\dots,\partial_{y_{0}}$ be the standard
basis. The push-forward of $g=\mathrm{susp}(h)$ with respect to this
coordinate system is uniquely determined by the function $h\colon
V\rightarrow\mathord{\mathbb{R}}^{d\times d}$, which is given by
$h_{\alpha\beta}(v)=g(v)(\partial_{y_{\alpha}},\partial_{y_{\beta}})$. Observe
that $h(0)=\mathrm{id}$ and $dh=0$. Applying the Gram-Schmidt algorithm to
$(\partial_{y_{-(d-1)}},\dots,\partial_{y_{0}})$ with respect to $h(t_{0})$,
we get an orthonormal frame $f_{-(d-1)},\dots f_{0}$ on $TM\times\\{t_{0}\\}$.
In geodesic coordinates, the metric satisfies
$h(v,t_{0})=\mathrm{id}+\mathcal{O}(||v||^{2})$ around the origin. Induction
over the Gram-Schmidt algorithm shows the orthonormal basis $(f_{j})$ is
tangent to $\partial_{y_{j}}$ at the origin, i.e.,
$f_{j}(v)-\partial_{y_{j}}=0+\mathcal{O}(||v||^{2})$.

There are unique smooth functions $a_{\alpha,j}\colon
V\rightarrow\mathord{\mathbb{R}}$ such that

$f_{j}(v)=\sum_{\alpha=1-d}^{0}a_{\alpha,j}(v)\partial_{\alpha}.$

Since the $(f_{j})$ and $(\partial_{y_{j}})$ are tangent to each other, we
have $a_{\alpha,j}=\delta_{\alpha,j}$ and $\mathrm{d}a_{j,\alpha}=0$.

In these local coordinates, we can extend the frame $f_{1-d},\dots,f_{0}$ from
$V=V\times{t_{0}}$ to a frame on $V\times\mathord{\mathbb{R}}^{n}$ via

$f_{j}(v,t)\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=h(v,t)^{-1/2}\cdot f_{j}(v,t_{0}).$

(To be precise, $h(v,t)$ denotes the push-forward of the metric
$h(t)|_{U}=g|_{TU\times\\{t\\}}$ with respect to the chosen geodesic
coordinates $(y_{(1-d)},\dots,y_{0}$)).

To calculate the Christoffel symbols of the Levi-Cevita connection of
$g=\mathrm{susp}(h)$ with respect to the frame $(e_{-(d-1)},\dots e_{n})=$
$(f_{-(d-1)},\dots,f_{0},\partial_{t_{1}},\dots,\partial_{t_{n}})$ we use the
Koszul formula for the Levi-Cevita connection

$\begin{split}2g(\nabla_{X}Y,Z)=\ &Xg(Y,Z)+Yg(Z,X)-Zg(X,Y)\\\
&-g(Y,[X,Z])-g(Z,[Y,X])+g(X,[Z,Y]).\end{split}$

Let $\Gamma_{ij}^{k}=g(\nabla_{e_{i}}e_{j},e_{k})$. We have chosen an
orthonormal frame, so $g(e_{i},e_{j})\equiv\delta_{ij}$ and the upper line in
the Koszul formula vanishes identically. Thus, only the lower line
contributes.

The commutators at $(0,t_{0})$ are given by:

$\displaystyle[f_{i},f_{j}]$ $\displaystyle=[h(t)^{-1/2}(\sum
a_{\alpha,i}\partial_{y_{\alpha}}),h(t)^{-1/2}(\sum
a_{\beta,j}\partial_{y_{\beta}})]$
$\displaystyle=\sum_{\alpha,\beta}[h(t)^{-1/2}(a_{\alpha,i}\partial_{y_{\alpha}}),h(t)^{-1/2}(a_{\beta,j}\partial_{y_{\beta}})]$
$\displaystyle=\sum_{\alpha,\beta}a_{\beta,j}\partial_{y_{j}}a_{\alpha,i}-a_{\alpha,i}\partial_{y_{i}}a_{\beta,j}+a_{\alpha,i}a_{\beta,j}[h(t)^{-1/2}\partial_{y_{\alpha}},h(t)^{-1/2}\partial_{y_{\beta}}]$
$\displaystyle=\sum_{\alpha,\beta}-1/2\cdot
a_{\alpha,i}a_{\beta,j}(0,t_{0})\bigl{(}\partial_{y_{\alpha}}h(0,t_{0})\cdot\partial_{y_{\beta}}-\partial_{y_{\beta}}h(0,t_{0})\cdot\partial_{y_{\alpha}}\bigr{)}=0,$
$\displaystyle[\partial_{t_{i}},f_{j}]$
$\displaystyle=\sum_{\alpha}[\partial_{t_{i}},h^{-1/2}(t)_{v}a_{\alpha,j}(v)\partial_{y_{\alpha}}]$
$\displaystyle=\sum_{\alpha}\partial_{t_{i}}a_{\alpha,j}(v)+a_{\alpha,j}[\partial_{t_{i}},h^{-1/2}(t)_{v}\partial_{y_{\alpha}}]$
$\displaystyle=0+\sum_{\alpha}a_{\alpha,j}(0,t_{0})\cdot(-1/2\partial_{t_{i}}h(t)|_{(0,t_{0})}\cdot\partial_{y_{\alpha}})$
$\displaystyle=-1/2(\partial_{t_{i}}h)(0,t_{0})\partial_{y_{j}},$

and, of course,

$[\partial_{t_{i}},\partial_{t_{j}}]=0.$

We can now calculate the Christoffel symbols $\Gamma_{ij}^{k}$ at $(0,t_{0})$.
If two of the three indices $i,j,k$ are bigger than zero, then
$\Gamma_{ij}^{k}=0$ because either the commutator vanishes, or the commutator
is orthogonal to all $\partial_{t_{m}}$. We deal with the remaining four cases
individually: For $i,j,k\leq 0$ we have $2\Gamma_{ij}^{k}=0$, because all
commutators vanish. For $i,j\leq 0$ and $k>0$, we have

$\displaystyle 2\Gamma_{ij}^{k}$
$\displaystyle=g(e_{i},[e_{k},e_{j}])-g(e_{j},[e_{i},e_{k}])=g(f_{i},[\partial_{t_{k}},f_{j}])-g(f_{j},[f_{i},\partial_{t_{k}}])$
$\displaystyle=g(\partial_{y_{i}},-1/2\partial_{t_{k}}h\cdot\partial_{y_{j}})+g(\partial_{y_{j}},-1/2\partial_{t_{k}}h\cdot\partial_{y_{i}})$
$\displaystyle=-1/2\partial_{t_{k}}h_{ij}-1/2\partial_{t_{k}}h_{ji}=-\partial_{t_{k}}h_{ij},$

where we used that $f_{\alpha}(0,t_{0})=\partial_{y_{\alpha}}$ in the second
line. For $i,k\leq 0$ and $j>0$ we derive analogously
$2\Gamma_{ij}^{k}=\partial_{t_{j}}h_{ik}$. For $j,k\leq 0$ and $i>0$ we have:
$2\Gamma_{ij}^{k}=1/2\partial_{t_{i}}h_{jk}-(1/2)\partial_{t_{i}}h_{kj}=0$.

This implies for the Christoffel symbol of the spinor connection with respect
to the frame $(e_{-(d-1)},\dots e_{n})=$
$(f_{-(d-1)},\dots,f_{0},\partial_{t_{1}},\dots,\partial_{t_{n}})$ in the case
$i>0$ the identity

$\displaystyle\Gamma_{\partial_{i}}^{\mathfrak{S}_{g}}$
$\displaystyle=\Gamma_{i}^{\mathfrak{S}_{g}}=\frac{1}{2}\sum_{1-d\leq j<k\leq
n}\Gamma_{ij}^{k}e_{j}e_{k}=0,$

and the lemma is proven. ∎

Let
$P\colon\mathord{\mathbb{R}}^{n}\rightarrow\Psi\mathrm{DO}^{1}(\mathfrak{S})$
be a block map with underlying block map $h$. Recall that $P^{\mathrm{ex}}$ is
the operator which corresponds under the decomposition
$\mathfrak{S}_{\mathrm{susp}\,h}\cong\mathfrak{S}_{h}\otimes Cl_{n,0}$ to
$P\otimes\mathrm{id}$. The commutators
$[P^{\mathrm{ex}},\nabla_{\partial_{t_{k}}}^{\mathfrak{S}_{\mathrm{susp}h}}]$
is
$\mathcal{C}^{\infty}_{c}(\mathord{\mathbb{R}}^{n},\mathord{\mathbb{R}})$-linear,
so it restricts to an operator on $\Gamma(M\times
t,\mathfrak{S}_{\mathrm{susp}h})=\Gamma(M\times t,\mathfrak{S}_{h_{t}}\otimes
Cl_{n,0})$ that we denote with
$[P^{\mathrm{ex}},\nabla_{\partial_{t_{k}}}^{\mathfrak{S}_{\mathrm{susp}h}}]\mathord{\upharpoonleft}_{M\times
t}$.

######  Lemma 4.5.2.

The assignment

$t\mapsto[P^{\mathrm{ex}},\nabla^{\mathfrak{S}_{\mathrm{susp}\,h}}_{\partial_{j}}]\mathord{\upharpoonleft}_{M\times
t}$

is the extension of a smooth block map
$\mathord{\mathbb{R}}^{n}\rightarrow\Psi\mathrm{DO}^{1}(\mathfrak{S})$ for all
$1\leq j\leq n$.

###### Proof.

Write $g$ for $\mathrm{susp}\,h$. Pick a Riemannian metric
$g_{00}\in\mathrm{Riem}(M)$ and denote the constant map with value $g_{00}$ by
the same letter. Set
$g_{0}=\mathrm{susp}(g_{00})=g_{00}\oplus\langle\cdot,\cdot\rangle_{\mathord{\mathbb{R}}^{n}}$.

We prove the equivalent statement that

$t\mapsto
p\mathcal{G}(g,g_{0})_{\ast}\left([P,\nabla^{\mathfrak{S}_{\mathrm{susp}\,h}}_{\partial_{j}}]\right)\mathord{\upharpoonleft}_{M\times
t}$

is a smooth block map
$\mathord{\mathbb{R}}^{n}\rightarrow\Psi\mathrm{DO}^{1}(\mathfrak{S}_{g_{0}})$.

Note that

$p\mathcal{G}(g,g_{0})_{\ast}P^{\mathrm{ex}}=\left(p\mathcal{G}(h,g_{00})_{\ast}P\right)^{\mathrm{ex}}$

because both operators agree on the dense subspace spanned by sections of the
form $u\otimes f$, with $u\in\Gamma(\mathfrak{S}_{g_{00}})$ and
$f\in\Gamma_{c}(Cl_{n,0})$. Denote $p\mathcal{G}(h,g_{00})_{\ast}P$ with $Q$.
Lemma 4.5.1 implies

$[Q^{\mathrm{ex}},p\mathcal{G}(g,g_{0})_{\ast}\left(\nabla^{\mathfrak{S}_{g}}\right)_{\partial_{j}}]\mathord{\upharpoonleft}_{M\times
t}=(\partial_{j}Q)^{\mathrm{ex}}\mathord{\upharpoonleft}_{M\times t},$

where $\partial_{j}Q$ denotes the partial differential of the smooth block map
$Q\colon\mathord{\mathbb{R}}^{n}\rightarrow\Psi\mathrm{DO}^{1}(\mathfrak{S}_{g_{00}})$.
Since the partial differential of a smooth block map is again a smooth block
map, the claim follows. ∎

From now on, we will not notationally distinguish
$[P^{\mathrm{ex}},\nabla_{\partial_{t_{k}}}^{\mathfrak{S}_{\mathrm{susp}h}}]$
from its restriction to $M\times\\{t\\}$ if it does not lead to confusion.
Also, we will write $M_{t}$ for $M\times\\{t\\}$.

######  Definition 4.5.3.

Let ${B_{\bullet}}$ be the following cubical subset of
$\Psi\mathrm{Dir}^{\times}_{\bullet}(M)$: A block map
$P\in\Psi\mathrm{Dir}^{\times}_{n}(M)$ with underlying block map of metrics
$h$ belongs to ${B_{n}}$ if the following conditions are satisfied

$\displaystyle\sum_{j=1}^{n}||(\partial_{k}h)^{\mathrm{op}}||_{\mathrm{op},h_{t}}<\min\\{d^{-1},1/8\cdot
2^{-(d+1)}\mathrm{low}(t)\\},$ (4.5)
$\displaystyle\sum_{j=1}^{n}||[P^{\mathrm{ex}},\nabla^{\mathfrak{S}_{\mathrm{susp}h}}_{\partial_{j}}]||_{1,0;M_{t}}<1/32\,\mathrm{low}(t)^{2}.$
(4.6)

Here, $d=\dim M$ and
$\mathrm{low}(t)=\mathrm{low}_{P}(t)\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\inf\\{||P_{t}v||_{0,h_{t}}\,:\,||v||_{1,h_{t}}=1\\}$. The function
$\mathrm{low}_{P}$ is uniformly positive because $t\mapsto P_{t}$ is a block
map with values in invertible operators.

######  Lemma 4.5.4.

${B_{\bullet}}$ is a cubical subset of
$\Psi\mathrm{Dir}^{\times}_{\bullet}(M)$.

###### Proof.

Given $U=\mathord{\mathbb{R}}_{<0}$ and $X=\Psi\mathrm{Dir}^{\times}(M)$,
which is an open subset of the affine Fréchet space $\Psi\mathrm{Dir}(M)$.
Then

$f_{n}(P)\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\sum_{k=1}^{n}||(\partial_{t_{k}}h)^{\mathrm{op}}||_{\mathrm{op},h_{t}}-\min\\{d^{-1},1/8\cdot
2^{-(d+1)}\mathrm{low}(t)\\}$

and

$g_{n}(P)\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\sum_{k=1}^{n}||[P^{\mathrm{ex}},\nabla_{\partial_{k}}^{\mathfrak{S}_{\mathrm{susp}h}}]||_{1,0;M_{t}}-1/32\,\mathrm{low}(t)^{2}$

are local and stable sequences of continuous maps
$\mathcal{C}^{\infty}(\mathord{\mathbb{R}}^{n},X)\rightarrow\mathcal{C}^{0}(\mathord{\mathbb{R}}^{n},\mathord{\mathbb{R}})$.
By definition, ${B_{n}}=f_{n}^{-1}(U)\cap g_{n}^{-1}(U)$. Since intersections
of two cubical subsets are again cubical subsets, the claim follows from Lemma
2.2.7. ∎

Next we will show that $\mathrm{susp}|_{{B_{\bullet}}}$ takes values in
$\widetilde{\Psi\mathrm{Dir}}^{\times}_{\bullet}(M)$ by showing that
$\mathrm{susp}(P)^{2}$ is invertible for all $P\in{B_{n}}$.

######  Lemma 4.5.5.

Let $P\colon\mathord{\mathbb{R}}^{n}\rightarrow\Psi\mathrm{Dir}(M)^{\times}$
be a smooth block map and set
$g\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\mathrm{susp}(h)$. For each
$u\in\Gamma_{c}(M\times\mathord{\mathbb{R}}^{n},\mathfrak{S}_{h})$, we have

$\displaystyle\mathrm{susp}(P)^{2}(u)=(P^{2})^{\mathrm{ex}}(u)$
$\displaystyle-\sum_{j=1}^{n}\partial_{j}\cdot[P^{\mathrm{ex}},\nabla^{\mathfrak{S}_{g}}_{\partial_{j}}](u)$
$\displaystyle-\sum_{j=1}^{n}\left(\nabla_{\partial_{j}}^{\mathfrak{S}_{g}}\right)^{2}(u)+\sum_{1\leq
j<k\leq n}\mathfrak{R}^{\mathfrak{S}_{g}}(\partial_{j},\partial_{k})(u).$

###### Proof.

We abbreviate $\nabla_{\partial_{j}}^{\mathfrak{S}_{g}}$ to $\nabla_{j}$ and
$\sum_{j=1}^{n}\partial_{j}\cdot\nabla_{j}$ to ${\not{\mathfrak{D}}}$.
Clearly,

$\mathrm{susp}(P)^{2}=(P^{\mathrm{ex}})^{2}+P^{\mathrm{ex}}{\not{\mathfrak{D}}}+{\not{\mathfrak{D}}}P^{\mathrm{ex}}+{\not{\mathfrak{D}}}^{2}$

and $(P^{2})^{\mathrm{ex}}=(P^{\mathrm{ex}})^{2}$. Recall that
$P^{\mathrm{ex}}$ is an odd operator. This yields

$\displaystyle P^{\mathrm{ex}}{\not{\mathfrak{D}}}u$
$\displaystyle=\sum_{j=1}^{n}P^{\mathrm{ex}}(\partial_{j}\cdot\nabla_{j}u)$
$\displaystyle=\sum_{j=1}^{n}(-1)\partial_{j}\cdot
P^{\mathrm{ex}}(\nabla_{j}u)$

implying

$\displaystyle
P^{\mathrm{ex}}{\not{\mathfrak{D}}}u+{\not{\mathfrak{D}}}P^{\mathrm{ex}}u$
$\displaystyle=\sum_{j=1}^{n}\partial_{j}\cdot\left(-P^{\mathrm{ex}}\nabla_{j}u+\nabla_{j}P^{\mathrm{ex}}u\right)$
$\displaystyle=-\sum_{j=1}^{n}\partial_{j}\cdot[P^{\mathrm{ex}},\nabla_{j}]u.$

The last summand is given by

$\displaystyle{\not{\mathfrak{D}}}^{2}$
$\displaystyle=\sum_{j=1}^{n}\sum_{k=1}^{n}\partial_{j}\cdot\nabla_{j}(\partial_{k}\cdot\nabla_{k}u)$
$\displaystyle=\sum_{j,k}\partial_{j}\partial_{k}\cdot\nabla_{j}\nabla_{k}u+\partial_{j}\cdot\underbrace{\nabla_{j}\partial_{k}}_{=0}\cdot\nabla
u$
$\displaystyle=\sum_{j=1}^{n}\partial_{j}^{2}\cdot\nabla_{j}\nabla_{j}u+\sum_{j<k}\partial_{j}\partial_{k}\cdot\left(\nabla_{j}\nabla_{k}-\nabla_{k}\nabla_{j}\right)u$
$\displaystyle=\sum_{j=1}^{n}-\nabla_{j}^{2}u+\sum_{j<k}\partial_{j}\partial_{k}\cdot
R^{\nabla^{\mathfrak{S}_{g}}}(\partial_{j},\partial_{k})u$
$\displaystyle=\sum_{j=1}^{n}-\nabla_{j}^{2}u+\sum_{j<k}\mathfrak{R}(\partial_{j},\partial_{k})u.$

Putting everything together gives the claim. ∎

######  Theorem 4.5.6.

For each $P\in{B_{n}}$, the operator $\mathrm{susp}(P)$ is invertible.
Consequently, the diagram

$\textstyle{{B_{\bullet}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{susp}}$$\textstyle{\Psi\mathrm{Dir}^{\times}_{\bullet}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{susp}}$$\textstyle{\widetilde{\Psi\mathrm{Dir}}^{\times}_{\bullet}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{\Psi\mathrm{Dir}}_{\bullet}(M)}$

commutes.

The strategy of the proof is actually quite easy. By Lemma 4.5.5,
$\mathrm{susp}(P)^{2}$ has a strictly positive summand, a non-negative one,
and two summands we need to control. We will show that the conditions we have
imposed on ${B_{\bullet}}$ guarantee that the two remaining terms have small
operator norm so that $\mathrm{susp}(P)^{2}$ remains strictly positive.

In order to prove this we need to derive several analytical results that allow
us to control the not necessarily non-negative summands of
$\mathrm{susp}(P)^{2}$.

