$\mathbb{R}\sigma_{xy}^{\mathcal{A}}$ (also called $st^{\mathcal{A}}_{xy}$):

$st^{\mathcal{A}}_{xy}:V_{x}\oplus
I_{xy}^{\mathcal{A}}\oplus\mathbb{R}\sigma_{xy}^{\mathcal{A}}\to\mathbb{R}^{a}\oplus\mathbb{R}\sigma_{xy}^{\mathcal{A}}\oplus
V_{y}$

sending $\sigma$ to $\sigma$.

Note that in either case, we do not switch the order of the factors of
$\mathbb{R}$.

######  Definition 65.

Let $\mathcal{M}_{xz}$ be a manifold with faces, equipped with a stable
isomorphism

$st^{\mathcal{M}}_{xz}:V_{x}\oplus I^{\mathcal{M}}_{xz}\to\mathbb{R}^{m}\oplus
V_{z}$

Assume that for each broken boundary face
$\mathcal{A}_{xy}\times\mathcal{B}_{yz}\to\mathcal{M}$, we are given stable
isomorphisms

$st^{\mathcal{A}}_{xy}:V_{x}\oplus I^{\mathcal{A}}_{xy}\to\mathbb{R}^{a}\oplus
V_{y}$

and

$st^{\mathcal{B}}_{yz}:V_{y}\oplus I^{\mathcal{B}}_{yz}\to\mathbb{R}^{b}\oplus
V_{z}$

such that $a+b=m$, and also that for each unbroken boundary face
$\mathcal{D}_{xz}\to\mathcal{M}_{xz}$, we are given a stable isomorphism

$st^{\mathcal{D}}_{xz}:V_{x}\oplus I^{\mathcal{D}}_{xz}\to\mathbb{R}^{d}\oplus
V_{z}$

with $d=m-1$.

We say that these form a _system of framings_ for $\mathcal{M}_{xz}$ if over
each boundary face, these stable framings (and their induced stable framings)
are compatible with each other via the abstract gluing isomorphisms.

More explicitly, this means that the following diagrams commute.

Over each broken boundary face
$\mathcal{A}_{xy}\times\mathcal{B}_{yz}\to\mathcal{M}_{xz}$:

($\star$) $\textstyle{V_{x}\oplus I^{\mathcal{A}}_{xy}\oplus
I^{\mathcal{B}}_{yz}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\scriptstyle{st^{\mathcal{B}}\circ
st^{\mathcal{A}}}$$\textstyle{V_{x}\oplus
I^{\mathcal{M}}_{xz}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{st^{\mathcal{M}}}$$\textstyle{\mathbb{R}^{m}\oplus
V_{z}}$

Over each unbroken boundary face $\mathcal{D}_{xz}\to\mathcal{M}_{xz}$,
viewing $\mathbb{R}\sigma^{\mathcal{D}}_{xz}=$ the final copy of $\mathbb{R}$
in $\mathbb{R}^{m}$ as the stabilising bundle:

$\textstyle{V_{x}\oplus
I^{\mathcal{D}}_{xz}\oplus\mathbb{R}\sigma^{\mathcal{D}}_{xz}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{st^{\mathcal{D}}}$$\scriptstyle{\psi}$$\textstyle{V_{x}\oplus
I^{\mathcal{M}}_{xz}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{st^{\mathcal{M}}}$$\textstyle{\mathbb{R}^{m}\oplus
V_{z}}$

######  Example 4.11.

Let $\mathcal{F}$ be a framed flow category. Then each
$st^{\mathcal{F}}_{xx^{\prime\prime}}$, along with the induced framings
$st^{\mathcal{F}}_{xx^{\prime}}\circ
st^{\mathcal{F}}_{x^{\prime}x^{\prime\prime}}$ on each boundary face, form a
system of framings.

######  Definition 66.

We define framings for every type of object introduced in Section 4.1.

  1. 1.

Let $\mathcal{W}:\mathcal{F}\to\mathcal{G}$ be a morphism of flow categories,
and assume that $\mathcal{F}$ and $\mathcal{G}$ are framed. A _framing_ on
$\mathcal{W}$ consists of stable isomorphisms

$st^{\mathcal{W}}_{xy}:V_{x}\oplus I^{\mathcal{W}}_{xy}\to\mathbb{R}\oplus
V_{y}$

for each $x\in\mathcal{F}$ and $y\in\mathcal{G}$, such that these, combined
with the framings for $\mathcal{F}$, $\mathcal{G}$ and each lower dimensional
$\mathcal{W}_{x^{\prime}y^{\prime}}$, form systems of framings for each
$\mathcal{W}_{xy}$.

  2. 2.