######  Lemma 4.5.7.

Let $U\subseteq\mathord{\mathbb{R}}^{n}$ be open. For a smooth map $g\colon
U\rightarrow\mathrm{Riem}(M)$ of Riemannian metrics, let $R$ be the Riemannian
curvature tensor of $\mathrm{susp}(g)$. If $\partial_{i}=\partial_{t_{i}}$ is
the standard basis of $\mathord{\mathbb{R}}^{n}$, then

$R(\partial_{i},\partial_{j})=\frac{1}{4}[(\partial_{j}g)^{\mathrm{op}},(\partial_{i}g)^{\mathrm{op}}].$

###### Proof.

We prove this formula in local coordinates. Pick a local holonomic frame of
$M\times\mathord{\mathbb{R}}^{n}$ that extends
$\partial_{1},\dots,\partial_{n}$. This “spacial” frame is indexed by $1-d\leq
j\leq 0$. Using Einstein’s sum convention, the Christoffel symbols of the
Levi-Cevita connection of $\mathrm{susp}(g)$ in these coordinates are given by

$\displaystyle 2\Gamma_{t_{i}j}^{k}$
$\displaystyle=\mathrm{susp}(g)^{k\alpha}(\mathrm{susp}(g)_{t_{i}\alpha,j}+\mathrm{susp}(g)_{\alpha
j,t_{i}}-\mathrm{susp}(g)_{t_{i}j,\alpha})$
$\displaystyle=\mathrm{susp}(g)^{k\alpha}\mathrm{susp}(g)_{\alpha j,t_{i}}$
$\displaystyle=\begin{cases}0&\text{if }\mathrm{max}\\{j,k\\}>0,\\\
\sum_{\alpha<0}g^{k\alpha}g_{j\alpha,t_{i}}&\text{if }\max\\{j,k\\}\leq
0.\end{cases}$

Thus, we can identify the endomorphism $2\Gamma_{t_{i}}$ with
$g^{-1}\partial_{t_{i}}g=(\partial_{t_{i}}g)^{\mathrm{op}}$. Abbreviating
$t_{i}$ and $t_{j}$ with $i$ and $j$, respectively, we get for the Riemannian
curvature of $\mathrm{susp}(g)$ the following expression:

$\displaystyle R(\partial_{i},\partial_{j})$
$\displaystyle=(\partial_{i}+\Gamma_{i})(\partial_{j}+\Gamma_{j})-(\partial_{j}+\Gamma_{j})(\partial_{i}+\Gamma_{i})$
$\displaystyle=\partial_{i}\Gamma_{j}-\partial_{j}\Gamma_{i}+\Gamma_{i}\Gamma_{j}-\Gamma_{j}\Gamma_{i}$
$\displaystyle\begin{split}&=-1/2g^{-1}(\partial_{i}g)g^{-1}(\partial_{j}g)+1/2g^{-1}(\partial_{j}g)g^{-1}\partial_{i}g\\\
&\qquad+1/4g^{-1}(\partial_{i}g)g^{-1}(\partial_{j}g)-1/4g^{-1}(\partial_{j}g)g^{-1}(\partial_{i}g)\end{split}$
$\displaystyle=1/4[g^{-1}\partial_{j}g,g^{-1}\partial_{i}g]=1/4[(\partial_{j}g)^{\mathrm{op}},(\partial_{i}g)^{\mathrm{op}}].$

and the lemma is proven. ∎

######  Corollary 4.5.8.

The operator norm of the bundle endomorphism
$\mathfrak{R}_{t}(\partial_{j},\partial_{k})\in\mathrm{End}(\mathfrak{S}|_{M_{t}})$
is bounded by

$2||\mathfrak{R}_{t}(\partial_{j},\partial_{k})||_{\mathrm{op},t}\leq
d^{2}||(\partial_{j}g)^{\mathrm{op}}||_{\mathrm{op},t}||(\partial_{k}g)^{\mathrm{op}}||_{\mathrm{op},t}$

for every $t\in U$, where $d=\dim M$. This implies

$\sum_{1\leq j<k\leq
n}\left|\left|\mathfrak{R}_{t}(\partial_{j},\partial_{k})\right|\right|_{\mathrm{op},t}\leq\frac{d^{2}}{2}\left(\sum_{j=1}^{n}||(\partial_{j}g)^{\mathrm{op}}||_{\mathrm{op},t}\right)^{2}.$

Here, $||\text{--}||_{\mathrm{op},t}$ denotes the operator norm on
$\mathfrak{S}_{g(t)}$ on the left hand side and the operator norm on $TM_{t}$
with respect to $g(t)$ on the right hand side.

###### Proof.

Pick a local orthonormal frame $e_{1-d},\dots,e_{0}$ on $TM$ that completes
$\partial_{1},\dots,\partial_{n}$ to a local orthonormal frame on
$TM\oplus\mathord{\mathbb{R}}^{n}$ Then, by [LawsonMichelsonSpin]*Theorem
II.4.15, the curvature operator of the spinor connection is given by

$\mathfrak{R}(\partial_{i},\partial_{j})=\frac{1}{2}\sum_{k<l\leq 0}\langle
R_{\partial_{i},\partial_{j}}(e_{k}),e_{l}\rangle e_{k}e_{l}\cdot.$

This yields

$\displaystyle 2||\mathfrak{R}(\partial_{i},\partial_{j})||_{\mathrm{op},t}$
$\displaystyle=\left|\left|\sum_{k<l\leq 0}\langle
R_{\partial_{i}\partial_{j}}(e_{k}),e_{l}\rangle
e_{k}e_{l}\right|\right|_{\mathrm{op},t}$ $\displaystyle\leq\sum_{k<l\leq
0}|\langle
R_{\partial_{i}\partial_{j}}(e_{k}),e_{l}\rangle|\underbrace{||e_{k}e_{l}\cdot||_{\mathrm{op},t}}_{=1}$
$\displaystyle\leq\sum_{k<l\leq
0}||R_{\partial_{i}\partial_{j}}||_{\mathrm{op},TM_{t}}$
$\displaystyle=\sum_{k<l\leq
0}1/4\cdot||[(\partial_{j}g)^{\mathrm{op}},(\partial_{i}g)^{\mathrm{op}}]||_{\mathrm{op},t}$
$\displaystyle\leq
2d^{2}||(\partial_{i}g)^{\mathrm{op}}||_{\mathrm{op},t}||(\partial_{j}g)^{\mathrm{op}}||_{\mathrm{op},t}.$

∎

Since $\mathfrak{S}_{\mathrm{susp}\,h}|_{M_{t}}\cong\mathfrak{S}_{h(t)}\otimes
Cl_{n,0}$ we have two canonical connections on this restriction, namely
$\nabla^{\mathfrak{S}_{\mathrm{susp}h}|_{M_{t}}}$ and
$\nabla^{\mathfrak{S}_{h(t)}\otimes\mathrm{id}}$. We denote the first
connection with $\nabla^{\mathrm{susp}}$ and the second with
$\nabla\otimes\mathrm{id}$. Of course, since $M$ is compact, these two
connections yield equivalent Sobolev norms
$||\text{--}||_{1,\nabla\otimes\mathrm{id}}$ and
$||\text{--}||_{1,\nabla^{\mathrm{susp}}}$ on
$\mathfrak{S}_{\mathrm{susp}h}|_{M_{t}}$. The next lemma helps to find
explicit constants.

######  Lemma 4.5.9.

The bundle endomorphism valued 1-form

$\mathcal{E}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\nabla\otimes\mathrm{id}-\nabla^{\mathrm{susp}}\colon
TM\rightarrow\mathrm{End}\left(\mathfrak{S}_{\mathrm{susp}\,h}|_{M_{t}}\right)$

induces a bounded operator $L^{2}(M_{t};TM_{t})\rightarrow
L^{2}(M_{t};\mathrm{End}(\mathfrak{S}_{\mathrm{susp}\,h}|_{M_{t}})$ that
satisfies the following operator norm estimation:

$||\nabla\otimes\mathrm{id}-\nabla^{\mathrm{susp}}||_{0,0}\leq
d/4\sum_{k=1}^{n}||\partial_{t_{k}}h^{\mathrm{op}}||_{\mathrm{op},h(t)}.$

###### Proof.

Let $e_{-(d-1)},\dots,e_{0}$ be a local orthonormal frame of $M$ with respect
to $h(t)$. With respect to this frame, $\mathcal{E}$ can be expressed in terms
of the Christoffel symbols of $\nabla\otimes\mathrm{id}$ and
$\nabla^{\mathrm{susp}}$, which we denote as $\Gamma^{h}$ and
$\Gamma^{\mathrm{susp}}$, respectively. Let $\Gamma_{ij}^{k}$ denote the
Christoffel symbols of $\mathrm{susp}(h)$ on
$M\times\mathord{\mathbb{R}}^{n}$, and denote $\partial_{t_{k}}$ by $e_{k}$
for $1\leq k\leq n$. For all $1-d\leq i\leq 0$ we have by
[LawsonMichelsonSpin]*Theorem II.4.14 and the proof of Lemma 4.5.1 the
following expression

$\displaystyle\mathcal{E}(e_{i})$
$\displaystyle=\Gamma_{i}^{h}-\Gamma^{\mathrm{susp}}_{i}=\frac{1}{2}\left(\sum_{j<k\leq
0}\Gamma_{ij}^{k}e_{j}e_{k}-\sum_{-(d-1)\leq j<k\leq
n}\Gamma_{ij}^{k}e_{j}e_{k}\right)$
$\displaystyle=\frac{1}{2}\sum_{j<k,\,k>0}-\Gamma^{k}_{ij}e_{j}e_{k}$
$\displaystyle=\frac{1}{4}\sum_{j\leq
0<k}\partial_{t_{k}}h_{ij}e_{j}\partial_{t_{k}}=\frac{1}{4}\sum_{k>0}(\partial_{t_{k}}h)^{\mathrm{op}}(e_{i})\partial_{t_{k}}.$

Using this identity, we can estimate the operator norm of $\mathcal{E}$ as
follows:

$\displaystyle 4||\mathcal{E}||_{0,0}$
$\displaystyle\leq\sum_{i=1-d}^{0}4||\mathcal{E}(e_{i})||_{\mathrm{End}(\mathfrak{S}_{\mathrm{susp}(h)})}$
$\displaystyle\leq\sum_{i=1-d}^{0}\sum_{k=1}^{n}||\partial_{t_{k}}h^{\mathrm{op}}(e_{i})\partial_{t_{k}}\cdot||_{\mathrm{End}(\mathfrak{S}_{\mathrm{susp}(h)})}$
$\displaystyle=\sum_{i=1-d}^{0}\sum_{k=1}^{n}||\partial_{t_{k}}h^{\mathrm{op}}(e_{i})||_{TM}$
$\displaystyle\leq
d\sum_{k=1}^{n}||\partial_{t_{k}}h^{\mathrm{op}}||_{\mathrm{op},h(t)}.$

Here, $||\text{--}||_{\mathrm{End}(\mathfrak{S}_{\mathrm{susp}(h)})}$ denotes
the usual $C^{\ast}$-algebra operator norm coming from the inner product on
the fibres of $\mathfrak{S}_{\mathrm{susp}(h)}$ and $||\text{--}||_{TM_{t}}$
denotes the norm on $TM$ induced by $h(t)$. Note that we used in the third
line that the endomorphim is given by the Clifford multiplication, so we could
use $C^{\ast}$-identity and the Clifford identity. ∎

######  Corollary 4.5.10.

If $||\text{--}||_{1,\nabla^{\mathrm{susp}}}$ and
$||\text{--}||_{1,\nabla\otimes\mathrm{id}}$ denote the Sobolev norms on
$\mathfrak{S}_{\mathrm{susp}h}|_{M_{t}}$ with respect to
$\nabla^{\mathrm{susp}}$ and $\nabla\otimes\mathrm{id}$, respectively, then
they are equivalent by the following constants:

$\displaystyle||\text{--}||_{1,\nabla^{\mathrm{susp}}}$
$\displaystyle\leq\left(1+d\sum_{k=1}^{n}||\partial_{t_{k}}h||_{\mathrm{op}}\right)||\text{--}||_{1,\nabla\otimes\mathrm{id}}$
$\displaystyle\leq\left(1+d\sum_{k=1}^{n}||\partial_{t_{k}}h||_{\mathrm{op}}\right)^{2}||\text{--}||_{1,\nabla^{\mathrm{susp}}}.$

###### Proof.

Using the generalised triangle inequality and Lemma 4.5.9, we estimate

$\displaystyle||\text{--}||_{1,\nabla^{\mathrm{susp}}}$
$\displaystyle\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\left(||\text{--}||_{0}^{2}+||\nabla^{\mathrm{susp}}(\text{--})^{2}||_{0}\right)^{1/2}$
$\displaystyle=\left(||\text{--}||_{0}^{2}+||\nabla\otimes\mathrm{id}(\text{--})+(\nabla^{\mathrm{susp}}-\nabla\otimes\mathrm{id})(\text{--})^{2}||_{0}\right)^{1/2}$
$\displaystyle\leq\left(||\text{--}||_{0}^{2}+||\nabla\otimes\mathrm{id}(\text{--})^{2}||_{0}\right)^{1/2}+||(\nabla^{\mathrm{susp}}-\nabla\otimes\mathrm{id})(\text{--})||_{0}$
$\displaystyle\leq||\text{--}||_{1,\nabla\otimes\mathrm{id}}+||\nabla^{\mathrm{susp}}-\nabla\otimes\mathrm{id}||_{0,0}\cdot||\text{--}||_{0}$
$\displaystyle\leq\left(1+d\sum_{k=1}^{n}||\partial_{t_{k}}h||_{\mathrm{op}}\right)||\text{--}||_{1,\nabla\otimes\mathrm{id}}.$

The proof of the second estimate is similar. ∎

###### Proof of Theorem 4.5.6.

Let $P\in{B_{n}}$ with underlying block map $h\in\mathcal{R}^{+}_{n}(M)$. Set
$g=\mathrm{susp}(h)$. We will prove the theorem by showing that
$\mathrm{susp}(P)^{2}$ is invertible. Recall from Lemma 4.5.5 that

$\displaystyle\mathrm{susp}(P)^{2}$
$\displaystyle=(P^{\mathrm{ex}})^{2}+\\{P^{\mathrm{ex}},{\not{\mathfrak{D}}}\\}+{\not{\mathfrak{D}}}^{2}$
$\displaystyle=(P^{2})^{\mathrm{ex}}-\sum_{j=1}^{n}\partial_{j}\cdot[P^{\mathrm{ex}},\nabla_{j}^{\mathfrak{S}_{g}}]-\Delta+\sum_{1\leq
j<k}\mathfrak{R}(\partial_{j},\partial_{k}).$

Note that

$-\Delta=\sum_{j=1}^{n}(\nabla^{\mathfrak{S}_{g}}_{\partial_{j}})^{2}$

is a non-negative operator. The proof is analogous to the one for the Dirac
operator presented in [LawsonMichelsonSpin]*Proposition II.5.3. Since $P$ is
invertible, $P^{2}$ is strictly positive. We have, for each compactly
supported smooth section, the following estimate

$\displaystyle\langle(P^{\mathrm{ex}})^{2}u,u\rangle$
$\displaystyle=||P^{\mathrm{ex}}(u)||_{0}^{2}=\int_{\mathord{\mathbb{R}}^{n}}||P_{t}u||^{2}_{0,\mathfrak{S}_{g}|_{M_{t}}}\mathrm{d}t$
$\displaystyle\geq\int_{\mathord{\mathbb{R}}^{n}}\mathrm{low}(t)^{2}||u||^{2}_{1,\mathfrak{S}_{g}|_{M_{t}},\nabla\otimes\mathrm{id}}\mathrm{d}t$
$\displaystyle>\frac{1}{4}\int\mathrm{low}(t)^{2}||u||^{2}_{1,\mathfrak{S}_{g}|_{M_{t}},\nabla^{\mathrm{susp}}}\mathrm{d}t.$

Here, we used in the first estimate that
$H^{1}(\mathfrak{S}_{g}|_{M_{t}},\nabla\otimes\mathrm{id})$ is isometric
isomorphic to $H^{1}(\mathfrak{S}_{h(t)},\nabla)\otimes Cl_{n,0}$, so that we
can apply $||P_{t}u||_{0}\geq\mathrm{low}(t)^{2}||u||_{1}$ component-wise. The
second estimate follows from the equivalence of norms (Corollary 4.5.10) and
condition (4.5) of ${B_{\bullet}}$.

In the following estimate we denote by $||\text{--}||_{1;t}$ the Sobolev norm
on $\mathfrak{S}_{g}|_{M_{t}}$ with respect to $\nabla^{\mathrm{susp}}$:

$\displaystyle\left|\langle\sum_{j=1}^{n}[P^{\mathrm{ex}},{\not{\mathfrak{D}}}]u,u\rangle\right|$
$\displaystyle\leq\int_{\mathord{\mathbb{R}}^{n}}\left|\sum_{j=1}^{n}\langle\partial_{j}\cdot[P^{\mathrm{ex}},\nabla^{\mathfrak{S}_{g}}_{\partial_{j}}]u_{t},u_{t}\rangle\right|_{0;t}\mathrm{d}t$
$\displaystyle\leq\int_{\mathord{\mathbb{R}}^{n}}||\sum_{j}\partial_{j}\cdot[P^{\mathrm{ex}},\nabla^{\mathfrak{S}_{g}}_{\partial_{j}}]_{t}u_{t}||_{0;t}||u_{t}||_{0;t}\mathrm{d}t$
$\displaystyle\leq\int_{\mathord{\mathbb{R}}^{n}}||\sum_{j}\partial_{j}\cdot[P^{\mathrm{ex}},\nabla^{\mathfrak{S}_{g}}_{\partial_{j}}]_{t}||_{1,0;t}||u_{t}||_{1;t}^{2}\mathrm{d}t$
$\displaystyle\leq\int_{\mathord{\mathbb{R}}^{n}}\sum_{j=1}^{n}\underbrace{||\partial_{j}\cdot||_{0,0;t}}_{=1}||[P^{\mathrm{ex}},\nabla^{\mathfrak{S}_{g}}_{\partial_{j}}]||_{1,0;t}||u_{t}||^{2}_{1;t}\mathrm{d}t$
$\displaystyle<\int_{\mathord{\mathbb{R}}^{n}}1/32\cdot\mathrm{low}(t)^{2}||u_{t}||^{2}_{1;t}\mathrm{d}t.$

We estimate the fourth summand with the help of Corollary 4.5.8 in a similar
manner and notation as before

$\displaystyle|\langle\sum_{1\leq j<k\leq
n}\mathfrak{R}(\partial_{j},\partial_{k})(u),u\rangle|$
$\displaystyle\leq\int_{\mathord{\mathbb{R}}^{n}}|\langle\sum_{j<k}\mathfrak{R}_{t}(\partial_{j},\partial_{k})(u_{t}),u_{t}\rangle|\mathrm{d}t$
$\displaystyle\leq\int_{\mathord{\mathbb{R}}^{n}}||\sum_{j<k}\mathfrak{R}_{t}(\partial_{j},\partial_{k})||_{0,0;t}||u_{t}||_{0;t}^{2}\mathrm{d}t$
$\displaystyle\leq\int_{\mathord{\mathbb{R}}^{n}}||\sum_{j<k}\mathfrak{R}_{t}(\partial_{j},\partial_{k})||_{\mathrm{op},t}||u_{t}||_{1;t}^{2}\mathrm{d}t$
$\displaystyle\leq\int_{\mathord{\mathbb{R}}^{n}}d^{2}/2\left(\sum_{j=1}^{n}||(\partial_{j}h)^{\mathrm{op}}||_{\mathrm{op},t}\right)^{2}||u||_{1;t}^{2}\mathrm{d}t$
$\displaystyle<\int_{\mathord{\mathbb{R}}^{n}}1/64\cdot 4^{-(d+1)}\cdot
d^{2}/2\cdot\mathrm{low}(t)^{2}||u_{t}||^{2}_{1;t}\mathrm{d}t$
$\displaystyle<\int_{\mathord{\mathbb{R}}^{n}}1/32\mathrm{low}(t)^{2}||u_{t}||^{2}_{1;t}\mathrm{d}t.$

Plugging everything together, we get:

$\displaystyle\quad\ |\langle\mathrm{susp}(P)^{2}u,u\rangle|$
$\displaystyle\geq\langle(P^{\mathrm{ex}})^{2}u,u\rangle-\langle\Delta
u,u\rangle-|\langle\\{P^{\mathrm{ex}},{\not{\mathfrak{D}}}\\}u,u\rangle|-|\langle\sum_{1\leq
j<k\leq n}\mathfrak{R}(\partial_{j},\partial_{k})u,u\rangle|$
$\displaystyle\geq\int_{\mathord{\mathbb{R}}^{n}}(1/4-2/32)\mathrm{low}(t)^{2}||u_{t}||_{1;t}^{2}\mathrm{d}t$
$\displaystyle\geq
3/16\min\\{\mathrm{low}(t)^{2}\,:\,t\in\mathord{\mathbb{R}}^{n}\\}||u||_{0}^{2},$

which is equivalent to

$\displaystyle||\mathrm{susp}(P)u||_{0}\geq\sqrt{3}/4\min\\{\mathrm{low}(t)\,:\,t\in\mathord{\mathbb{R}}^{n}\\}||u||_{0}.$

Since $\mathrm{susp}(P)$ is a self-adjoint operator, this inequality implies
that $\mathrm{susp}(P)$ is invertible. ∎

Finally, we wish to show that ${B_{\bullet}}$ is a Kan set and that the
inclusion is a weak homotopy equivalence. To this end, we need to know how the
anti-commutator behaves under rescaling.

######  Lemma 4.5.11.

Let $D\subseteq\mathord{\mathbb{R}}^{n}$ be an open subset of
$\mathord{\mathbb{R}}^{n}$ and let $\varphi\colon D\rightarrow D$ and $P\colon
D\rightarrow\Psi\mathrm{Dir}(M)$ be a smooth maps. The map $P$ comes with an
underlying smooth map $h\colon D\rightarrow\mathrm{Riem}(M)$. Then we have

$\displaystyle[(P\circ\varphi)^{\mathrm{ex}},\nabla_{\partial_{t_{i}}}^{\mathfrak{S}_{\mathrm{susp}(h\circ\varphi)}}]=\sum_{k=1}^{n}[P^{\mathrm{ex}},\nabla_{\partial_{t_{k}}}^{\mathfrak{S}_{\mathrm{susp}(h)}}]\circ\varphi\cdot\partial_{t_{i}}\varphi_{k},$
$\displaystyle\partial_{t_{i}}(h\circ\varphi)=\sum_{k=1}^{n}(\partial_{t_{k}}h)\circ\varphi\cdot\partial_{t_{i}}\varphi_{k}.$

###### Proof.

The second equation is simply the chain rule for differentials, so it remains
to prove the first equation. Let $g_{00}\colon D\rightarrow\mathrm{Riem}(M)$
be the constant map with value $g_{00}$. Set further
$g_{0}=\mathrm{susp}(g_{00})$. Recall from the proof of Lemma 4.5.2 the
identity

$p\mathcal{G}(\mathrm{susp}(h),g_{0})_{\ast}(P^{\mathrm{ex}})=(p\mathcal{G}(h,g_{00})_{\ast}P)^{\mathrm{ex}}.$

In the following, we will abbreviate $p\mathcal{G}(\mathrm{susp}(h),g_{0})$ to
$p\mathcal{G}(\mathrm{susp}(h))$ and $p\mathcal{G}(h,g_{00})$ to
$p\mathcal{G}(h)$. Using Lemma 4.5.1, we calculate

$\displaystyle\qquad
p\mathcal{G}(\mathrm{susp}(h\circ\varphi))_{\ast}\left([(P\circ\varphi)^{\mathrm{ex}},\nabla_{\partial_{t_{i}}}^{\mathfrak{S}_{\mathrm{susp}(h\circ\varphi)}}]\right)$
$\displaystyle=[(p\mathcal{G}(\mathrm{susp}(h\circ\varphi))_{\ast}(P\circ\varphi))^{\mathrm{ex}},(p\mathcal{G}(h\circ\varphi)_{\ast}\nabla^{\mathfrak{S}_{\mathrm{susp}(h\circ\varphi)}})_{\partial_{t_{i}}}]$
$\displaystyle=[(p\mathcal{G}(h\circ\varphi)_{\ast}(P\circ\varphi))^{\mathrm{ex}},\partial_{t_{i}}]=[\bigl{(}(p\mathcal{G}(h)_{\ast}(P))\circ\varphi\bigr{)}^{\mathrm{ex}},\partial_{t_{i}}]$
$\displaystyle\underset{\mathrm{rule}}{\overset{\mathrm{chain}}{=}}\Bigl{(}\partial_{t_{i}}\bigl{(}(p\mathcal{G}(h)_{\ast}P)\circ\varphi\bigr{)}\Bigr{)}^{\mathrm{ex}}=\sum_{j=1}^{n}((\partial_{t_{k}}p\mathcal{G}(h)_{\ast}P)\circ\varphi)^{\mathrm{ex}}\cdot\partial_{t_{i}}\varphi_{k}.$

A similar calculation shows

$\displaystyle p\mathcal{G}(\mathrm{susp}(h\circ\varphi))_{\ast}$
$\displaystyle\left(\sum_{j=1}^{n}([P^{\mathrm{ex}},\nabla_{\partial_{t_{k}}}^{\mathfrak{S}_{\mathrm{susp}h}}])\circ\varphi\cdot\partial_{t_{i}}\varphi_{k}\right)$
$\displaystyle=\sum_{j=1}^{n}((\partial_{t_{k}}p\mathcal{G}(h)_{\ast}P)\circ\varphi)^{\mathrm{ex}}\cdot\partial_{t_{i}}\varphi_{k},$

so the lemma follows. ∎

######  Proposition 4.5.12.

The cubical set ${B_{\bullet}}$ is a Kan set.

inline]Draw a picture and give a rough proof idea. The rough idea of the proof
is the following one: A given cubical $n$-horn $\sqcap^{n}_{(i,\varepsilon)}$
gives rise to a well defined map $P_{0}$ on the complement of an infinte half-
cylinder $Q_{K}$ with base $K$, where $K$ is a compact, convex set with smooth
boundary. By the block form of the operator-valued maps in the given horn, the
operator-valued map $P_{0}$ is the elongation of $P_{0}|_{\\{\varepsilon
x_{i}<S\\}}$ for some sufficiently large $S$. We need to "extend"
$P_{0}|_{\\{\varepsilon x_{i}<S\\}}$ to $Q_{K}$ and we do this by modifying
$P_{0}|_{\\{\varepsilon x_{i}<S\\}}$ on a compact set of $\\{\varepsilon
x_{i}\leq S\\}$. More precisely, we do this by pushing down the argument of
$P_{0}$ in $\varepsilon x_{i}$-direction as closer we get to $Q_{K}$ so that
we can extend the modification to $Q_{K}$ by $P_{(i,-\varepsilon)}\circ
p_{i}$.

###### Proof.

Given an arbitrary cubical $n$-horn

$\sqcap^{n}_{(i,\varepsilon)}=\\{P_{(j,\omega)}\in{B_{n-1}}\,:\,\partial^{\omega}_{j}P_{(k,\eta)}=\partial^{\eta}_{k-1}P_{(j,\omega)};\,j<k;(j,\omega),(k,\eta)\neq(i,\varepsilon)\\}.$

Since that horn contains only finitely many block maps, there is a
sufficiently large $\rho>0$ such that all block maps $P_{(j,\omega)}$
decompose outside of $\rho I^{n-1}$. We restrict $\sigma_{j}P_{(j,\omega)}$ to
a map $\\{\omega x_{j}>\rho\\}\rightarrow\Psi\mathrm{Dir}^{\times}(M)$. By the
compatibility requirement, these restrictions agree on the intersection of
their domains and we can therefore glue them together to a map

$P_{0}\colon\left({\mathrm{Cub}_{n,i,\varepsilon}^{\rho}}\right)^{c}\rightarrow\Psi\mathrm{Dir}^{\times}(M),$

where
${\mathrm{Cub}_{n,i,\varepsilon}^{\rho}}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\\{x\in\mathord{\mathbb{R}}^{n}\,:\,x_{j}\in[-\rho,\rho]\text{ if }j\neq
i,\,\varepsilon x_{i}\geq-\rho\\}$ is the cuboid with diameter $\rho$. Let
$K\subseteq\mathord{\mathbb{R}}^{n-1}$ be a compact, convex subset that
contains $\rho I^{n-1}$, that is point symmetric at the origin, and that has a
smooth boundary. Let
$Q_{K}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\\{x\in\mathord{\mathbb{R}}^{n}\,:\,p_{i}(x)\in K,\varepsilon
x_{i}\geq-\rho\\}$ be the cuboid with base $K$. Restricting $P_{0}$ yields

$P_{0}\colon Q_{K}^{c}\cap\\{\varepsilon
x_{i}<\rho+R_{1}\\}\rightarrow\Psi\mathrm{Dir}^{\times}(M),$

for a fixed choice of $R_{1}>1$.

For the chosen $K$, there is a unique norm $||\text{--}||_{K}$ whose unit ball
is $K$. Since $\partial K$ is smooth, the norm is smooth on
$\mathord{\mathbb{R}}^{n-1}\setminus\\{0\\}$. For a sufficiently large
$R_{2}=R_{2}(\rho,R_{1})$ to be determined later, pick
$\chi\colon\mathord{\mathbb{R}}\rightarrow[0,1]$ that is identically $1$ on
$\mathord{\mathbb{R}}_{\leq 1.2}$, vanishes on $\mathord{\mathbb{R}}_{\geq
R_{2}-1}$, and whose first and second derivative satisfy $|\chi^{\prime}|\leq
1.3/R_{2}$ and $|\chi^{\prime\prime}|\leq 10/R_{2}^{2}$. Furthermore, pick a
monotonically increasing function
$q\colon\mathord{\mathbb{R}}\rightarrow\mathord{\mathbb{R}}$ that is the
identity on $\mathord{\mathbb{R}}_{\leq\rho}$, the constant map with value
$\rho+R_{1}-1$ on $\mathord{\mathbb{R}}_{\rho+R_{1}}$, and whose derivatives
satisfy $q^{\prime}\leq 1$ and $|q^{\prime\prime}|\leq 1.2/R_{1}$. The map $q$
defines a map

$\displaystyle q$
$\displaystyle\colon\mathord{\mathbb{R}}^{n}\rightarrow\mathord{\mathbb{R}}^{n}$
$\displaystyle(x_{1},\dots,x_{n})$
$\displaystyle\mapsto(x_{1},\dots,x_{i-1},\varepsilon q(\varepsilon
x_{i}),x_{i+1},\dots,x_{n}).$

Finally, we define

$\displaystyle H\colon\left(Q_{K}^{c}\cap\\{\varepsilon
x_{i}<\rho+R_{1}\\}\right)\times[0,1]$ $\displaystyle\rightarrow
Q_{K}^{c}\cap\\{\varepsilon x_{i}<\rho+R_{1}\\},$ $\displaystyle(x,t)$
$\displaystyle\mapsto x-2\varepsilon(\rho+R_{1})te_{i}$

and set

$\displaystyle P\colon\mathord{\mathbb{R}}^{n}$
$\displaystyle\rightarrow\Psi\mathrm{Dir}^{\times}(M),$ $\displaystyle x$
$\displaystyle\mapsto\begin{cases}P_{0}\circ
H\bigl{(}q(x),\chi(||p_{i}(x)||_{K})\bigr{)},&\text{if }x\in Q_{K}^{c},\\\
P_{(i,-\varepsilon)}(p_{i}(x)),&\text{if }x\in Q_{K}.\end{cases}$

Note that $P$ is well defined and smooth because $P_{0}$ agrees with
$P_{(i,-\varepsilon)}\circ p_{i}$ on the set $\\{-\varepsilon x_{i}>\rho\\}$
and $\partial Q_{K}$ lies in the preimage of $\\{-\varepsilon x_{i}>\rho\\}$
under the map $H\circ(q(\text{--}),\chi(||p_{i}(\text{--})||_{K}))$.

The map $P$ is a block map. Indeed, $\chi(||p_{i}(x)||_{K})=0$ on $\\{\omega
x_{j}>R_{2}\\}$ for all $j\neq i$, which yields

$P(x)=P_{0}(H(q(x),0))=P_{0}(q(x))=P_{0}(x)=P_{(j,\omega)}(p_{j}(x))$

because $P_{(j,\omega)(p_{j}(\text{--}))}$ is independent of $x_{i}$ if
$|x_{i}|>\rho$. For all $x$ with $-\varepsilon x_{i}>R_{2}$, we have

$\displaystyle P(x)$
$\displaystyle=P_{0}(q(x)-2\varepsilon(\rho+R_{1})\chi(||p_{i}(x)||_{K})e_{i})$
$\displaystyle=P_{0}(x-2\varepsilon(\rho+R_{1})\chi(||p_{i}(x)||_{K})e_{i})$
$\displaystyle=P_{(i,-\varepsilon)}(p_{i}(x))$

because, in this case, $-\varepsilon x_{i}+2\varepsilon(\rho+R_{1})t>R_{2}$
for every $t\geq 0$. Lastly, for all $x$ with $\varepsilon
x_{i}>R_{2}>\rho+R_{1}$ we have

$P(x)=P(x_{1},\dots,x_{i-1},\varepsilon(\rho+R_{1}-1),x_{i+1},\dots,x_{n})$

by the definition of $q$.

The proof also shows that $\partial^{\omega}_{j}P=P_{(j,\omega)}$ for all
$(j,\omega)\neq(i,\varepsilon)$. This means that $P$ is a filler for
$\sqcap^{n}_{(i,\varepsilon)}$ in $\Psi\mathrm{Dir}^{\times}_{n}(M)$.

It remains to show that $P$ lies in $B_{n}$. It is clear that $P$ satisfies
the defining conditions of $B_{n}$ in an open neighbourhood of $Q_{K}$, for
example on $\chi^{-1}(\mathord{\mathbb{R}}_{<1.2})$. We will show that $P$
satisfies the defining conditions of $B_{n}$ on
$\chi^{-1}(\mathord{\mathbb{R}}_{>1.1})$, too. Clearly, we have

$\mathrm{low}_{P}(t)=\mathrm{low}_{P_{0}}(H(q(t),\chi(||p_{i}(t)||_{K})).$

Actually, this equation holds for every smooth self map $\varphi\colon
Q_{K}^{c}\rightarrow Q_{K}^{c}$. For every such self map $\varphi$ and each
smooth map $Q\colon Q_{K}^{c}\rightarrow\Psi\mathrm{Dir}^{\times}(M)$ the
underlying map of Riemannian metrics $h_{Q}$ satisfies
$h_{Q\circ\varphi}=h_{Q}\circ\varphi$. By Lemma 4.5.11, we find the following
upper bound for its derivatives

$\displaystyle||(\partial_{j}h_{Q\circ\varphi})^{\mathrm{op}}||_{\mathrm{op},h_{Q\circ\varphi}}$
$\displaystyle=||\sum_{k=1}^{n}(\partial_{k}h_{Q})^{\mathrm{op}}\circ\varphi\cdot\partial_{j}\varphi_{k}||_{\mathrm{op},h_{Q\circ\varphi}}$
$\displaystyle\leq\sum|\partial_{j}\varphi_{k}|\cdot||(\partial_{k}h_{Q})^{\mathrm{op}}\circ\varphi||_{\mathrm{op},h\circ\varphi}$
$\displaystyle=\sum_{k=1}^{n}|\partial_{j}\varphi_{k}|\cdot||(\partial_{k}h_{Q})^{\mathrm{op}}||_{\mathrm{op},h}\circ\varphi.$

We would like to apply this estimate to $Q=P_{0}$ and
$\varphi=H\circ(q,\chi(||p_{i}(\text{--})||_{K})$. For $j\neq i$, the
derivatives of $\varphi_{k}$ are given by

$\displaystyle\partial_{j}\varphi_{k}=\begin{cases}0,&\text{if
}k\notin\\{i,j\\},\\\ 1,&\text{if }k=j,\\\
-2\varepsilon(\rho+R_{1})\chi^{\prime}\cdot(\partial_{j}||\text{--}||_{K})\circ
p_{i},&\text{if }j<i,k=i,\\\
-2\varepsilon(\rho+R_{1})\chi^{\prime}\cdot(\partial_{j-1}||\text{--}||_{K})\circ
p_{i},&\text{if }j>i,k=i.\end{cases}$

For $j=i$, they are given by

$\displaystyle\partial_{i}\varphi_{k}=\begin{cases}q^{\prime}(\varepsilon\cdot\text{--}),&\text{if
}k=i,\\\ 0,&\text{if }k\neq i.\end{cases}$

Plugging these results into the previous estimate yields

$\displaystyle||(\partial_{j}h_{P})^{\mathrm{op}}||_{\mathrm{op},h_{P}}\leq\begin{cases}||(\partial_{j}h_{P_{0}})^{\mathrm{op}}||_{\mathrm{op},P_{0}}\circ\varphi+|\partial_{j}\varphi_{i}|\cdot||(\partial_{t_{i}}h)^{\mathrm{op}}||_{\mathrm{op},h_{P_{0}}}\circ\varphi,&\text{if
}j\neq i,\\\
q^{\prime}(\varepsilon\cdot\text{--})\cdot||(\partial_{t_{i}}h_{P_{0}})^{\mathrm{op}}||_{\mathrm{op},h_{P_{0}}}\circ\varphi,&\text{if
}j=i,\end{cases}$

which gives

$\displaystyle\begin{split}\sum_{j=1}^{n}||(\partial_{j}h_{P})^{\mathrm{op}}||_{\mathrm{op},h_{P}}\leq&\sum_{j=1}^{n}||\partial_{j}h_{P_{0}}^{\mathrm{op}}||_{\mathrm{op},h_{P_{0}}}\circ\varphi\\\
&+\left(\sum_{j=1}^{n-1}\frac{2(\rho+R_{1})\cdot
1.3}{R_{2}}\cdot|\partial_{j}||\text{--}||_{K}|\circ
p_{i}\cdot||\partial_{i}h_{P_{0}}^{\mathrm{op}}||_{\mathrm{op},h_{P_{0}}}\right)\circ\varphi.\end{split}$

Applying Lemma 4.5.11 to $[P^{\mathrm{ex}},\nabla_{\partial_{j}}]$ we obtain,
by a similar calculation, the estimate

$\displaystyle\begin{split}&\quad\
\sum_{j=1}^{n}||[P^{\mathrm{ex}},\nabla_{j}^{\mathfrak{S}_{\mathrm{susp}(h_{P})}}]||_{1,0,h_{P}}\\\
&\leq\sum_{j=1}^{n}||[P_{0}^{\mathrm{ex}},\nabla_{j}^{\mathfrak{S}_{\mathrm{susp}(h_{P_{0}})}}]||_{1,0,h_{P_{0}}}\circ\varphi\\\
&\quad\
+\left(\sum_{j=1}^{n-1}\frac{2.6(\rho+R_{1})}{R_{2}}|\partial_{j}||\text{--}||_{K}|\circ
p_{i}\cdot||[P_{0}^{\mathrm{ex}},\nabla_{i}^{\mathfrak{S}_{\mathrm{susp}(h_{P_{0}})}}]||_{1,0,h_{P_{0}}}\right)\circ\varphi.\end{split}$

Since $\partial_{i}h^{\mathrm{op}}_{P}$ and
$[P^{\mathrm{ex}},\nabla_{\partial_{i}}]$ have smooth extensions from
$\chi^{-1}(\mathord{\mathbb{R}}_{>1.1})$ to
$\chi^{-1}(\mathord{\mathbb{R}}_{\geq 1.05})$ their norms are uniformly
bounded by $B_{1}$, say. Also all $|\partial_{j}||\text{--}||_{K}|$ are
uniformly bounded by $B_{2}$, say. Thus, if we choose
$R_{2}>2.6(\rho+R_{1})B_{1}B_{2}\mathcal{I}^{-1}$, where
$\mathcal{I}=\min\\{\mathcal{I}_{1},\mathcal{I}_{2}\\}$ is the minimum of

$\mathcal{I}_{1}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\mathrm{inf}\left\\{\frac{1}{32}\,\mathrm{low}_{P_{0}}(t)^{2}-\frac{1}{2}\sum_{j=1}^{n}||[P_{0}^{\mathrm{ex}},\nabla_{j}^{\mathfrak{S}_{\mathrm{susp}h_{P_{0}}}}]||_{1,0,h_{P_{0}}}\,:\,t\in
Q_{K}^{c}\right\\}$

and

$\mathcal{I}_{2}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\left\\{\mathrm{min}\\{d^{-1},2^{-(d+4)}\mathrm{low}(t)\\}-\sum_{k=1}^{n}||\partial_{k}h^{\mathrm{op}}||_{\mathrm{op},h_{P_{0}}(t)}\,:\,t\in
Q_{K}^{c}\right\\}$

then $P$ satisfies the defining equations of ${B_{\bullet}}$. Note that
$\mathcal{I}>0$ because $P_{0}|_{Q_{K}^{c}}$ is a union of restriction of
degenerate elements of $B_{n}$ and the defining conditions of $B_{n}$ are
local.

Thus, $P\in{B_{n}}$ is a filler for the given cubical $n$-horn and
${B_{\bullet}}$ is therefore a Kan set. ∎

######  Corollary 4.5.13.

The inclusion
${B_{\bullet}}\hookrightarrow\Psi\mathrm{Dir}^{\times}_{\bullet}(M)$ is a weak
homotopy equivalence.