Let $\mathcal{R}$ be a bordism from $\mathcal{W}$ to $\mathcal{W}^{\prime}$,
where $\mathcal{W},\mathcal{W}^{\prime}:\mathcal{F}\to\mathcal{G}$ are framed
morphisms between framed flow categories. A _framing_ on $\mathcal{R}$
consists of stable isomorphisms

$st^{\mathcal{R}}_{xy}:V_{x}\oplus I^{\mathcal{R}}_{xy}\to\mathbb{R}^{2}\oplus
V_{y}$

such that these, combined with the framings for
$\mathcal{F},\mathcal{G},\mathcal{W}$, $\mathcal{W}^{\prime}$ and each lower
dimensional $\mathcal{R}_{x^{\prime}y^{\prime}}$, form systems of framings for
each $\mathcal{R}_{xy}$.

  3. 3.

Let $\mathcal{W}:\mathcal{F}\to\mathcal{G}$ and
$\mathcal{V}:\mathcal{G}\to\mathcal{H}$ be framed morphisms between framed
flow categories. We form the manifolds
$\tilde{\mathcal{Q}}_{xz}:=\tilde{\mathcal{Q}}_{xz}(\mathcal{W},\mathcal{V})$
as in Section 4.1. A _framing_ on the collection of these manifolds
$\tilde{\mathcal{Q}}$ consists of stable isomorphisms

$st^{\tilde{\mathcal{Q}}}_{xz}:V_{x}\oplus
I^{\tilde{\mathcal{Q}}}_{xz}\to\mathbb{R}^{2}\oplus V_{z}$

such that, combined with the framings for $\mathcal{F}$, $\mathcal{G}$,
$\mathcal{W}$, $\mathcal{V}$ and each lower dimensional
$\tilde{\mathcal{Q}}_{x^{\prime}y^{\prime}}$, form systems of framings for
each $\tilde{\mathcal{Q}}_{xz}$.

  4. 4.

Let $\mathcal{W},\mathcal{W}^{\prime}:\mathcal{F}\to\mathcal{G}$ and
$\mathcal{V}:\mathcal{G}\to\mathcal{H}$ be framed morphisms between framed
flow categories. Assume that we are given framings on
$\tilde{\mathcal{Q}}:=\tilde{\mathcal{Q}}(\mathcal{W},\mathcal{V})$ and
$\tilde{\mathcal{Q}}^{\prime}:=\tilde{\mathcal{Q}}(\mathcal{W}^{\prime},\mathcal{V})$.
We form the manifolds
$\tilde{\mathcal{S}}_{xz}:=\tilde{\mathcal{S}}_{xz}(\mathcal{R},\mathcal{V})$
as in Lemma 56. A _framing_ on the collection of these manifolds
$\tilde{\mathcal{S}}$ consists of stable isomorphisms

$st^{\tilde{\mathcal{S}}}_{xz}:V_{x}\oplus
I^{\tilde{\mathcal{S}}}_{xz}\to\mathbb{R}^{3}\oplus V_{z}$

such that, when combined with all other framings in sight, form systems of
framings for each manifold in sight.

We spell out what this means more explicitly in the simplest cases.

######  Example 4.12.

Let $\mathcal{W}:\mathcal{F}\to\mathcal{G}$ be a morphism between two framed
flow categories. Note all boundary faces of each $\mathcal{W}_{xy}$ are
broken. Given stable isomorphisms

$st^{\mathcal{W}}_{xy}:V_{x}\oplus I^{\mathcal{W}}_{xy}\to\mathbb{R}\oplus
V_{y}$

the compatibility requirement for them to form a framing on $\mathcal{W}$ is
that the following diagrams commute.

Over each $\mathcal{W}_{xx^{\prime};y}$:

$\textstyle{V_{x}\oplus I^{\mathcal{F}}_{xx^{\prime}}\oplus
I^{\mathcal{W}}_{x^{\prime}y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{st^{\mathcal{F}}}$$\scriptstyle{\psi}$$\textstyle{V_{x^{\prime}}\oplus
I^{\mathcal{W}}_{x^{\prime}y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{st^{\mathcal{W}}}$$\textstyle{V_{x}\oplus
I^{\mathcal{W}}_{xy}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{st^{\mathcal{W}}}$$\textstyle{\mathbb{R}\oplus
V_{y}}$

and over each $\mathcal{W}_{x;y^{\prime}y}$:

$\textstyle{V_{x}\oplus I^{\mathcal{W}}_{xy^{\prime}}\oplus
I^{\mathcal{G}}_{y^{\prime}y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{st^{\mathcal{W}}}$$\scriptstyle{\psi}$$\textstyle{\mathbb{R}\oplus
V_{y^{\prime}}\oplus
I^{\mathcal{G}}_{y^{\prime}y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{st^{\mathcal{G}}}$$\textstyle{V_{x}\oplus
I^{\mathcal{W}}_{xy}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{st^{\mathcal{W}}}$$\textstyle{\mathbb{R}\oplus
V_{y}}$

######  Definition 67.

Let $\mathcal{W}:\mathcal{F}\to\mathcal{G}$ be a framed morphism of framed
flow categories. We define $\overline{\mathcal{W}}$ to be the morphism
$\mathcal{W}$, equipped with the framing

$st^{\overline{\mathcal{W}}}:V_{x}\oplus
I^{\mathcal{W}}_{xy}\to\mathbb{R}\oplus V_{y}$

given by $st^{\mathcal{W}}$ postcomposed with the map
$-1:\mathbb{R}\to\mathbb{R}$.

######  Example 4.13.

Let $\mathcal{R}$ be a bordism between two framed morphisms
$\mathcal{W},\mathcal{W}^{\prime}:\mathcal{F}\to\mathcal{G}$ of framed flow
categories. Given stable isomorphisms

$st^{\mathcal{R}}_{xy}:V_{x}\oplus I^{\mathcal{R}}_{xy}\to\mathbb{R}\oplus
V_{y}$

the compatibility requirement for them to form a framing on $\mathcal{R}$ is
that the following diagrams commute.

Over each $\mathcal{R}_{xx^{\prime};y}$ and $\mathcal{R}_{x;y^{\prime}y}$,
diagrams similar to those in Example 4.12 commute (note that some factors of
$\mathbb{R}$ must be replaced with factors of $\mathbb{R}^{2}$ though).

Furthermore, over each $\mathcal{W}_{xy}$:

$\textstyle{V_{x}\oplus
I^{\mathcal{W}}_{xy}\oplus\mathbb{R}\sigma^{\mathcal{W}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{st^{\mathcal{W}}}$$\scriptstyle{\psi}$$\textstyle{V_{x}\oplus
I^{\mathcal{R}}_{xy}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{st^{\mathcal{R}}}$$\textstyle{\mathbb{R}^{2}\oplus
V_{y}}$

Note that the diagonal arrow sends the line $\mathbb{R}\sigma^{\mathcal{W}}$
to the first factor in $\mathbb{R}^{2}$.

Over each $\mathcal{V}_{xy}$:

$\textstyle{V_{x}\oplus
I^{\mathcal{V}}_{xy}\oplus\mathbb{R}\sigma^{\mathcal{V}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{st^{\mathcal{V}}}$$\scriptstyle{\psi}$$\textstyle{V_{x}\oplus
I^{\mathcal{R}}_{xy}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{st^{\mathcal{R}}}$$\textstyle{\mathbb{R}^{2}\oplus
V_{y}}$

The boundary faces $\mathcal{W}_{xy}$ and $\mathcal{V}_{xy}$ are the unbroken
boundary faces of $\mathcal{R}_{xy}$.

######  Example 4.14.

Let $\mathcal{W}:\mathcal{F}\to\mathcal{G}$ be a framed morphism between
framed flow categories, and $\mathcal{R}$ the bordism from $\mathcal{W}$ to
itself constructed in Lemma 52. We equip $\mathcal{R}$ with a framing as
follows.

We write $\mathcal{R}_{xy}=\mathcal{W}_{xy}\times[0,1]_{s}$, with $s$ the
coordinate on the interval. Then define

$st^{\mathcal{R}}_{xy}:V_{x}\oplus I^{\mathcal{R}}_{xy}=V_{x}\oplus
I^{\mathcal{W}}_{xy}\oplus\mathbb{R}\partial_{s}\xrightarrow{st^{\mathcal{W}}}\mathbb{R}^{2}\oplus
V_{y}$

where $\partial_{s}$ is sent to $(0,1)\in\mathbb{R}^{2}$.

It is straightforward to check this is a stable framing; we spell out the
details over the unbroken outgoing boundary face
$\mathcal{W}_{xy}\times\\{1\\}$. Note that the inwards normal along this face
is $-\partial_{s}$.

We must check that the following diagram commutes over
$\mathcal{W}_{xy}\times\\{1\\}$:

$\textstyle{V_{x}\oplus
I^{\mathcal{W}}_{xy}\oplus\mathbb{R}\sigma\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\scriptstyle{st^{\mathcal{W}}}$$\textstyle{V_{x}\oplus
I^{\mathcal{R}}_{xy}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{st^{\mathcal{R}}}$$\textstyle{\mathbb{R}^{2}\oplus
V_{y}}$

Both ways around the diagram send $\sigma$ to $(0,1)\in\mathbb{R}^{2}$, and
agree on the other factors by construction of $st^{\mathcal{R}}$.