###### Proof.

We first prove that the inclusion induces a surjection on homotopy groups. Let
$[P]\in\pi_{n}(\Psi\mathrm{Dir}^{\times}_{\bullet}(M),P_{0})$ be represented
by a smooth block map
$P\colon\mathord{\mathbb{R}}^{n}\rightarrow\Psi\mathrm{Dir}^{\times}(M)$ that
is constant with value $P_{0}$ outside some cube $rI^{n}$. For each $\rho\geq
1$ the rescaled map
${}_{\rho}P(x)\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=P(\rho^{-1}x)$ is still constant near infinity with value $P_{0}$ and
thus represents an element in
$\pi_{n}(\Psi\mathrm{Dir}^{\times}_{\bullet}(M))$, too. By Lemma 4.5.11 there
is a $\rho_{0}\geq 1$ such that ${}_{\rho}P\in\pi_{n}({B_{\bullet}})$ for all
$\rho\geq\rho_{0}$ because the left hand side of the defining inequalities for
${B_{\bullet}}$ rescale with $1/\rho$, while the right hand side is scaling
invariant.

We need to show ${}_{\rho}P$ and $P={}_{1}P$ are homotopic relative boundary.
A homotopy $H\in\Psi\mathrm{Dir}^{\times}_{n+1}(M)$ relating ${}_{\rho}P$ and
${}_{1}P$ is given by

$H(t,x)\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt={}_{(1-\chi(t))\rho+\chi(t)}P(x),$

where $\chi\colon\mathord{\mathbb{R}}\rightarrow[0,1]$ is a smooth function
that is identically zero on $\mathord{\mathbb{R}}_{\leq-1}$ and identically
$1$ on $\mathord{\mathbb{R}}_{\geq 1}$. Indeed, $H$ satisfies

$\displaystyle\partial^{-1}_{1}H(x)$
$\displaystyle=\lim_{R\to\infty}H(-R,x)={}_{\rho}P(x),$
$\displaystyle\partial^{1}_{1}H(x)$
$\displaystyle=\lim_{R\to\infty}H(R,x)={}_{1}P(x),$
$\displaystyle\partial^{\varepsilon}_{i}H(t,x)$
$\displaystyle=\lim_{R\to\infty}H(t,x_{1},\dots,x_{i-1},\varepsilon
R,x_{i},\dots,x_{n-1})$
$\displaystyle=\lim_{R\to\infty}P\left(((1-\chi(t))\rho+\chi(t))^{-1}\cdot(x_{1},\dots,\varepsilon
R,\dots,x_{n-1})\right)$
$\displaystyle=P_{0}=\sigma_{1}\partial^{\varepsilon}_{i-1}P(t,x).$

To prove injectivity, we follow the common strategy that uniqueness is just a
relative form of existence. For each $[P]\in\pi_{n}({B_{\bullet}},P_{0})$ with
$0=[P]\in\pi_{n}(\Psi\mathrm{Dir}^{\times}_{\bullet}(M),P_{0})$ there exists
an element $H\in\Psi\mathrm{Dir}^{\times}_{n+1}(M)$ that satisfies

$\displaystyle\partial^{1}_{1}H$ $\displaystyle=P_{0},$
$\displaystyle\partial^{-1}_{1}H$ $\displaystyle=P,$
$\displaystyle\partial^{\varepsilon}_{j}H$
$\displaystyle=\sigma_{1}\partial^{\varepsilon}_{j-1}P=P_{0}.$

As before, there is a sufficiently large $\rho\geq 1$ such that
${}_{\rho}H\in{B_{n+1}}$. Note that ${}_{\rho}H$ is a homotopy between
${}_{\rho}P$ and $P_{0}$ in ${B_{\bullet}}$, so it remains to find a homotopy
between $P$ and ${}_{\rho}P$ in ${B_{\bullet}}$. An example for such a
homotopy is
$\mathcal{H}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\sigma_{1}P\circ{}_{R}\varphi$, where

$\displaystyle{}_{R}\varphi\colon\mathord{\mathbb{R}}^{n+1}$
$\displaystyle\rightarrow\mathord{\mathbb{R}}^{n+1}$ $\displaystyle(t,x)$
$\displaystyle\mapsto\left(t,((1-\chi(R^{-1}t)\rho+\chi(R^{-1}t))^{-1}x\right),$

provided $R$ is sufficiently large. Abbreviate $(1-\chi(t))\rho+\chi(t)$ to
$\mu(t)$ and set ${}_{R}\mu(t)=\mu(R^{-1}t)$. Furthermore, abbreviate $(t,x)$
to $y$ so that $t=y_{1}$ and $x_{i}=y_{i+1}$. Lemma 4.5.11, applied to
$y_{1}=t$ yields

$\displaystyle\left[\mathcal{H}^{\mathrm{ex}},\nabla_{\partial_{y_{1}}}^{\mathfrak{S}_{\mathrm{susp}(g_{\mathcal{H}})}}\right](y)$
$\displaystyle=\left[\sigma_{1}P^{\mathrm{ex}}\circ{}_{R}\varphi,\nabla_{\partial_{y_{1}}}^{\mathfrak{S}_{\mathrm{susp}(\sigma_{1}g_{P}\circ{}_{R}\varphi)}}\right](y)$
$\displaystyle=\sum_{k=1}^{n+1}[\sigma_{1}P^{\mathrm{ex}},\nabla_{\partial_{y_{k}}}^{\mathfrak{S}_{\mathrm{susp}(\sigma_{1}g_{P})}}]\circ{}_{R}\varphi\cdot\partial_{y_{1}}{}_{R}\varphi_{k}(y)$
$\displaystyle=\frac{1}{R}\sum_{k=2}^{n+1}[\sigma_{1}P^{\mathrm{ex}},\nabla_{\partial_{y_{k}}}^{\mathfrak{S}_{\mathrm{susp}(\sigma_{1}g_{P})}}]\circ{}_{R}\varphi\cdot{}_{R}\mu^{-2}\cdot((\rho-1)\cdot{}_{R}(\chi^{\prime}))y_{k}$

where we used
$[\sigma_{1}P^{\mathrm{ex}},\nabla^{\mathfrak{S}_{\mathrm{susp}(\sigma_{1}g_{P})}}_{\partial_{y_{1}}}]=[\sigma_{1}P^{\mathrm{ex}},\partial_{y_{1}}]=(\partial_{y_{1}}\sigma_{1}P)^{\mathrm{ex}}=0$.
Applied to $y_{i}=x_{i-1}$ for $i>1$, Lemma 4.5.11 yields

$\displaystyle\left[\mathcal{H}^{\mathrm{ex}},\nabla_{\partial_{y_{i}}}^{\mathfrak{S}_{\mathrm{susp}(g_{\mathcal{H}})}}\right](y)=\dots$
$\displaystyle=\sum_{k=2}^{n+1}[\sigma_{1}P^{\mathrm{ex}},\nabla_{\partial_{y_{k}}}^{\mathfrak{S}_{\mathrm{susp}(\sigma_{1}g_{P})}}]\circ{}_{R}\varphi\cdot{}_{R}\mu^{-1}\delta_{ik}$
$\displaystyle={}_{R}\mu^{-1}\cdot[\sigma_{1}P^{\mathrm{ex}},\nabla_{\partial_{y_{i}}}^{\mathfrak{S}_{\mathrm{susp}(\sigma_{1}g_{P})}}]\circ{}_{R}\varphi$
$\displaystyle={}_{R}\mu^{-1}\cdot\sigma_{1}\left([P^{\mathrm{ex}},\nabla_{\partial_{x_{i-1}}}^{\mathfrak{S}_{\mathrm{susp}(g_{P})}}]\right)\circ{}_{R}\varphi.$

We assumed that $P=P_{0}$ outside of $rI^{n}$, so
$[\sigma_{1}P^{\mathrm{ex}},\nabla_{\partial_{y_{k}}}]\circ{}_{R}\varphi$ is
supported within $\mathord{\mathbb{R}}\times\rho rI^{n}$ for $k>1$ and
therefore it is uniformly bounded. Thus, we can choose $R$ sufficiently large
such that $[\mathcal{H}^{\mathrm{ex}},\nabla_{\partial_{t}}]$ is arbitrarily
small. Since
$\mathrm{low}_{\mathcal{H}}=\mathrm{low}_{\sigma_{1}P}\circ{}_{R}\varphi=\mathrm{low}_{P}\circ
p_{1}\circ{}_{R}\varphi=\sigma_{1}(\mathrm{low}_{P})\circ{}_{R}\varphi$ and
since ${}_{R}\mu^{-1}\leq 1$, we have

$\sum_{k=2}^{n+1}\left|\left|\left[\mathcal{H}^{\mathrm{ex}},\nabla_{\partial_{y_{k}}}^{\mathfrak{S}_{\mathrm{susp}(g_{\mathcal{H}})}}\right](y)\right|\right|_{1,0,M_{y}}<1/32\mathrm{low}_{\mathcal{H}}(y)^{2},$

so we can choose $R>1$ so large that $\mathcal{H}$ satisfies the second
defining inequality (4.6) of ${B_{\bullet}}$.

The same strategy shows that the underlying map of metrics $g_{\mathcal{H}}$
satisfies the first defining inequality (4.5) after enlarging $R$, if
necessary. ∎

###  4.6 Kan Property of the Operator Concordance Set

The main theorem of this section is the analogous result of Theorem 3.3.1 for
invertible block Dirac operators.

######  Theorem 4.6.1.

The operator concordance set
$\widetilde{\Psi\mathrm{Dir}}^{\times}_{\bullet}(M)$ is a Kan set for all
closed, spin manifolds $M$ that admit at least one invertible pseudo Dirac
operator.

The proof of this theorem relies on an operator theoretic analog of Lemma
3.3.2.

######  Lemma 4.6.2 (Operator Modification Lemma).

Let $P$ be a block Dirac operator on $M\times\mathord{\mathbb{R}}^{n}$ with
underlying block metric $g$. Assume that the core of $P$ is contained in
$M\times\rho/2\cdot I^{n}$ for $\rho>20$ (in particular, $g$ decomposes on the
complement). Assume further that $\partial^{\omega}_{j}P$ is invertible if
$(j,\omega)\neq(1,1)$. Let $H$ and $\Phi$ as in Theorem 3.2.2 and denote
${}_{R}\Phi\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=R\Phi\circ(\mathrm{id}\times R^{-1})$. Then

  * (o)

the operator $P$ restricts to $M\times R\Phi(H\times(a,b))$ for all $1\leq
a<b\leq 2$ and all $R\geq 1$.

For all $R\geq 1$, there is a Riemannian metric $G_{R}$ on $M\times
R\Phi(H\times[1,2])$, a compact subset $K=K(R)$ of
$M\times\mathord{\mathbb{R}}^{n}$, a relative compact, open subset
$L=L(R)\supseteq K$ of $M\times\mathord{\mathbb{R}}^{n}$, and a pseudo Dirac
operator
$Q\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=Q(R)$ on $\Gamma_{c}(R\Phi(H\times(1,2)),\mathfrak{S}_{G_{R}})$ that
satisfy:

  1. (i)

The operator ${}_{R}\Phi^{\ast}Q$ restricts to $M\times H\times(a,b)$ for all
$R\leq a<b\leq 2R$ such that ${}_{R}\Phi^{\ast}Q$ restricts to
${}_{R}\Phi^{-1}(L)$ and to the complement of ${}_{R}\Phi^{-1}(K)$.

  2. (ii)

The restriction of ${}_{R}\Phi^{\ast}Q$ and ${}_{R}\Phi^{\ast}P$ to $M\times
H\times(1.8R,2R)$ and to the complement of $K$ agree.

  3. (iii)

The metric ${}_{R}\Phi^{\ast}G_{R}$ is a product metric on $M\times
H\times[R,1.2R]$. Under the decomposition
$\mathfrak{S}_{{}_{R}\Phi^{\ast}G_{R}}|_{M\times
H\times[R,1.2R]}\cong\mathfrak{S}_{{}_{R}\Phi^{\ast}G_{R}}\mathord{\upharpoonleft}_{M\times
H\times{1.1R}}\boxtimes Cl_{1,0}$, the operator ${}_{R}\Phi^{\ast}Q$
decomposes into

${}_{R}\Phi^{\ast}Q={}_{R}\Phi^{\ast}Q\mathord{\upharpoonleft}_{{M\times
H\times{1.1R}}}\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}_{\mathord{\mathbb{R}}}.$

  4. (iv)

There is an $R_{0}\geq 1$ and a constant $c>0$, such that $Q$ is bounded from
below with lower bound $\geq c$ for all $R\geq R_{0}$.

  5. (v)

If $P-{\not{\mathfrak{D}}}_{g_{P}}$ is bounded, then the operator
$Q-{\not{\mathfrak{D}}}_{G_{R}}$ is bounded.

###### Proof.

We start with the proof of $(o)$. The block form of $P$ implies that it
restricts to $V_{R\rho}({\bm{\alpha}})$, see Lemma 4.2.6, and that
$P|_{V_{R\rho({\bm{\alpha}})}}$ can be identified with
$P(\hat{{\bm{\alpha}}})\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}_{\mathord{\mathbb{R}}^{\hat{{\bm{\alpha}}}}}$
via a permutation of coordinates. The same permutation of coordinates gives a
diffeomorphism between $R\Phi(H\times(1,2))\cap V_{R\rho}({\bm{\alpha}})$ and
$(-R\rho,R\rho)^{|\mathrm{Null\,{\bm{\alpha}}}|}\times
R\Phi^{\hat{{\bm{\alpha}}}}(H^{\hat{{\bm{\alpha}}}}\times(1,2))$. The first
factor contains the core of $P(\hat{{\bm{\alpha}}})$. On the second factor,
$P$ acts as the differential operator
${\not{\mathit{D}}}_{\mathord{\mathbb{R}}^{\hat{{\bm{\alpha}}}}}$. Thus, $P$
restricts to $M\times V_{R\rho}({\bm{\alpha}})\cap R\Phi(H\times(a,b))$ for
all $1\leq a<b\leq 2$.

As
$\\{V_{R\rho}({\bm{\alpha}})\,:\,{\bm{\alpha}}\neq\mathbf{0},(1,0\dots,0)\\}$
covers $R\Phi(H\times(1,2))$, the operator $P$ restricts to a symmetric
operator on $M\times R\Phi(H\times(a,b))$.

The same strategy shows that $P$ also restricts to a symmetric operator on the
complement of $V_{R\rho}(\mathbf{0})$. This proves $(o)$.

For all $R\geq 1$, let $G_{R}$ be the metric from the Metric Modification
Lemma 3.3.2. Define the operator $Q$ on $\Gamma_{c}(M\times
R\Phi(H\times(1,2)),\mathfrak{S}_{G_{R}})$ via

$Q\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=Q(R)\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\mathcal{G}(g_{P},G_{R})_{\ast}(P-{\not{\mathfrak{D}}}_{g_{P}})+{\not{\mathfrak{D}}}_{G_{R}}.$

Since $Q$ is obtained from $P$ by pushing it forward with a vector bundle map
and adding a differential operator to $P$, we conclude
$Q\in\overline{\Psi\mathrm{DO}}^{1}(\mathfrak{S}_{G_{R}})$. The operator $Q$
also restricts to $M\times R\Phi(H\times(a,b))$ for all $1\leq a<b\leq 2$, to
the intersection of
$L(R)\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=V_{R(\rho+5)}(\mathbf{0})$ and $M\times R\Phi(H\times(a,b))$, and to the
complement of
$K(R)\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\overline{V_{R(\rho+4)}(\mathbf{0})}$ in $M\times R\Phi(H\times(a,b))$,
which proves $(i)$.

The gauge map $\mathcal{G}(g_{P},G_{R})$ is an isometry between spaces of
square integrable spinors, so $Q$ is symmetric. We will calculate below that
the principal symbol of $Q$ and ${\not{\mathfrak{D}}}_{G_{R}}$ agree, thus $Q$
is a symmetric pseudo Dirac operator.

To prove $(ii)$, recall that $g_{P}$ and $G_{R}$ agree on $M\times
R\Phi(H\times(1.8,2))$, so that $\mathcal{G}(g_{P},G_{R})=\mathrm{id}$ and
${\not{\mathfrak{D}}}_{g_{P}}={\not{\mathfrak{D}}}_{G_{R}}$ on this subset.
Thus, $P=Q$ on this subset. The same is true on the complement of
$K(R)=\overline{V_{R(\rho+4)}(\mathbf{0})}$.

Next we prove $(iii)$. Consider the restriction of ${}_{R}\Phi^{\ast}Q$ to
${}_{R}\Phi^{-1}(V_{R\rho}({\bm{\alpha}}))$. Recall from the Metric
Modification Lemma 3.3.2 that, on these subsets, the metrics decompose into

$\displaystyle{}_{R}\Phi^{\ast}g_{P}=R^{\ast}\partial^{\hat{{\bm{\alpha}}}}g\oplus({}_{R}\Phi^{\hat{{\bm{\alpha}}}})^{\ast}\langle\cdot,\cdot\rangle\quad\text{and}\quad{}_{R}\Phi^{\ast}G_{R}=R^{\ast}\partial^{\hat{{\bm{\alpha}}}}g\oplus
k_{R}^{\hat{{\bm{\alpha}}}}.$

Thus, the gauge map
$\mathcal{G}({}_{R}\Phi^{\ast}g_{P},{}_{R}\Phi^{\ast}G_{R})$ decomposes into
$\mathrm{id}\boxtimes\mathcal{G}(({}_{R}\Phi^{\hat{{\bm{\alpha}}}})^{\ast}\langle\cdot,\cdot\rangle,k_{R}^{\hat{{\bm{\alpha}}}})$.
Since the gauge map is natural with respect to linear maps, we get

$\displaystyle{}_{R}\Phi^{\ast}Q$
$\displaystyle=\mathcal{G}({}_{R}\Phi^{\ast}g_{P},{}_{R}\Phi^{\ast}\langle\cdot,\cdot\rangle)_{\ast}({}_{R}\Phi^{\ast}P-{\not{\mathfrak{D}}}_{{}_{R}\Phi^{\ast}g_{P}})+{\not{\mathfrak{D}}}_{{}_{R}\Phi^{\ast}G_{R}}$
$\displaystyle\begin{split}\hfil\displaystyle\begin{split}&=\mathrm{id}\boxtimes\mathcal{G}(({}_{R}\Phi^{\hat{{\bm{\alpha}}}})^{\ast}\langle\cdot,\cdot\rangle,k_{R}^{\hat{{\bm{\alpha}}}})_{\ast}\bigl{(}P(\hat{{\bm{\alpha}}})\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathfrak{D}}}_{({}_{R}\Phi^{\hat{{\bm{\alpha}}}})^{\ast}\langle\cdot,\cdot\rangle}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad-{\not{\mathfrak{D}}}_{R^{\ast}\partial^{\hat{{\bm{\alpha}}}}g}\boxtimes\mathrm{id}-\mathrm{e/o}\boxtimes{\not{\mathfrak{D}}}_{({}_{R}\Phi^{\hat{{\bm{\alpha}}}})^{\ast}\langle\cdot,\cdot\rangle}\bigr{)}\end{split}\\\
&\quad+{\not{\mathfrak{D}}}_{R^{\ast}\partial^{\hat{{\bm{\alpha}}}}g}\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathfrak{D}}}_{k_{R}^{\hat{{\bm{\alpha}}}}}\end{split}$
$\displaystyle=P(\hat{{\bm{\alpha}}})\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathfrak{D}}}_{k_{R}^{\hat{{\bm{\alpha}}}}}.$

Since
$k_{R}^{\hat{{\bm{\alpha}}}}=k_{R}^{\hat{{\bm{\alpha}}}}\mathord{\upharpoonleft}_{H\times
1.1R}\oplus\mathrm{d}t^{2}$ on $H^{{\bm{\alpha}}}\times(R,1.2R)$, the Dirac
operator ${\not{\mathfrak{D}}}_{k_{R}^{\hat{{\bm{\alpha}}}}}$ decomposes
accordingly on this set, hence $Q$ does, too. These decompositions are
compatible on the intersections of different $V_{R\rho}({\bm{\alpha}})$, so we
get a global decomposition

${}_{R}\Phi^{\ast}Q={}_{R}\Phi^{\ast}Q\mathord{\upharpoonleft}_{M\times
H\times
1.1R}\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}_{\mathord{\mathbb{R}}},$

which proves $(iii)$.

We can read off the principal symbol of ${}_{R}\Phi^{\ast}Q$ from the (local)
decompositions. On ${}_{R}\Phi^{-1}(V_{R\rho}({\bm{\alpha}}))$, we have

$\displaystyle\mathrm{symb}_{1}({}_{R}\Phi^{\ast}Q)$
$\displaystyle=i\mathbf{c}_{\partial^{\hat{{\bm{\alpha}}}}g}(\partial^{\hat{{\bm{\alpha}}}}g_{\sharp}(\text{--}))\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes
i\mathbf{c}_{k_{R}^{\hat{{\bm{\alpha}}}}}({k_{R}^{\hat{{\bm{\alpha}}}}}_{\sharp}(\text{--}))$
$\displaystyle=\mathrm{symb}_{1}({\not{\mathfrak{D}}}_{{}_{R}\Phi^{\ast}G_{R}}),$

so $Q$ also has the same principal symbol as ${\not{\mathfrak{D}}}_{G_{R}}$.

We continue with the proof of $(iv)$. Since the gauge map
$\mathcal{G}(g_{P},G_{R})$ induces an isometry between
$H^{0}(\mathfrak{S}_{g_{P}})$ and $H^{0}(\mathfrak{S}_{G_{R}})$, the operator
$\mathcal{G}(g_{P},G_{R})_{\ast}(P)$ is bounded from below with the same lower
bound as $P$. Being bounded from below is an open condition for elliptic
operators in
$\mathrm{Hom}(H^{1}(\mathfrak{S}_{G_{R}}),H^{0}(\mathfrak{S}_{G_{R}}))$, see
Lemma 4.3.13, so it suffices to show that $Q$ and
$\mathcal{G}(g_{P},G_{R})_{\ast}P$ are sufficient close.

The difference of these two operators is a differential operator that is
supported within $M\times R\Phi(H\times[1,1.8])$.

The previous calculation shows that

$\displaystyle(R\Phi)^{\ast}(Q-\mathcal{G}(g_{P},G_{R})_{\ast}(P))=\left(\mathrm{e/o}\boxtimes{\not{\mathfrak{D}}}_{R^{2}k^{\hat{{\bm{\alpha}}}}}-\mathcal{G}((R\Phi^{\hat{{\bm{\alpha}}}})^{\ast}\langle\cdot,\cdot\rangle,R^{2}k^{\hat{{\bm{\alpha}}}})_{\ast}{\not{\mathfrak{D}}}_{(R\Phi^{\hat{{\bm{\alpha}}}})^{\ast}\langle\cdot,\cdot\rangle}\right)$

on $R\Phi^{-1}(V_{R\rho}({\bm{\alpha}}))$, which we abbreviate to
$\mathrm{e/o}\boxtimes T$. Recall that
$R\Phi^{-1}(V_{R\rho}({\bm{\alpha}}))=M\times(-\rho,\rho)^{\mathrm{Null}({\bm{\alpha}})}\times
H^{\hat{{\bm{\alpha}}}}\times(1,2)$ for
${\bm{\alpha}}\neq\mathbf{0},(1,0\dots,0)$. Hence, for each
$\sigma\in\Gamma_{c}(R\Phi^{-1}(V_{R\rho}({\bm{\alpha}})),\mathfrak{S}_{\partial^{\hat{{\bm{\alpha}}}}g}\boxtimes\mathfrak{S}_{R^{2}k^{{\bm{\alpha}}}})$,
using Fubini’s theorem we have

$\displaystyle\qquad||\mathrm{e/o}\boxtimes
T(\sigma)||_{0}^{2}=\langle\mathrm{id}\boxtimes T^{2}(\sigma),\sigma\rangle$
$\displaystyle=\int_{M\times(-R\rho,R\rho)^{|\mathrm{Null}({\bm{\alpha}})|}}\int_{H^{\hat{{\bm{\alpha}}}}\times(1,2)}||\mathrm{id}\boxtimes
T^{2}\sigma(t,x)||^{2}_{\mathfrak{S}_{\partial^{\hat{{\bm{\alpha}}}}g}\boxtimes\mathfrak{S}_{R^{2}k^{{\bm{\alpha}}}}}\mathrm{d}\mathrm{vol}_{R^{2}k^{\hat{{\bm{\alpha}}}}}(x)\mathrm{d}\mathrm{vol}_{\partial^{\hat{{\bm{\alpha}}}}g}(t)$
$\displaystyle\leq\int\int||T||_{1,0}^{2}\left(||\sigma(t,x)||^{2}+||\nabla^{\mathfrak{S}_{R^{2}k^{\hat{{\bm{\alpha}}}}}}\sigma(t,x)||^{2}\right)\mathrm{d}\mathrm{vol}(x)\mathrm{d}\mathrm{vol}(t)$
$\displaystyle\leq||T||_{1,0}^{2}||\sigma||_{1}^{2},$

where the first inequality is the operator norm inequality applied to the
section $\sigma(t,\text{--})$. This gives $||\mathrm{e/o}\boxtimes
T||_{1,0}\leq||T||_{1,0}$.

It follows from [hijazi1986conformal]*Prop 4.3.1 and 4.2.1 that

$\mathcal{G}(R^{2}g,g)_{\ast}{\not{\mathfrak{D}}}_{R^{2}g}=R^{-1}{\not{\mathfrak{D}}}_{g}\quad\text{and}\quad\mathcal{G}(R^{2}g,g)_{\ast}\nabla^{\mathfrak{S}_{R^{2}g}}=\nabla^{\mathfrak{S}_{g}}$

for all Riemannian metrics $g$ and all values $R>0$. The latter equation
implies that the Gauge map induces an isometry between all Sobolev spaces.
Furthermore, we have
$\mathcal{G}\bigl{(}R^{2}k^{\hat{{\bm{\alpha}}}},k^{\hat{{\bm{\alpha}}}}\bigr{)}\circ\mathcal{G}\bigl{(}(R\Phi^{\hat{{\bm{\alpha}}}})^{\ast}\langle\cdot,\cdot\rangle,R^{2}k^{\hat{{\bm{\alpha}}}}\bigr{)}=\mathcal{G}\bigl{(}(R\Phi^{\hat{{\bm{\alpha}}}})^{\ast}\langle\cdot,\cdot\rangle,k^{\hat{{\bm{\alpha}}}}\bigr{)}$.
Putting these facts together yields

$\displaystyle||T||_{1,0}$
$\displaystyle=||{\not{\mathfrak{D}}}_{R^{2}k^{\hat{{\bm{\alpha}}}}}-\mathcal{G}((R\Phi^{\hat{{\bm{\alpha}}}})^{\ast}\langle\cdot,\cdot\rangle,R^{2}k^{\hat{{\bm{\alpha}}}})_{\ast}{\not{\mathfrak{D}}}_{(R\Phi^{\hat{{\bm{\alpha}}}})^{\ast}\langle\cdot,\cdot\rangle}||_{1,0}$
$\displaystyle=||\mathcal{G}(R^{2}k^{\hat{{\bm{\alpha}}}},k^{\hat{{\bm{\alpha}}}})_{\ast}\left({\not{\mathfrak{D}}}_{R^{2}k^{\hat{{\bm{\alpha}}}}}-\mathcal{G}((R\Phi^{\hat{{\bm{\alpha}}}})^{\ast}\langle\cdot,\cdot\rangle,R^{2}k^{\hat{{\bm{\alpha}}}})_{\ast}{\not{\mathfrak{D}}}_{(R\Phi^{\hat{{\bm{\alpha}}}})^{\ast}\langle\cdot,\cdot\rangle}\right)||_{1,0}$
$\displaystyle=||R^{-1}{\not{\mathfrak{D}}}_{k^{\hat{{\bm{\alpha}}}}}-\mathcal{G}((R\Phi^{\hat{{\bm{\alpha}}}})^{\ast}\langle\cdot,\cdot\rangle,k^{\hat{{\bm{\alpha}}}})_{\ast}({\not{\mathfrak{D}}}_{(R\Phi^{\hat{{\bm{\alpha}}}})^{\ast}\langle\cdot,\cdot\rangle})||_{1,0}$
$\displaystyle=||{\not{\mathfrak{D}}}_{k^{\hat{{\bm{\alpha}}}}}-\mathcal{G}({\Phi^{\hat{{\bm{\alpha}}}}}^{\ast}\langle\cdot,\cdot\rangle,k^{\hat{{\bm{\alpha}}}})_{\ast}({\not{\mathfrak{D}}}_{{\Phi^{\hat{{\bm{\alpha}}}}}^{\ast}\langle\cdot,\cdot\rangle})||_{1,0}/R\xrightarrow{R\to\infty}0.$

It follows that $Q$ and $\mathcal{G}(g_{P},G_{R})_{\ast}(P)$ are arbitrarily
close if $R$ is sufficient large. Together with Lemma 4.3.13, we find an
$R_{0}\geq 1$ such that for all $R\geq R_{0}$, the operator $Q$ is bounded
from below with lower bound $c>0$, that only depends on the lower bound of $P$
and some chosen upper bound of the distance of $Q$ to
$\mathcal{G}(g_{P},G_{R})_{\ast}(P)$. In fact, the proof of Lemma 4.3.13
shows, that the constant $c$ can be made arbitrarily close to the lower bound
of $P$ at the cost of increasing $R_{0}$.

It remains to prove $(v)$. On $V_{R\rho}({\bm{\alpha}})\cap M\times
R\Phi(H\times(1,2))$, we have

$\displaystyle{}_{R}\Phi^{\ast}(Q-{\not{\mathfrak{D}}}_{G_{R}})=(P(\hat{{\bm{\alpha}}})-{\not{\mathfrak{D}}}_{\partial^{\hat{{\bm{\alpha}}}}g})\boxtimes\mathrm{id},$

which is bounded by assumption of $P$. Hence,
${}_{R}\Phi^{\ast}(Q-{\not{\mathfrak{D}}}_{G_{R}})$ is bounded by a partition
of unity argument. ∎

###### Proof of Theorem 4.6.1.

By assumption, $M$ has at least one invertible pseudo Dirac operator so that
$\widetilde{\Psi\mathrm{Dir}}^{\times}_{0}(M)$ (and hence
$\widetilde{\Psi\mathrm{Dir}}^{\times}_{\bullet}(M)$) is non-empty. A given
cubical $n$-horn
$\sqcap^{n}_{(i,\varepsilon)}\rightarrow\widetilde{\Psi\mathrm{Dir}}^{\times}_{\bullet}(M)$
is represented by the following set of invertible block Dirac operators

$\\{P_{(j,\omega)}\in\widetilde{\Psi\mathrm{Dir}}^{\times}_{n-1}(M)\,:\,\partial^{\omega}_{j}P_{(k,\eta)}=\partial^{\eta}_{k-1}P_{(j,\omega)}\text{
for }j<k;(j,\omega),(k,\eta)\neq(i,\varepsilon)\\}.$

The problem is symmetric in $(i,\varepsilon)$, so we assume
$(i,\varepsilon)=(1,1)$.

We pick a sufficient large $\rho>20$ such that the cores of all
$P_{(j,\omega)}$ are contained in $M\times\rho/2I^{n-1}$. The operators
$\sigma_{j}P_{(j,\omega)}|_{\\{\omega x_{j}>\rho\\}}$ restrict to each
$\\{\eta x_{k}>\rho\\}$ if the intersection is not empty and the restrictions
agree there, see Lemma 4.4.12. Thus, the union of these operators form an
operator $P$ on the complement of the half-infinite cuboid
$\mathord{\mathbb{R}}^{n}\setminus{\mathrm{Cub}_{n,1,1}^{\rho}}=\bigcup_{((j,\omega)\neq(i,\varepsilon))}\\{\omega
x_{j}>\rho\\}$ that restricts to
$\mathord{\mathbb{R}}^{n}\setminus{\mathrm{Cub}_{n,1,1}^{\rho^{\prime}}}$ for
all $\rho^{\prime}>\rho$.

Applying Corollary 4.3.22 inductively, we conclude that the restriction of $P$
to $\mathord{\mathbb{R}}^{n}\setminus{\mathrm{Cub}_{n,1,1}^{\rho^{\prime}}}$
is bounded from below for all sufficient large $\rho^{\prime}$.

To meet the assumptions of the Operator Modification Lemma 4.6.2, we extend
$P$ to a block Dirac operator $\mathtt{P}$ on
$M\times\mathord{\mathbb{R}}^{n}$. By increasing $\rho$ if necessary, we may
assume that the core of $\mathtt{P}$ is contained in $M\times\rho/2I^{n}$.

Let $H$, $\Phi$, and
$U_{left}\subseteq\mathord{\mathbb{R}}^{n}\setminus{\mathrm{Cub}_{n,1,1}^{\rho}}$
be as in the Embedding Theorem 3.2.2. By the Operator Modification Lemma
4.6.2, $\mathtt{P}$ restricts to $P$ on $M\times R\Phi(H\times(a,b))$ for all
$1\leq a<b\leq 2$ and acts like a differential operator on the cylinder
variable; more precisely, it is a
${(R\Phi)^{-1}}^{\ast}\mathord{\mathrm{pr}}_{2}^{\ast}\mathcal{C}_{c}^{\infty}(a,b)$-derivation.
Thus, $\mathtt{P}$ also restricts to $P$ on $M\times\bigl{(}U_{left}\cup
R\Phi(H\times(1.8,2))\bigr{)}$.

By the Operator Modification Lemma 4.6.2, there is a $Q=Q(R)$ on
$\Gamma_{c}(M\times R\Phi(H\times(1,2);\mathfrak{S}_{G_{R}})$ for all $R\geq
1$ that agrees with $P$ on $M\times R\Phi(H\times(1.8,2))$ and decomposes on
$M\times R\Phi(H\times(1,1.2))$ as follows

$\Phi_{R}^{\ast}Q\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\Phi_{R}^{\ast}Q\mathord{\upharpoonleft}_{M\times H\times
1.1R}\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}_{\mathord{\mathbb{R}}}.$

Let ${}_{R}\Theta$ and ${}_{R}\Psi_{n}$ be as in the proof of Theorem 3.3.1.
To ease the notation, we abbreviate $\mathrm{id}\times{}_{R}\Theta$ and
$\mathrm{id}\times{}_{R}\Psi_{n}$ to ${}_{R}\Theta$ and ${}_{R}\Psi_{n}$,
respectively.

The operator $({}_{R}\Theta\circ{}_{R}\Psi_{n})_{\ast}(Q)$ is again a
symmetric pseudo Dirac operator. We chose ${}_{R}\Theta$ in the proof of
Theorem 3.3.1 such that
${}_{R}\Theta\circ{}_{R}\Psi_{n}\circ{}_{R}\Phi(m,h,s)=(m,R\Phi(h,1)-(s-R)e_{1})$
on $M\times H\times R[1,1+\epsilon))$ for all $1/10>\epsilon>0$. Since
${}_{R}\Theta\circ{}_{R}\Psi_{n}\circ{}_{R}\Phi(H\times
R[1,1+\epsilon))=\\{R(\rho+1-\epsilon)<x_{1}\leq R(\rho+1)\\}$, we obtain the
decomposition

$({}_{R}\Theta{}_{R}\Psi_{n})_{\ast}(Q)=Q\mathord{\upharpoonleft}_{\\{x_{1}=R(\rho+1)\\}}\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}_{\mathord{\mathbb{R}}}$

on $\\{R(\rho+1-\epsilon)<x_{1}\leq R(\rho+1)\\}$.

By $(iv)$ of the Operator Modification Lemma 4.6.2, the operator $Q$ and hence
$({}_{R}\Theta\circ{}_{R}\Psi_{n})_{\ast}(P)$ is bounded from below with lower
bound $c>0$ that is independent of $R$, provided $R$ is sufficiently large.
Lemma 4.4.9 implies that the restriction
$Q\mathord{\upharpoonleft}_{\\{x_{1}=R(\rho+1)\\}}$ is bounded from below with
lower bound bigger than $c/2$, for all sufficient large $R$. Thus, the
constant extension of
$Q\mathord{\upharpoonleft}_{\\{x_{1}=R(\rho+1)\\}}\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}_{\mathord{\mathbb{R}}}$
to $\\{R(\rho+1-\epsilon)<x_{1}\\}$ is bounded from below with lower bound
bigger than $c/2$.

It follow from Corollary 4.3.22 and Lemma 4.3.20, that

$\overline{Q}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=({}_{R}\Theta\circ{}_{R}\Psi_{n})_{\ast}(Q)\cup
Q\mathord{\upharpoonleft}_{\\{x_{1}=R(\rho+1)\\}}\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}_{\mathord{\mathbb{R}}}$

is a pseudo Dirac operator on
$M\times\bigl{(}{}_{R}\Theta\circ{}_{R}\Psi_{n}\circ{}_{R}\Phi(H\times
R(1,2))\cup\\{R(\rho+1-\epsilon)<x_{1}\\}\bigr{)}$ that is symmetric and
bounded from below, provided $R$ is sufficiently large.

Of course, $\overline{Q}$ restricts to $M\times R\Phi(H\times(1.8,2))$ and
agrees there with $P$. Applying Corollary 4.3.22 and 4.3.20 a second time, the
operator
$P_{\mathrm{fill}}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=P\cup\overline{Q}$ is a symmetric pseudo Dirac operator on
$M\times\mathord{\mathbb{R}}^{n}$ that is bounded from below, provided $R$ is
sufficiently large.

It remains to show that $P_{\mathrm{fill}}$ is a block Dirac operator. Let
$\mathtt{R}_{1}=\mathtt{R}_{1}(R)>R(\rho+1)$ be sufficiently large such that
$M\times\mathtt{R}_{1}I^{n}$ contains $L(R)$ and
${}_{R}\Theta\circ{}_{R}\Psi_{n}(L(R))$ and such that ${}_{R}\Theta$ and
${}_{R}\Psi_{n}$ restrict to the identity on the complement of
${\mathrm{Cub}_{n,1,1}^{\mathtt{R}_{1}}}$. Then $M\times\mathtt{R}_{1}I^{n}$
serves as a core for $P_{\mathrm{fill}}$. Since $P_{\mathrm{fill}}$ restricts
to $P$ on
$M\times\mathord{\mathbb{R}}^{n}\setminus{\mathrm{Cub}_{n,1,1}^{\mathtt{R}_{1}}}$
and since, by construction,

$P_{\\{x_{1}>R(\rho+1)\\}}=({}_{R}\Theta\circ{}_{R}\Psi_{n})_{\ast}(Q)\mathord{\upharpoonleft}_{\\{x_{1}=R(\rho+1)\\}}\boxtimes\mathrm{id}+\mathrm{e/o}\boxtimes{\not{\mathit{D}}}_{\mathord{\mathbb{R}}},$

we deduce that $P_{\mathrm{fill}}$ has block form. Finally,
$P_{\mathrm{fill}}-{\not{\mathfrak{D}}}_{g_{P_{\mathrm{fill}}}}$ is bounded,
because it is true for all the three summands whose union forms
$P_{\mathrm{fill}}$.

Thus, $P_{\mathrm{fill}}$ a filler for the given $n$-horn, so
$\widetilde{\Psi\mathrm{Dir}}^{\times}_{\bullet}(M)$ is a Kan set. ∎

###  4.7 The Symbol Map is a Kan Fibration

Recall that every pseudo Dirac operator $P$ comes with an underlying
Riemannian metric $g_{P}$ that determines the principal symbol of $P$ (and
also determines the spinor bundle on which $P$ acts). We demand that $g_{P}$
is a block metric and that the decompositions of $P$ and $g_{P}$ are
compatible. In Section 4.4 we have established that the map that assigns to a
pseudo Dirac operator its underlying metric assembles to a cubical map
$\widetilde{\Psi\mathrm{Dir}}_{\bullet}(M)\rightarrow\widetilde{\mathcal{R}_{\bullet}}(M)$
which we refer to as the symbol map.

We use the Kan property of
$\widetilde{\Psi\mathrm{Dir}}^{\times}_{\bullet}(M)$ from the previous
subsection to establish the main result of this section.

######  Theorem 4.7.1.

The symbol map
$\widetilde{\Psi\mathrm{Dir}}^{\times}_{\bullet}(M)\rightarrow\widetilde{\mathcal{R}_{\bullet}}(M)$
is a Kan fibration.

This result is remarkable because it implies that all index theoretic
properties are stored in a single fibre
$\widetilde{\Psi\mathrm{Dir}}^{\times}_{\bullet}(M)_{g_{0}}$ because
$\widetilde{\mathcal{R}_{\bullet}}(M)$ is combinatorially contractible, see
Proposition 2.1.33. The fibre
$\widetilde{\Psi\mathrm{Dir}}^{\times}_{\bullet}(M)_{g_{0}}$ is a Kan set by
formal reasons, see Lemma 2.1.23. When we are determining the homotopy groups
of $\widetilde{\Psi\mathrm{Dir}}^{\times}_{\bullet}(M)$, we will make explicit
use of the Kan property of the fibre
$\widetilde{\Psi\mathrm{Dir}}^{\times}_{\bullet}(M)_{g_{0}}$, because we will
compare the fibre to the space of Fredholm operators on a _single_ Hilbert
space determined by $g_{0}$.

Of course, Theorem 4.7.1 implies Theorem 4.6.1 by formal reasons. However, the
proof of Theorem 4.7.1 relies on
$\widetilde{\Psi\mathrm{Dir}}^{\times}_{\bullet}(M)$ being a Kan set. It goes
roughly as follows: For a given $i$-horn of invertible block Dirac operator
and a given block metric, we find a filler by the Kan property of
$\widetilde{\Psi\mathrm{Dir}}^{\times}_{\bullet}(M)$. So far, the underlying
metric may disagree with the given one. To correct this mismatch, we will
modify the filler so that they do agree.

The modification procedure will rely on two tools. The first tool is a
fibration-kind result inspired from the result of [ebert2017indexdiff]*Section
4.2 although the methods are quite different. It allows us to “rotate” the
principal symbol of the filler to the required one. The second tool is based
on the gluing constructions from Section 4.3.

The first tool is based on the following analytical result and its
consequences.

######  Lemma 4.7.2.

Let $P\in\widetilde{\Psi\mathrm{Dir}}_{n}(M)$ such that all faces
$\partial^{\varepsilon}_{i}P$ are invertible. Then there are constants
$c,\texttt{C}>0$, each depending only on the faces
$\partial^{\varepsilon}_{i}P$, and a compact subset $K\subseteq
M\times\mathord{\mathbb{R}}^{n}$, such that all sections $0\neq u\in
L^{2}(M\times\mathord{\mathbb{R}}^{n};\mathfrak{S}_{g_{P}})$ with
$||Pu||_{0}\leq\gamma||u||_{0}$ for $\gamma<c$ satisfy

$\frac{||u||_{0,K}}{||u||_{0}}\geq\frac{10}{11}\left(\frac{9}{10}-\mathtt{C}\gamma\right)>0,$
(4.7)

where
$||u||_{0,K}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=||1_{K}\cdot u||_{0}$.

###### Proof.

For this proof, it will be more convenient to use the following variant of the
Sobolev $1$-norm

$||u||_{1}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=||u||_{0}+||\nabla u||_{0},$

which is equivalent to the (usual) Sobolev 1-norm. More generally, for each
measurable $K\subset M\times\mathord{\mathbb{R}}^{n}$ we will denote, just for
this proof,
$||u||_{1,K}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=||u||_{0,K}+||\nabla u||_{0,K}$. For each compactly supported smooth
function $\varphi$, we can estimate $||\varphi u||_{1}$ against
$||u||_{1,\mathrm{supp}\,\varphi}$ as follows:

$\displaystyle||\varphi u||_{1}$ $\displaystyle=||\varphi
u||_{0}+||\nabla\varphi u||_{0}$ $\displaystyle=||\varphi
u||_{0}+||\mathrm{d}\varphi\otimes u+\varphi\nabla u||_{0}$
$\displaystyle\leq||\varphi
u||_{0}+||\mathrm{grad}(\varphi)||_{\infty}\cdot||u||_{0,\mathrm{supp}\,\varphi}+||\varphi\nabla
u||_{0}$
$\displaystyle\leq||\varphi||_{\infty}||u||_{1,\mathrm{supp}\,\varphi}+||\mathrm{grad}(\varphi)||_{\infty}\cdot||u||_{0,\mathrm{supp}\,\varphi}.$

Let $\chi\colon\mathord{\mathbb{R}}\rightarrow[0,1]$ be a smooth, even
function that is identically $1$ on $(-r,r)$ and such that
$\chi^{-1}(\mathord{\mathbb{R}}_{>0})=(-r^{\prime},r^{\prime})$ for some
$r^{\prime}>r>0$. We assume that $r$ and $r^{\prime}$ are sufficiently large
such that the interior of $M\times rI^{n}$ contains the core of $P$. Let
$\\{\chi^{\bm{\alpha}}\\}$ be the associated partition of unity from
Definition 4.2.5 and set
$K\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\mathrm{supp}\,\chi^{\mathbf{0}}$. Recall from the proof of Proposition
4.3.7 that invertibility of $\partial^{\varepsilon}_{i}P$ implies, for all
sections $u$ supported within $\\{\varepsilon x_{i}>R\\}$, the inequality

$||Pu||_{0}\geq c(i,\varepsilon)||u||_{1},$

where $c(i,\varepsilon)$ is the product of the operator norm of
$(\partial^{\varepsilon}_{i}P)^{-1}\colon
L^{2}(M\times\mathord{\mathbb{R}}^{n-1};\mathfrak{S}_{g_{\partial^{\varepsilon}_{i}P}})\rightarrow
H^{1}(M\times\mathord{\mathbb{R}}^{n-1};\mathfrak{S}_{g_{\partial^{\varepsilon}_{i}P}})$
and a constant that comes from the equivalence of the two variants of the
Sobolev 1-norms on
$H^{1}(M\times\mathord{\mathbb{R}}^{n-1};\mathfrak{S}_{g_{\partial^{\varepsilon}_{i}P}})$.
The second constant only depends on $g_{\partial^{\varepsilon}_{i}P}$, too.

Recall further that the block form implies
$\mathrm{supp}\,P\chi^{\bm{\alpha}}u\subseteq\mathrm{supp}\,\chi^{\bm{\alpha}}$
by Corollary 4.2.7. Indeed, if we use the notation from Corollary 4.2.7, then
$\mathrm{supp}\,\chi^{{\bm{\alpha}}}$ is the closure of
$V_{r<r^{\prime}}({\bm{\alpha}})$ and the claim follows from applying
Corollary 4.2.7 to $V_{r-\epsilon<r^{\prime}+\epsilon}$ for all sufficiently
small $\epsilon>0$.

For ${\bm{\alpha}}\neq\mathbf{0}$, this implies the inequality

$\displaystyle||\chi^{\bm{\alpha}}u||_{1}$
$\displaystyle\leq\max\\{c(i,\varepsilon)^{-1}\\}||P\chi^{\bm{\alpha}}u||_{0,\mathrm{supp}\,\chi^{\bm{\alpha}}}$
$\displaystyle\leq\max\\{c(i,\varepsilon)^{-1}\\}\left(||\chi^{\bm{\alpha}}Pu||_{0}+||\mathrm{grad}(\chi)^{\bm{\alpha}}||_{\infty}||u||_{0,\mathrm{supp}\,\chi^{\bm{\alpha}}}\right)$
$\displaystyle\leq\max\\{c(i,\varepsilon)^{-1}\\}\left(||Pu||_{0}+||\mathrm{grad}(\chi)^{\bm{\alpha}}||_{\infty}||u||_{0}\right).$

Putting these inequalities together yields

$\displaystyle||u||_{1}$
$\displaystyle\leq||\chi^{\mathbf{0}}u||_{1}+\sum_{{\bm{\alpha}}\neq\mathbf{0}}||\chi^{\bm{\alpha}}u||_{1}$
$\displaystyle\begin{split}&\leq||u||_{1,K}+||\mathrm{grad}(\chi^{\mathbf{0}})||_{\infty}||u||_{0,K}\\\
&\qquad\qquad\
+\underbrace{3^{n}\max\\{c(i,\varepsilon)^{-1}\\}}_{=:\mathtt{C}}\left(||Pu||_{0}+\max\\{||\mathrm{grad}(\chi^{\bm{\alpha}})||_{\infty}\\}||u||_{0}\right).\end{split}$

We now assume that $||Pu||_{0}\leq\gamma||u||_{0}$. The previous estimate then
yields

$\displaystyle||u||_{0}$ $\displaystyle\leq||u||_{0}+||\nabla u||_{0}-||\nabla
u||_{0,K}=||u||_{1}-||\nabla u||_{0,K}$
$\displaystyle\begin{split}&\leq||u||_{1,K}+||\mathrm{grad}(\chi^{\mathbf{0}})||_{\infty}||u||_{0,K}-||\nabla
u||_{0,K}\\\ &\qquad\qquad\
+\mathtt{C}\left(\gamma||u||_{0}+\max\\{||\mathrm{grad}(\chi^{\bm{\alpha}})||_{\infty}\\}||u||_{0}\right)\end{split}$
$\displaystyle\leq\left(1+||\mathrm{grad}(\chi^{\mathbf{0}})||_{\infty}\right)||u||_{0,K}+\mathtt{C}\left(\gamma||u||_{0}+\max\\{||\mathrm{grad}(\chi^{\bm{\alpha}})||_{\infty}\\}||u||_{0}\right),$

or equivalently,

$\bigl{(}1-\mathtt{C}(\gamma+\max\\{||\mathrm{grad}(\chi^{\bm{\alpha}})||_{\infty}\\})\bigr{)}||u||_{0}\leq(1+||\mathrm{grad}(\chi^{\mathbf{0}})||_{\infty})||u||_{0,K}.$

By replacing $\chi^{\bm{\alpha}}$ with
$\chi^{\bm{\alpha}}(\rho^{-1}\,\cdot\,)$, where $\rho$ is sufficiently large,
we may assume777Keep in mind that rescaling the partition of unity replaces
$K$ by $\rho K$.
$||\mathrm{grad}(\chi^{{\bm{\alpha}}})||_{\infty}<\min\\{1/10,1/10\mathtt{C}^{-1}\\}$.
For $u\neq 0$, we end up with the desired inequality:

$\frac{||u||_{0,K}}{||u||_{0}}\geq\frac{10}{11}\left(\frac{9}{10}-\mathtt{C}\gamma\right).$

This lower bound is positive if and only if

$\gamma<3^{2-n}\frac{\min\\{c(i,\varepsilon)\\}}{10}=:c.\qed$

This inequality has important implications concerning the spectrum of $P$.
Recall that a sequence of elements $(v_{n})_{n\in\mathord{\mathbb{N}}}$ in a
Hilbert space $\mathcal{H}$ is a sequence of _approximate eigenvectors for_
$P$ if $||v_{n}||=1$ and there is a (necessarily unique)
$\lambda\in\mathord{\mathbb{C}}$ such that the sequence
$(P-\lambda\mathrm{id})v_{n}$ converges to zero. In that case, $\lambda$ is
called an _approximate eigenvalue_ (associated to
$(v_{n})_{n\in\mathord{\mathbb{N}}}$). Note that the spectrum of a self-
adjoint operator consists of approximate eigenvalues only.

######  Corollary 4.7.3.

Let $P$ be as in the previous lemma. Then each sequence of approximate
eigenvectors of $P$ with approximate eigenvalue $\lambda\in(-c,c)$ has a
convergent subsequence. In particular, $\lambda$ is an eigenvalue.

###### Proof.

For a given spectral value $\lambda\in(-c,c)$ of $P$ take a sequence of
approximate eigenvalues $(v_{n})_{n\in\mathord{\mathbb{N}}}$ associated to it.
Since $||Pv_{n}||_{0}<c||v_{n}||_{0}=c$ for almost all
$n\in\mathord{\mathbb{N}}$, the fundamental elliptic estimate implies
$v_{n}\in H^{1}(\mathfrak{S}_{g_{P}})$ and that
$(||v_{n}||_{1})_{n\in\mathord{\mathbb{N}}}$ is bounded.

Pick a smooth compactly supported function $\varphi$ that is identically $1$
on $K$, the compact set from Lemma 4.7.2. Then $||v_{n}||_{1,K}\leq||\varphi
v_{n}||_{1}\leq||\varphi||_{1,1}||v_{n}||_{1}$ and the classical Rellich lemma
implies that $(\varphi v_{n})_{n\in\mathord{\mathbb{N}}}$ has a convergent
subsequence with respect to $||\text{--}||_{0}$. Thus
$(v_{n})_{n\in\mathord{\mathbb{N}}}$ has a convergent subsequence with respect
to $||\text{--}||_{0,K}$. But inequality 4.7 implies that the convergent
subsequence also converges with respect to $||\text{--}||_{0}$. This limit is
an eigenvector and $\lambda$ its associated eigenvalue. ∎

As an application, we prove the for us very important

######  Proposition 4.7.4 (Sausage Lemma).

Let $P$ and $c$ as in Lemma 4.7.2. Then $\mathrm{spec}(P)\cap(-c,c)$ consists
of eigenvalues only and the linear hull
$\bigoplus_{|\lambda|<c_{0}}\mathrm{Eig}(P,\lambda)$ of eigenspaces forms a
finite dimensional subvector space for all $c_{0}<c$.

###### Proof.

The first claim was part of the previous corollary.

To prove the second claim, we first show that every eigenspace is finite
dimensional. Obviously, $\mathrm{Eig}(P,\lambda)$ is a closed subspace of
$L^{2}(\mathfrak{S}_{g_{P}})$, hence a Hilbert space. For a given sequence of
normalised eigenvectors $(v_{n})_{n\in\mathord{\mathbb{N}}}$, the previous
corollary implies the existence of a convergent subsequence. Thus, the unit
ball of $\mathrm{Eig}(P,\lambda)$ is compact, so $\mathrm{Eig}(P,\lambda)$ is
finite dimensional.

We now prove that $P$ has only finitely many eigenvalues in $[-c_{0},c_{0}]$
for all $c_{0}<c$. Assume the contrary. Then there is a sequence
$(\lambda_{n})_{n\in\mathord{\mathbb{N}}}$ of pairwise different eigenvalues
lying in $[-c_{0},c_{0}]$. For each $\lambda_{n}$, pick a normalised
eigenvector $v_{n}$. For
$\lambda\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\limsup_{n}\lambda_{n}\in[-c_{0},c_{0}]$, we can pick a subsequence of
$(v_{n})_{n\in\mathord{\mathbb{N}}}$ that serves as a sequence of approximate
eigenvectors for $\lambda$. But Corollary 4.7.3 implies that this subsequence
has a convergent subsequence $(v_{n_{k}})_{k\in\mathord{\mathbb{N}}}$. The
limit $w$ is an eigenvector corresponding to $\lambda$ and eigenvectors
corresponding to different eigenvalues are perpendicular, so we end up with

$0=\langle v_{n_{k}},w\rangle\xrightarrow{k\to\infty}\langle w,w\rangle=1,$

a contradiction. ∎

The Sausage Lemma offers the possibility to use spectral-flow-style arguments.

######  Lemma 4.7.5.

Let $P\colon X\rightarrow\widetilde{\Psi\mathrm{Dir}}_{n}(M)$ be a map, such
that ${\mathcal{G}_{g_{P},g_{0}}}_{\ast}(P)\colon
X\rightarrow\mathrm{Hom}(H^{1}(\mathfrak{S}_{g_{0}}),H^{0}(\mathfrak{S}_{g_{0}}))$
is continuous and such that $\partial^{\varepsilon}_{i}P$ is a constant map of
invertible operators. Assume that there is a constant $c>0$ such that

$\mathrm{spec}(P_{y})\cap(-c,c)=:\\{\lambda_{0}(y)\leq\dots\leq\lambda_{N_{y}}(y)\\}$

consists of finitely many eigenvalues for all $y\in X$. Then, for each $x\in
X$, there is an open neighbourhood $U$ of $x$ such that $N_{(\text{--})}$ is
constant on $U$ and all $\lambda_{j}\colon U\rightarrow\mathord{\mathbb{R}}$
are continuous functions.

###### Proof.

For every $c_{0}$ that lies strictly between
$\max\\{|\lambda_{0}(x)|,|\lambda_{N}(x)|\\}$ and $c$, choose a smooth,
monotonically increasing function
$b\colon\mathord{\mathbb{R}}\rightarrow[-2c,2c]$ that is the identity on
$[-c_{0},c_{0}]$. The bounded transform
$\hat{P}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=b(\mathcal{G}_{g_{P},g}\circ P\circ\mathcal{G}_{g_{0},g_{P}})$ is a
continuous map of bounded, self-adjoint operators $X\rightarrow
B(L^{2}(\mathfrak{S}_{g_{0}}))$ [ebert2017indexdiff]*Proposition 3.7 whose
values have the same eigenvalues between $(-c_{0},c_{0})$ as the values of
$P$. The statement now follows from the fact that eigenvalues of bounded
operators depend continuously on the operator with respect to the operator
norm [booss1993elliptic]*Lemma 17.1. ∎

######  Remark 4.7.6.

The functions of eigenvalues $\lambda_{j}$ are only defined locally near $x$
because they may leave the interval $(-c,c)$. Also, the number of eigenvalues
may not be locally constant.

Recall from Theorem B.0.30 that a smooth path
$P\colon(0,1)\rightarrow\overline{\Psi\mathrm{DO}}^{1}(\mathfrak{S})$ with
underlying path $g_{P}$ of Riemannian metrics has an extension
$P^{\mathrm{ex}}\in Op^{1}(L^{2}(\mathfrak{S}_{\mathrm{susp}(g_{P})}))$ whose
values are not necessarily in the subspace
$\overline{\Psi\mathrm{DO}}^{1}(\mathfrak{S}_{\mathrm{susp}(g_{P})})$.
However, this is a nice property to have, so we will give it a name.

######  Definition 4.7.7.

Let $X\subseteq\mathord{\mathbb{R}}^{n}$ be open and let $P\colon
X\rightarrow\overline{\Psi\mathrm{DO}}^{1}(\mathfrak{S})$ be a smooth map with
underlying map $g_{P}$ of Riemannian metrics. Then $P$ called _suspensionable_
if the extension $P^{\mathrm{ex}}$ given by

$(P^{\mathrm{ex}}\sigma)(n,x)=P_{x}(\sigma(\text{--},x))$

is an element of
$\overline{\Psi\mathrm{DO}}^{1}(\mathfrak{S}_{\mathrm{susp}(g_{P})})$.

The first tool is the following fibration-like result.

######  Proposition 4.7.8 (Path Lifting Property).

Consider $\widetilde{\mathcal{R}_{n}}(M)$ as a topological space equipped with
the smooth Fréchet topology and $\widetilde{\Psi\mathrm{Dir}}^{\times}_{n}(M)$
as a topological space equipped with the subspace topology of
$\overline{\Psi\mathrm{DO}}^{1}(\mathfrak{S})$. Let
$\gamma\colon[0,1]\rightarrow\widetilde{\mathcal{R}_{n}}(M)$ be a smooth curve
that is constant near the end points and satisfies

$\gamma_{t}|_{\\{\varepsilon
x_{i}>R\\}}=\sigma_{i}\partial^{\varepsilon}_{i}\gamma_{0}|_{\\{\varepsilon
x_{i}>R\\}}$

for some $R>0$, all
$(i,\varepsilon)\in\\{1,\dots,n\\}\times\mathord{\mathbb{Z}}_{2}$, and all
$t\in[0,1]$.

Then each commutative outer square

can be filled with a smooth map that is constant near the boundary and such
that the upper triangle commutes and the lower right triangle commutes up to
homotopy relative to the boundary. In fact $g_{Q_{(\,\cdot\,)}}$ and
$\gamma_{(\,\cdot\,)}$ differs by a (non-diffeomorphical) reparametrisation.
Furthermore, we may assume that the lift is suspensionable and satisfies
$\partial^{\omega}_{j}Q_{t}=\partial^{\omega}_{j}Q_{0}$ for all
$(j,\omega)\in\\{1,\dots,n\\}\times\mathord{\mathbb{Z}}_{2}$.

In contrast to the proof in [ebert2017indexdiff]*Section 4.2 our argument is
quite geometrical. The idea is the following one: For an ordered pair of two
Riemannian metric $(g_{1},g_{2})$ and a non-zero covector $\xi\in T^{\vee}M$,
there is a unique $A\in\mathord{\mathrm{GL}}^{+}(d)$ that stretches and
rotates ${g_{1}}_{\sharp}(\xi)$ to ${g_{2}}_{\sharp}(\xi)$ in the plane
spanned by the two vectors and that acts identically on the orthogonal
complement. We find a lift $b_{g_{1},g_{2}}(\xi)$ in $Cl_{d,0}$ such that its
induced linear map
$b_{g_{1},g_{2}}(\xi)(\text{--})b_{g_{1},g_{2}}(\xi)^{\ast}$ on
$\mathord{\mathbb{R}}^{d}$ agrees with $A$. We quantise the symbol to get a
pseudo differential operator $B_{g_{1},g_{2}}$. If
$B_{\gamma_{0},\gamma_{t}}PB_{\gamma_{0},\gamma_{t}}^{\ast}$ were a curve of
invertible operators, then it would be the desired lift. Unfortunately, it is
not clear that this curve is invertible, so we need to stop the curve from
time to time and “push the spectral values away from zero”.

Let us now introduce the required concepts.

######  Definition 4.7.9.

For $h\in\widetilde{\mathcal{R}_{n}}(M)$ and $R>0$, denote by
$\widetilde{\mathcal{R}_{n}}(M)_{h,R}$ the space of all block metrics that
agree with $h$ outside of $M\times RI^{n}$ equipped with the smooth compact
open topology.

######  Lemma 4.7.10.

For all $h\in\widetilde{\mathcal{R}_{n}}(M)$ and all $R>0$, there is a smooth
map
$B\colon\widetilde{\mathcal{R}_{n}}(M)_{h,R}\times\widetilde{\mathcal{R}_{n}}(M)_{h,R}\rightarrow\Psi\mathrm{DO}^{0}(\mathfrak{S},\mathfrak{S})$
that satisfies the following three conditions:

  1. (i)

The following diagram commutes

$\textstyle{\widetilde{\mathcal{R}_{n}}(M)_{h,R}\times\widetilde{\mathcal{R}_{n}}(M)_{h,R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{B}$$\scriptstyle{\mathrm{id}}$$\textstyle{\Psi\mathrm{DO}^{0}(\mathfrak{S},\mathfrak{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{target}}$$\scriptstyle{\mathrm{source}}$$\textstyle{\widetilde{\mathcal{R}_{n}}(M)_{h,R}\times\widetilde{\mathcal{R}_{n}}(M)_{h,R}.}$

  2. (ii)

For all $g_{1},g_{2}\in\widetilde{\mathcal{R}_{n}}(M)_{h,R}$, the operator
$B_{g_{1},g_{2}}-p\mathcal{G}_{g_{1},g_{2}}$ is compactly supported.

  3. (iii)

For all $g_{1},g_{2}\in\widetilde{\mathcal{R}_{n}}(M)_{h,R}$, we have

$\mathrm{symb}_{0}(B_{g_{1},g_{2}})\circ
g_{1\,\sharp}(\,\cdot\,)\cdot_{g_{1}}\circ\,\mathrm{symb}_{0}(B_{g_{1},g_{2}}^{\ast})=g_{2\,\sharp}(\,\cdot\,)\cdot_{g_{2}},$

where $\cdot_{g_{j}}$ is the Clifford left action on $\mathfrak{S}_{g_{j}}$.

###### Proof.

Let $\pi\colon\mathrm{Pos}\rightarrow M\times\mathord{\mathbb{R}}^{n}$ be the
bundle whose fibres consist of positive definite, symmetric bilinear forms and
$\mathfrak{S}\rightarrow\mathrm{Pos}$ be the fibre bundle whose fibre of
$g_{x}$ is $\mathfrak{S}_{g_{x}}$. Similarly, we define
$Cl(T(M\times\mathord{\mathbb{R}}^{n}),\text{--})\rightarrow\mathrm{Pos}$ to
be the bundle whose fibre of $g_{x}$ is the Clifford algebra
$Cl(T_{x}(M\times\mathord{\mathbb{R}}^{n}),g_{x})$. We denote with
$||\text{--}||\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=||\text{--}||_{g_{x}}$ the norm on
$Cl(T_{x}(M\times\mathord{\mathbb{R}}^{n}),g_{x})$ induced by $g_{x}$ and we
denote with $\cdot_{g_{x}}$ the Clifford multiplication and left action of
$Cl(M,g_{x})$ on $\mathfrak{S}_{g_{x}}$. Recall furthermore that

$\Phi_{g_{1,x},g_{2,x}}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=Cl(\tau_{g_{1,x},g_{2,x}})\colon
Cl(T_{x}(M\times\mathord{\mathbb{R}}^{n}),g_{1,x})\rightarrow
Cl(T_{x}(M\times\mathord{\mathbb{R}}^{n}),g_{2,x})$

denotes the algebra isomorphism induced by the pre-gauge map on the tangent
spaces. Using this notation, we set

$\displaystyle\begin{split}b_{g_{1},g_{2}}(\xi):=&\frac{||\Phi_{g_{2},g_{1}}g_{2\,\sharp}(\xi)||^{1/2}}{||g_{1\,\sharp}(\xi)||^{1/2}}\cdot\\\
&\quad\frac{\Phi_{g_{2},g_{1}}g_{2\,\sharp}(\xi)/||\Phi_{g_{2},g_{1}}g_{2\,\sharp}(\xi)||+g_{1\,\sharp}(\xi)/||g_{1\,\sharp}(\xi)||}{||\Phi_{g_{2},g_{1}}g_{2\,\sharp}(\xi)/||\Phi_{g_{2},g_{1}}g_{2\,\sharp}(\xi)||+g_{1\,\sharp}(\xi)/||g_{1\,\sharp}(\xi)||\,||}\cdot_{g_{1}}\frac{g_{1\,\sharp}(\xi)}{||g_{1\,\sharp}(\xi)||}\cdot_{g_{1}}.\end{split}$

The map $b$ takes values in even and $Cl_{n,0}$-right linear symbols, more
precisely,

$b_{g_{1},g_{2}}\in\\{\sigma\in\mathrm{Symb}_{0}(\mathfrak{S}_{g_{1}},\mathfrak{S}_{g_{2}})\,:\,\sigma\text{
is even and }Cl_{d,0}\text{-right linear}\\},$

see Definition B.0.12 for a reminder of
$\mathrm{Symb}_{0}(\mathfrak{S}_{g_{1}},\mathfrak{S}_{g_{2}})$.

Denote by $\mathrm{Pos}^{2}=\mathrm{Pos}\times_{M}\mathrm{Pos}$ the fibre
bundle of pairs of positive definite, symmetric, bilinear forms on
$T(M\times\mathord{\mathbb{R}}^{n})$. We consider it as fibre bundle over
$\mathrm{Pos}$, whose bundle projection is the projection to the first
component. Furthermore, let

$F\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\\{(v,w)\in T(M\times\mathord{\mathbb{R}}^{n})\setminus\\{0\\}\oplus
T(M\times\mathord{\mathbb{R}}^{n})\setminus\\{0\\}\,:\,v\neq-\lambda w\text{
for all }\lambda>0\\}$

be the bundle of pairs of tangent vectors whose cones are not antipodal half
lines.

The map
$b\colon\widetilde{\mathcal{R}_{n}}(M)_{h,R}\times\widetilde{\mathcal{R}_{n}}(M)_{h,R}\rightarrow\mathrm{Symb}_{0}^{\times}(\mathfrak{S},\mathfrak{S})$
is continuous because it is induced by the following sequence of bundle maps

$\textstyle{\pi^{\ast}T^{\vee}(M\times\mathord{\mathbb{R}}^{n})\setminus\\{0\\}\times_{M}\mathrm{Pos}^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi^{\ast}F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Cl_{,}(M\times\mathord{\mathbb{R}}^{n},\text{--})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Hom}(\mathfrak{S},\mathfrak{S}).}$

The first map is given by
$(g_{1},g_{2})\mapsto(g_{1\,\sharp}(\xi),\Phi_{g_{2},g_{1}}(g_{2\,\sharp}(\xi)))$.
Notice that this map indeed takes values in $\pi^{\ast}F$ because
$g_{1\,\sharp}(\xi)=-\Phi_{g_{2},g_{1}}(\lambda g_{2\,\sharp}(\xi))$ for some
$\lambda>0$ implies that $g_{1}$ is not positive definite:

$\displaystyle g_{1}\bigl{(}g_{1\,\sharp}(\xi),g_{1\,\sharp}(\xi)\bigr{)}$
$\displaystyle=-\lambda
g_{1}\bigl{(}g_{1\,\sharp}(\xi),\Phi_{g_{2},g_{1}}(g_{2\,\sharp}(\xi))\bigr{)}$
$\displaystyle=-\lambda
g_{1}\bigl{(}{g_{1}}_{\sharp}(\xi),\Phi_{g_{2},g_{1}}({g_{2}}_{\sharp}\circ
g_{1}^{\flat}\,{g_{1}}_{\sharp}(\xi)\bigr{)}$ $\displaystyle=-\lambda
g_{1}\bigl{(}{g_{1}}_{\sharp}(\xi),\Phi_{g_{2},g_{1}}^{3}({g_{1}}_{\sharp}(\xi))\bigr{)}<0$

because
$\Phi_{g_{2},g_{1}}|_{T(M\times\mathord{\mathbb{R}}^{n})}=\tau_{g_{1},g_{2}}$
is the pre-gauge map, which is positive definite and self-adjoint with respect
to $g_{1}$. The second map is given by

$(v,w)\mapsto\frac{||w||^{1/2}_{g_{1}}}{||v||^{1/2}_{g_{1}}}\frac{w/||w||_{g_{1}}+v/||v||_{g_{1}}}{||w/||w||_{g_{1}}+v/||v||_{g_{1}}||_{g_{1}}}\cdot_{g_{1}}\frac{v}{||v||_{g_{1}}}\cdot_{g_{1}},$

which is clearly continuous. The third map is just the action of the Clifford
algebra bundle on the spinor bundle, which is even isometric. Note, in
particular, that $b$ takes values in invertible symbols.

Away from $M\times RI^{n}$, the map $b$ is the constant map with value
$\mathrm{id}_{\mathfrak{S}_{h_{x}}}$. Recall from Lemma B.0.23 the continuous
(with respect to the amplitude topology) section
$\rho\colon\mathrm{Symb}_{0}(\mathfrak{S}_{g_{1}})\rightarrow\Psi\mathrm{DO}^{0}(\mathfrak{S}_{g_{1}})$.
We set
$\tilde{B}_{g_{1},g_{2}}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\rho(b_{g_{1},g_{2}})-\rho(b_{g_{1},g_{1}})+\mathrm{id}$. So far,
$\tilde{B}_{g_{1},g_{2}}-\mathrm{id}$ might not have compact support. To
achieve this, we pick smooth compactly supported functions $\varphi,\psi\colon
M\times\mathord{\mathbb{R}}^{n}\rightarrow[0,1]$ such that $\varphi\equiv 1$
on $M\times RI^{n}$ and $\psi\equiv 1$ on $\mathrm{supp}\,\varphi$. The
operator
$\hat{B}_{g_{1},g_{2}}\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\psi\tilde{B}_{g_{1},g_{2}}\varphi+\mathrm{id}\cdot(1-\varphi)$ has the
same principal symbol as $\tilde{B}_{g_{1},g_{2}}$ and differs from the
identity by a compactly supported operator. Recall the function
$\alpha_{g_{2},g_{1}}^{2}=\mathrm{d}\mathrm{vol}(g_{2})/\mathrm{d}\mathrm{vol}(g_{1})$.
Finally, we define the desired map

$\displaystyle
B\colon\widetilde{\mathcal{R}_{n}}(M)_{h,R}\times\widetilde{\mathcal{R}_{n}}(M)_{h,R}$
$\displaystyle\rightarrow\Psi\mathrm{DO}^{0}(\mathfrak{S},\mathfrak{S}),$
$\displaystyle(g_{1},g_{2})$ $\displaystyle\mapsto
p\mathcal{G}_{g_{1},g_{2}}\circ\alpha_{g_{1},g_{2}}\hat{B}_{g_{1},g_{2}}\in\Psi\mathrm{DO}^{0}(\mathfrak{S}_{g_{1}},\mathfrak{S}_{g_{2}}).$

By the definition of the topology of
$\Psi\mathrm{DO}^{0}(\mathfrak{S},\mathfrak{S})$, the map $B$ is smooth if and
only if $\hat{B}$ is smooth, which follows from the construction. It is clear,
that $B$ satisfies $(i).$

The composition of compactly supported pseudo differential operators with
vector bundle morphisms are still compactly supported. Furthermore,
$g_{1}=g_{2}$ on the complement of $M\times RI^{n}$ so that
$\alpha_{g_{1},g_{2}}=1$ on this set. Hence,
$B_{g_{1},g_{2}}-p\mathcal{G}_{g_{1},g_{2}}$ is compactly supported, which is
$(ii)$.

We now prove $(iii)$. Using the relation $(v+w)\cdot w=v\cdot(v+w)$ for unit-
length vectors $v,w$ in the Clifford algebra, we derive

$b_{g_{1},g_{2}}(\xi)\cdot_{g_{1}}g_{1\,\sharp}(\xi)\cdot_{g_{1}}b_{g_{1},g_{2}}(\xi)^{\ast}\cdot_{g_{1}}=g_{2\,\sharp}(\xi)\cdot_{g_{1}}.$

Recall that the pre-Gauge map is an isometric vector bundle map and
$Cl(M,g_{1})$-$Cl(M,g_{2})$-equivariant. It induces a linear map
$Op(p\mathcal{G}_{g_{1},g_{2}})\colon L^{2}(\mathfrak{S}_{g_{1}})\rightarrow
L^{2}(\mathfrak{S}_{g_{2}})$. The adjoints of the two maps are related by the
formula

$Op(p\mathcal{G}_{g_{1},g_{2}})^{\ast}=Op(\alpha_{g_{2},g_{1}}^{2}p\mathcal{G}_{g_{1},g_{2}}^{\ast})=Op(\alpha_{g_{2},g_{1}}^{2}p\mathcal{G}_{g_{2},g_{1}})$

because the underlying volume induced by the metrics is different. Symbol
calculus now implies the third claim

$\mathrm{symb}_{0}(B_{g_{1},g_{2}})\circ
g_{1\,\sharp}(\,\cdot\,)\cdot_{g_{1}}\circ\,\mathrm{symb}_{0}(B_{g_{1},g_{2}}^{\ast})=g_{2\,\sharp}(\,\cdot\,)\cdot_{g_{2}}.\qed$

Since compactly supported pseudo differential operators have compactly
supported adjoint, the previous proposition has the following important
consequence.

######  Corollary 4.7.11.

For every $g\in\widetilde{\mathcal{R}_{n}}(M)_{h,R}$ and every block Dirac
operator $P$ with underlying metric
$g_{P}\in\widetilde{\mathcal{R}_{n}}(M)_{h,R}$ the operator $B_{g_{P},g}\circ
P\circ B_{g_{P},g}^{\ast}$ is a block Dirac operator with underlying metric
$g$.

###### Proof.

The composition $B_{g_{P},g}PB_{g_{P},g}^{\ast}$ lies in
$\overline{\Psi\mathrm{DO}}^{1}(\mathfrak{S}_{g})$ because $B_{g_{P},g}$ is an
actual pseudo differential operator of order zero,
$\overline{\Psi\mathrm{DO}}^{1}(\mathfrak{S}_{g})$ is local, and compositions
on actual pseudo differential operators are actual pseudo differential
operators. The composition is clearly a symmetric operator.

Next we show that the composition is of block form. Let $r_{2}>r_{1}>R$ be a
real numbers such that $M\times r_{1}I^{n}$ contains the core of $P$, the
support of $B_{g_{P},g}-p\mathcal{G}_{g_{P},g}$ and
$B_{g_{P},g}^{\ast}-p\mathcal{G}_{g_{P},g}^{\ast}$ and such that $M\times
rI^{n}$ and such that $P$ decomposes away from $M\times r_{1}I^{n}$. Lemma
4.2.6 applied to $P$ now implies that $M\times r_{2}I^{n}$ serves as a core
for $B_{g_{P},g}PB_{g_{P},g}^{\ast}$.

Since $g_{P}=g=h$ away from $M\times r_{1}I^{n}$, we deduce that
$B_{g_{P},g}PB_{g_{P},g}^{\ast}$ decomposes there as $P$.

The operator
$B_{g_{P},g}{\not{\mathfrak{D}}}_{g_{P}}B_{g_{P},g}^{\ast}-{\not{\mathfrak{D}}}_{g}$
is a pseudo differential operator of order zero with compact support, hence a
bounded operator. The calculation

$\displaystyle B_{g_{P},g}PB_{g_{P},g}^{\ast}-{\not{\mathfrak{D}}}_{g}$
$\displaystyle=B_{g_{P},g}PB_{g_{P},g}^{\ast}-B_{g_{P},g}{\not{\mathfrak{D}}}_{g_{P}}B_{g_{P},g}^{\ast}+B_{g_{P},g}{\not{\mathfrak{D}}}_{g_{P}}B_{g_{P},g}^{\ast}-{\not{\mathfrak{D}}}_{g}$
$\displaystyle=B_{g_{P},g}\left(P-{\not{\mathfrak{D}}}_{P}\right)B_{g_{P},g}^{\ast}+\left(B_{g_{P},g}{\not{\mathfrak{D}}}_{g_{P}}B_{g_{P},g}^{\ast}-{\not{\mathfrak{D}}}_{g}\right)$

implies that $B_{g_{P},g}PB_{g_{P},g}^{\ast}-{\not{\mathfrak{D}}}_{g}$ is the
sum of two bounded operators, hence bounded.

In conclusion, $B_{g_{P},g}PB_{g_{P},g}^{\ast}$ is a block Dirac operator with
underlying metric $g$. ∎

Unfortunately, the information that the eigenvalues of a continuous family of
block Dirac operators are continuous functions is not good enough for the
arguments to come. More refined information for the case $g\mapsto
B_{g_{P},g}PB_{g_{P},g}^{\ast}$ is provided by the next lemma. To formulate
it, we recall some notations taken from [kato1966perturbation]*Section IV.2.

######  Definition 4.7.12.

Let $S,T\colon X\rightarrow Y$ be two closed, unbounded operators between the
Banach spaces $X$ and $Y$. We define $\delta(S,T)$ to be the smallest number
$\delta$ that satisfies for all $u\in\mathrm{dom}(S)$ the inequality

$\mathrm{dist}\bigl{(}(u,Su);\mathrm{Graph}(T)\bigr{)}\leq\delta(||u||_{X}+||Su||_{Y})\overset{\text{def}}{=}\delta\cdot||(u,Su)||_{X\oplus
Y}$

and
$\hat{\delta}(S,T)\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=\max\\{\delta(S,T);\delta(T,S)\\}$.

It is shown in $\mathrm{loc}.$ $\mathrm{cit}.$ that $\hat{\delta}$ is a
(incomplete) metric on the set $C(X,Y)$ of all closed, unbounded operators
from $X$ to $Y$. The distance $\hat{\delta}$ behaves nicely with compositions
of invertible bounded operators.

######  Lemma 4.7.13.

Let $T\in C(X,Y)$ and let $A\colon X\rightarrow X$, $B\colon Y\rightarrow Y$
be bounded invertible operators such that $A$ and $A^{-1}$ preserve the domain
of $T$.

Then

$\displaystyle\delta(T,BT)\leq||\mathrm{id}-B||,$
$\displaystyle\delta(BT,T)\leq||\mathrm{id}-B||,$
$\displaystyle\delta(T,TA)\leq||\mathrm{id}-A^{-1}||,$
$\displaystyle\delta(TA,T)\leq||\mathrm{id}-A||.$

###### Proof.

Let $u\in{\mathrm{dom}\,}T$ be non-zero. For $v=u\in{\mathrm{dom}\,}BT$ we
have

$\displaystyle\mathrm{dist}\bigl{(}(u,Tu);\mathrm{Graph}(BT)\bigr{)}\leq||(u,Tu)-(v,BTv)||$
$\displaystyle=||(0,(\mathrm{id}-B)Tu)||$
$\displaystyle\leq||\mathrm{id}-B||\cdot||u||.$

If we choose $v=A^{-1}u\in{\mathrm{dom}\,}T$, then

$\displaystyle\mathrm{dist}\bigl{(}(u,Tu);\mathrm{Graph}(TA)\bigr{)}\leq||(u,Tu)-(v,TAv)||$
$\displaystyle=||((\mathrm{id}-A^{-1})u,0)||$
$\displaystyle\leq||\mathrm{id}-A^{-1}||\cdot||u||.$

The other two inequalities are derived in a similar manner. ∎

Important for us is the fact that spectral gaps of operators are, in a more
general sense, lower semi-continuous (they cannot suddenly decrease but they
may suddenly increase) with respect to the topology generated by
$\hat{\delta}$.

######  Theorem 4.7.14.

[kato1966perturbation]*Theorem IV.3.1 Let $T\in C(X,Y)$ be a closed operator
and let $R\colon\zeta\mapsto(T-\zeta)^{-1}\in B(Y,X)$ be its resolvent
function. For every compact subset $\Gamma$ of the resolvent set $\rho(T)$ of
$T$, there is a $\delta>0$ such that if $\hat{\delta}(S,T)<\delta$, then
$\Gamma$ is a subset of $\rho(S)$. An explicit choice for $\delta$ is

$\delta=\min_{\zeta\in\Gamma}\frac{1}{2}(1+|\zeta|^{2})^{-1}(1+||R(\zeta)||^{2})^{-1/2}.$

###### Proof of the Path Lifting Property.

Let the following commutative square be given

By assumption, $\gamma$ factors through
$\widetilde{\mathcal{R}_{\bullet}}(M)_{\gamma_{0},R}$ for some $R>0$. Let
$0<c<1$ be a constant smaller than the corresponding constant from the Sausage
Lemma for the faces $\\{\partial^{\varepsilon}_{i}Q_{0}\\}$. Choose a positive
number $\mu_{0}<c/8$. Since $\gamma$ is uniformly continuous there is an
$N\in\mathord{\mathbb{N}}$ such that all $s,t\in[0,1]$ with $|s-t|<1/N$
satisfy

$||B(\gamma_{s},\gamma_{t})-\mathcal{G}_{\gamma_{s},\gamma_{t}}||_{0,0}=||B(\gamma_{s},\gamma_{t})\mathcal{G}_{\gamma_{s},\gamma_{t}}^{\ast}-\mathrm{id}||_{0,0}<\mu_{0}/8.$
(4.8)

We will prove the following slightly stronger statement. For every
$P\in\widetilde{\Psi\mathrm{Dir}}^{\times}_{n}(M)$ that contains
$(-2\mu_{0},2\mu_{0})$ within its resolvent set, we construct a smooth path
$Q\colon[0,1/N]\rightarrow\widetilde{\Psi\mathrm{Dir}}^{\times}_{n}(M)$ that
is constant near the boundary, lifts the restriction of $\gamma|_{[0,1/N]}$
with initial value $Q_{0}=P$ up to reparametrisation, satisfies
$\partial^{\omega}_{j}Q_{t}=\partial^{\omega}_{j}Q_{0}$, and whose endpoint
contains $(-2\mu_{0},2\mu_{0})$ in its resolvent set.

We will apply this construction iteratively over the partition $[k/N,(k+1)/N]$
starting with the given operator $Q_{0}$, which satisfies our construction
assumptions. The union of all these curves will then be the desired homotopy
lift.

Let us now describe the construction in more detail. Fix, once and for all, a
smooth, odd, injective function
$\varphi\colon\mathord{\mathbb{R}}\setminus\\{0\\}\rightarrow\mathord{\mathbb{R}}\setminus[-3\mu_{0},3\mu_{0}]$
that is the identity on the complement of $(-c/2,c/2)$. The construction is
divided into two steps. The first step is to extend the inital value $Q_{0}$
to a path, the second step is adjusting the eigenvalues of the endpoint
$Q_{1/N}$.

Let us start with the first step: Abbreviate $B_{\gamma_{0},\gamma_{s}}$ to
$B_{s}$ and consider on $[0,1/N]$ the map $t\mapsto B_{t}PB_{t}^{\ast}$. By
Lemma 4.7.10 and Corollary 4.7.11 this map is a smooth map with values in
$\widetilde{\Psi\mathrm{Dir}}_{n}(M)$.

To see that $B_{t}PB_{t}^{\ast}$ is invertible, we argue as follows. The
distance between the spectrum of $Q_{0}$ and
$\Gamma\mathrel{\vbox{\offinterlineskip\hbox{.}\vskip-3.44444pt\hbox{.}}}\joinrel\hskip
1.0pt=[-\mu_{0},\mu_{0}]$ is bigger than $\mu_{0}$. Thus we can estimate the
constant $\delta$ in Theorem 4.7.14 by

$\displaystyle\delta$
$\displaystyle=\min_{\zeta\in\Gamma}\frac{1}{2}(1+|\zeta|^{2})^{-1}(1+||R(\zeta)||^{2})^{-1/2}$
$\displaystyle\geq\frac{1}{2}(1+\mu_{0}^{2})^{-1}(1+\mu_{0}^{-2})^{-1/2}$
$\displaystyle\geq\frac{1}{2}\mu_{0}(1+\mu_{0}^{2})^{-3/2}>\frac{1}{3}\mu_{0}.$

The last inequality holds because $\mu_{0}<1/8$.

On the other hand Lemma 4.7.13 and inequality (4.8) together with the estimate
from the geometric series yield for all $t\in[0,1/N]$ the estimate

$\displaystyle\hat{\delta}(\mathcal{G}_{t}^{\ast}B_{t}PB_{t}^{\ast}\mathcal{G}_{t},P)$
$\displaystyle\leq\hat{\delta}(\mathcal{G}_{t}^{\ast}B_{t}PB_{t}^{\ast}\mathcal{G}_{t},\mathcal{G}_{t}^{\ast}B_{t}P)+\hat{\delta}(\mathcal{G}_{t}^{\ast}B_{t}P,P)$
$\displaystyle\leq\max\\{||B^{\ast}_{t}-\mathcal{G}_{\gamma_{t},\gamma_{0}}||_{0,0},||(B^{\ast}_{t})^{-1}-\mathcal{G}_{\gamma_{0},\gamma_{t}}||_{0,0}\\}+||B_{t}-\mathcal{G}_{\gamma_{0},\gamma_{t}}||_{0,0}$
$\displaystyle\leq\mu_{0}/8\cdot(1-\mu_{0}/8)^{-1}+\mu_{0}/8<\mu_{0}/3<\delta,$

so all $B_{t}PB_{t}^{\ast}$ have $[-\mu_{0},\mu_{0}]$ in their resolvent set.

Compose $t\mapsto B_{t}PB_{t}^{\ast}$ with a smooth, monotone increasing,
surjective self map on $[0,1/N]$ such that the composition is a map
$[0,1/N]\rightarrow\widetilde{\Psi\mathrm{Dir}}^{\times}_{n}(M)$ that is
constant near the boundary. By abuse of notation, we will denote the
composition again with $BPB^{\ast}$.

Now comes the second step. Let
$\lambda_{0}^{+}\leq\dots\leq\lambda_{m^{+}}^{+}$ be all positive eigenvalues
and let $\lambda_{0}^{-}\geq\dots\geq\lambda^{-}_{m^{-}}$ be all negative
eigenvalues of $BQ_{0}B^{\ast}$ inside $(-c,c)$. Pick eigenvectors
$v_{j}^{\pm}$ corresponding to $\lambda_{j}^{\pm}$ such that they form an
orthonormal system and pick smooth compactly supported sections $w_{j}^{\pm}$
that satisfy $||v_{j}^{\pm}-w_{j}^{\pm}||_{0}<\mu_{0}/C\cdot 10^{-j}$, where
$C\geq 1$ is a constant to be determined later. Let
$\Phi\colon[1/N,1+1/N]\times\mathord{\mathbb{R}}\setminus\\{0\\}\rightarrow\mathord{\mathbb{R}}\setminus\\{0\\}$
be the homotopy between $\mathrm{id}$ and $\varphi$ given by convex
combination and compose the first entry with a smooth, monotonically
increasing, surjective self-map on $[1/N,1+1/N]$ such that
$\Phi(\,\cdot\,,x)=x$ near $1/N$ and $\Phi(\,\cdot\,,x)=\varphi(x)$ near
$1+1/N$. Denote the result again with $\Phi$. Finally, define

$\displaystyle\Theta\colon[1/N,1+1/N]\rightarrow\widetilde{\Psi\mathrm{Dir}}^{\times}_{n}(\mathfrak{S}_{g_{B_{1/N}Q_{0}B_{1/N}^{\ast}}}),$
$\displaystyle t\mapsto
B_{1/N}Q_{0}B_{1/N}^{\ast}+\sum_{j=0}^{m^{\pm}}(\Phi(t,\lambda_{j}^{\pm})-\lambda_{j}^{\pm})\langle\cdot,w_{j}^{\pm}\rangle
w_{j}^{\pm}$

and the auxiliary map

$\displaystyle\widetilde{\Theta}\colon[1/N,1+1/N]\rightarrow\overline{\Psi\mathrm{DO}}^{1}(\mathfrak{S}_{g_{B_{1/N}Q_{0}B_{1/N}^{\ast}}}),$
$\displaystyle t\mapsto
B_{1/N}Q_{0}B_{1/N}^{\ast}+\sum_{j=0}^{m^{\pm}}(\Phi(t,\lambda_{j}^{\pm})-\lambda_{j}^{\pm})\langle\cdot,v_{j}^{\pm}\rangle
v_{j}^{\pm}.$

For all $t\in[1/N,1+1/N]$, the two operators $\Theta_{t}$ and
$\widetilde{\Theta}_{t}$ differ from $B_{1/N}Q_{0}B_{1/N}^{\ast}$ by a self-
adjoint infinitely smoothing operator of finite rank. They are therefore self-
adjoint pseudo differential operators of order $1$ with the same principal
symbol as the Dirac operator of the underlying metric. They also differ from
this Dirac operator by a bounded operator, because
$B_{1/N}Q_{0}B_{1/N}^{\ast}$ does it. Since all $w_{j}^{\pm}$ are compactly
supported, $\Theta_{t}$ is of block form and has the same faces as
$B_{1/N}Q_{0}B_{1/N}^{\ast}$; hence, $\Theta_{t}$ is a block Dirac operator.

We prove invertibility of $\Theta$ by showing that $\widetilde{\Theta}$ is
invertible and that $\Theta$ is a sufficiently small perturbation of
$\widetilde{\Theta}$. On the positive subspace of $B_{1/N}Q_{0}B_{1/N}$, i.e.,
the one on which $B_{1/N}Q_{0}B_{1/N}^{\ast}$ is positive definite, we have,
by construction,

$\widetilde{\Theta_{t}}\geq\Phi(t,\lambda_{0}^{+})\mathrm{id}.$

A similar result holds for the negative subspace, so the interval
$(\Phi(t,\lambda_{0}^{-}),\Phi(t,\lambda_{0}^{+}))$ does not intersect the
spectrum of $\widetilde{\Theta_{t}}$. In particular, each
$\widetilde{\Theta_{t}}$ is invertible.

It remains to show that $\Theta$ is a sufficiently small perturbation of
$\widetilde{\Theta}$. First observe

$\displaystyle\quad\ ||\langle\,\cdot\,,w_{j}^{\pm}\rangle
w_{j}^{\pm}-\langle\,\cdot\,,v_{j}^{\pm}\rangle v_{j}^{\pm}||_{0,0}$
$\displaystyle\leq||\langle\,\cdot\,,w_{j}^{\pm}-v_{j}^{\pm}\rangle||_{\mathrm{dual}}||w_{j}^{\pm}||_{0}+||\langle\,\cdot\,,v_{j}^{\pm}\rangle||_{\mathrm{dual}}||v_{j}^{\pm}-w_{j}^{\pm}||_{0}$
$\displaystyle\leq\left(2+||v_{j}-w_{j}||_{0}\right)||v_{j}+w_{j}||_{0}<2.5\mu_{0}/C10^{-j}.$

This implies

$\displaystyle||\Theta_{t}-\widetilde{\Theta_{t}}||_{0,0}$
$\displaystyle\leq\sum_{j=0}^{M,N}\underbrace{|\Phi(t,\lambda_{j}^{\pm})-\lambda_{j}^{\pm}|}_{\leq
c}2.5\mu_{0}/C10^{-j}$ $\displaystyle\leq c\cdot 2.5\cdot 2\cdot
10/9\cdot\mu_{0}/C$ $\displaystyle\leq 6\mu_{0}/C.$

Thus, the spectrum of $\Theta_{t}$ does not intersect
$(\Phi(t,\lambda_{0}^{-})+6\mu_{0}/C,\Phi(t,\lambda_{0}^{+})-6\mu_{0}/C)$. If
we choose $C$ sufficiently large, for example $C=12$ and estimate
$\tilde{\Theta}_{t}$ against its lowest positive eigenvalue
$\Phi(t,\lambda_{0}^{+})$ on its positive subspace, we derive the following
chain of inequalities

$\displaystyle\Theta_{t}$
$\displaystyle=\Theta_{t}+\tilde{\Theta}_{t}-\tilde{\Theta}_{t}\geq\left(\Phi(t,\lambda_{0}^{+})-||\tilde{\Theta}_{t}-\Theta_{t}||_{0,0}\right)\mathrm{id}$
$\displaystyle\geq\left(\lambda_{0}^{+}-||\tilde{\Theta}_{t}-\Theta_{t}||_{0,0}\right)\mathrm{id}\geq\left(2\mu_{0}-||\tilde{\Theta}_{t}-\Theta_{t}||_{0,0}\right)\mathrm{id}$
$\displaystyle\geq(2\mu_{0}-6\mu_{0}/C)\mathrm{id}\geq\mu_{0}\mathrm{id}.$

A similar result holds on the negative subspace, so $\Theta_{t}$ is invertible
for all $t\in[1/N,1+1/N]$. Using
$\Phi(1+1/N,\lambda_{0}^{+})=\varphi(\lambda_{0}^{+})\geq 3\mu_{0}$ and

$\displaystyle\Theta_{1+1/N}$
$\displaystyle=\Theta_{1+1/N}-\tilde{\Theta}_{1+1/N}+\tilde{\Theta}_{1+1/N}$
$\displaystyle\geq\left(3\mu_{0}-||\Theta_{1+1/N}-\tilde{\Theta}_{1+1/N}||_{0,0}\right)\mathrm{id}$
$\displaystyle\geq(3\mu_{0}-\mu_{0}/2)\mathrm{id}>2\mu_{0}\mathrm{id},$

we see that the spectrum of $\Theta_{1/N}$ does not intersect
$(-2\mu_{0},2\mu_{0})$.

The desired homotopy lift is now given by

$\textstyle{Q\colon[0,1/N]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t\mapsto(N+1)t}$$\textstyle{[0,1+1/N]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{B_{(\cdot)}Q_{0}B_{(\cdot)}\cup\Theta}$$\textstyle{\widetilde{\Psi\mathrm{Dir}}^{\times}_{n}(M).}$

Since $Q_{1/N}$ satisfies the assumption of the described construction, we can
construct a lift on $[1/N,2/N]$ starting with $Q_{1/N}$ instead of $Q_{0}$
(but with the same $\mu_{0}$). If we repeat this construction $(N-2)$-times
further and glue the results together, we end up with the desired (homotopy)
lift $Q\colon[0,1]\rightarrow\widetilde{\Psi\mathrm{Dir}}^{\times}_{n}(M)$.
Since the homotopy between $g_{Q}$ and $\gamma$ is given by reparametrisation
relative endpoints, the map $Q$ is indeed a homotopy lift.

It remains to prove that our construction is a suspensionable path. More
precisely, if $Q$ also denotes the constant extension of the constructed lift,
we need to show that
$Q^{\mathrm{ex}}\in\overline{\Psi\mathrm{DO}}^{1}(M\times\mathord{\mathbb{R}}^{n}\times\mathord{\mathbb{R}};\mathfrak{S}_{\mathrm{susp}(\gamma)})$.
Since $\overline{\Psi\mathrm{DO}}^{1}(\mathfrak{S})$ is local, see Proposition
B.0.27, it suffices to prove the statement only for each piece of the
construction.

To verify
$\bigl{(}B_{(\cdot)}Q_{0}B_{(\cdot)}^{\ast}\bigr{)}^{\mathrm{ex}}\in\overline{\Psi\mathrm{DO}}^{1}(\mathfrak{S})$,
take a sequence of actual pseudo differential operators
$P_{n}\in\Psi\mathrm{DO}^{1}(\mathfrak{S}_{\gamma_{0}})$ that approximates
$Q_{0}$ with respect to the Fréchet structure on $Op^{1}$ (the one that
induces the Atiyah-Singer topology on
$\overline{\Psi\mathrm{DO}}^{1}(\mathfrak{S}_{\gamma_{0}})$). Since
$B-p\mathcal{G}$ is compactly supported, Proposition B.0.15 and a partition of
unity argument shows that $P\mapsto BPB^{\ast}$ is continuous with respect to
the amplitude topology. But $(\text{--})^{\mathrm{ex}}$, which is denoted with
$\text{--}\tilde{\boxtimes}\mathrm{id}$ in the appendix, is a continuous
linear map, see Theorem B.0.30, so we conclude

$\displaystyle\lim_{n\to\infty}\bigl{(}BP_{n}B^{\ast}\bigr{)}^{\mathrm{ex}}$
$\displaystyle=B^{\mathrm{ex}}\lim_{n\to\infty}P_{n}^{\mathrm{ex}}(B^{\ast})^{\mathrm{ex}}$
$\displaystyle=\bigl{(}BQ_{0}B^{\ast}\bigr{)}^{\mathrm{ex}}\in\overline{\Psi\mathrm{DO}}^{1}(M\times\mathord{\mathbb{R}}^{n}\times\mathord{\mathbb{R}},\mathfrak{S}_{g_{\mathrm{susp}(BQ_{0}B^{\ast})}}).$

For $\tilde{\Theta}$ it is even easier to verify because it is the sum of a
block Dirac operator (interpreted as constant curve) and a smooth curve $W$ of
infinitely smoothing operators. We conclude

$\tilde{\Theta}^{\mathrm{ex}}=\bigl{(}B_{1/N}Q_{0}B_{1/N}^{\ast}\bigr{)}^{\mathrm{ex}}+W^{\mathrm{ex}}\in\overline{\Psi\mathrm{DO}}^{1}(M\times\mathord{\mathbb{R}}^{n}\times\mathord{\mathbb{R}};\mathfrak{S}_{\mathrm{susp}(\gamma_{1/N})})$

from the Theorem B.0.28 and Theorem B.0.30. ∎

The next tool allows to glue two matching invertible pseudo Dirac operators
together. To introduce this tool, we need to discuss shift operators on pseudo
differential operators.

######  Definition 4.7.15.

Let $E\rightarrow M\times\mathord{\mathbb{R}}^{n}$ be a vector bundle. For
$v\in\mathord{\mathbb{R}}^{n}$, define the _shift operator_

$\displaystyle\mathrm{Sh}_{v}\colon\Gamma_{c}(M\times\mathord{\mathbb{R}}^{n};E)$
$\displaystyle\rightarrow\Gamma_{c}(M\times\mathord{\mathbb{R}}^{n};E),$
$\displaystyle\mathrm{Sh}_{v}(\sigma)(m,x)$ $\displaystyle=\sigma(m,x-v).$

The shift operators induce linear maps on the spaces of pseudo differential
operators via push-forward

$\displaystyle\mathrm{Sh}_{v\,\ast}\colon\Psi\mathrm{DO}^{k}(M\times\mathord{\mathbb{R}}^{n};E)$
$\displaystyle\rightarrow\Psi\mathrm{DO}^{k}(M\times\mathord{\mathbb{R}}^{n};E),$
$\displaystyle P$ $\displaystyle\mapsto\mathrm{Sh}_{v}\circ
P\circ\mathrm{Sh}_{-v}.$

In case that $E$ is the tangent or the spinor bundle, there is a less ad-hoc
way to define the shift operator. The translation
$\mathrm{trans}_{v}\colon(m,x)\mapsto(m,x+v)$ is a diffeomorphism on
$M\times\mathord{\mathbb{R}}^{n}$. Its differential gives an isometry

$T\mathrm{trans}_{v}\colon(T(M\times\mathord{\mathbb{R}}^{n}),g)\rightarrow(T(M\times\mathord{\mathbb{R}}^{n}),\mathrm{trans}_{-v}^{\ast}g).$

Clearly, $\mathrm{trans}_{-v}^{\ast}g$ and (the ad-hoc version of)
$\mathrm{Sh}_{v}g$ agree. Since the spinor bundle construction is functorial
with respect to Pin structure preserving isometries, we obtain an isomorphism
of Clifford-module bundles
$\mathfrak{S}_{g}\rightarrow\mathfrak{S}_{\mathrm{Sh}_{v}g}$ over
$\mathrm{trans}_{v}$. The fibres $(\mathfrak{S}_{g})_{(m,x-v)}$ and
$(\mathfrak{S}_{\mathrm{Sh}_{v}(g)})_{(m,x)}$ are the same, so the push
forward with this Clifford bundle isomorphism on the level of sections is
again given by

$\displaystyle\mathrm{Sh}_{v}\colon\Gamma_{c}(M\times\mathord{\mathbb{R}}^{n};\mathfrak{S}_{g})$
$\displaystyle\rightarrow\Gamma_{c}(M\times\mathord{\mathbb{R}}^{n};\mathfrak{S}_{\mathrm{Sh}_{v}g}),$
$\displaystyle\mathrm{Sh}_{v}(\sigma)(m,x)$ $\displaystyle=\sigma(m,x-v).$