######  Example 4.15.

Homotopic framings on the same morphism are bordant.

######  Definition 68.

Let $\mathcal{F}$ and $\mathcal{G}$ be framed flow categories. We define
$[\mathcal{F},\mathcal{G}]^{fr}$ to be the set of framed morphisms
$\mathcal{F}\to\mathcal{G}$, modulo the equivalence relation generated by

  1. 1.

$\mathcal{W}\sim\mathcal{V}$ if $\mathcal{W}$, $\mathcal{V}$ are framed
bordant.

  2. 2.

$\mathcal{W}\sim\mathcal{V}$ if $\mathcal{W}$ and $\mathcal{V}$ are
diffeomorphic, via a diffeomorphism compatible with the framings.

Similarly to $[\mathcal{F},\mathcal{G}]$, it is clear that
$[\mathcal{F},\mathcal{G}]^{fr}$ forms a commutative monoid under disjoint
union, with unit given by the empty morphism.

######  Lemma 69.

Inverses exist in $[\mathcal{F},\mathcal{G}]^{fr}$, i.e. it’s a group.

######  Proof 4.16.

The framed bordism constructed in Example 4.14 can alternatively be viewed as
a framed nullbordism of $\mathcal{W}\sqcup\overline{\mathcal{W}}$, implying
that the inverse of $\mathcal{W}$ is $\overline{\mathcal{W}}$.

######  Lemma 70.

Let $\mathcal{W},\mathcal{W}^{\prime}:\mathcal{F}\to\mathcal{G}$ and
$\mathcal{V}:\mathcal{G}\to\mathcal{H}$ be framed morphisms of framed flow
categories. Let
$\tilde{\mathcal{Q}}:=\tilde{\mathcal{Q}}(\mathcal{W},\mathcal{V})$, let
$\tilde{\mathcal{Q}}^{\prime}:=\tilde{\mathcal{Q}}(\mathcal{W}^{\prime},\mathcal{V})$
and let $\tilde{\mathcal{S}}:=\tilde{\mathcal{S}}(\mathcal{R},\mathcal{V})$.

Then $\tilde{\mathcal{Q}},\tilde{\mathcal{Q}}^{\prime}$ and
$\tilde{\mathcal{S}}$ are naturally framed.

######  Proof 4.17.

We first frame $\tilde{\mathcal{Q}}$; framing $\tilde{\mathcal{Q}}^{\prime}$
is identical.

We define

$st^{\tilde{\mathcal{Q}}}_{xz}:V_{x}\oplus
I^{\tilde{\mathcal{Q}}}_{xz}\to\mathbb{R}^{2}\oplus V_{z}$

as follows. On each
$\mathcal{W}_{xy}\times\mathcal{V}_{yz}\times[0,\varepsilon)_{s}\subseteq\tilde{\mathcal{Q}}_{xz}$,
we define it as the composition

$st^{\tilde{\mathcal{Q}}}_{xz}:V_{x}\oplus
I^{\tilde{\mathcal{Q}}}_{xz}\xrightarrow{\psi^{-1}}V_{x}\oplus
I^{\mathcal{W}}_{xy}\oplus
I^{\mathcal{V}}_{yz}\xrightarrow{st^{\mathcal{V}}\circ
st^{\mathcal{W}}}\mathbb{R}^{2}\oplus V_{z}$

noting that the first copy of $\mathbb{R}$ comes from $st^{\mathcal{W}}$ and
the second from $st^{\mathcal{V}}$. Note that $\psi$ isn’t a priori defined
over all of this open subset but we can extend it from the boundary to the
rest in a canonical way, since $\partial_{s}$ is naturally identified with
$\nu_{y}$.

We must check these agree on overlaps for this to define a stable framing over
the entirety of $\tilde{\mathcal{Q}}_{xz}$. Noting that on this subset, we
have that $T\tilde{\mathcal{Q}}=\mathbb{R}\partial_{s}\oplus
T\mathcal{W}_{xy}\oplus T\mathcal{V}_{yz}$, we can define an isomorphism of
vector bundles

$\bar{\psi}:\mathbb{R}\partial_{s}\oplus\mathbb{R}\tau_{z}\oplus
T\mathcal{W}_{xy}\oplus T\mathcal{V}_{yz}\to I^{\mathcal{W}}_{xy}\oplus
I^{\mathcal{V}}_{yz}$

by sending $\partial_{s}$ to $\tau_{y}$ and $\tau_{z}$ to $\tau_{z}$. Then
noting that we can identify the domain with $I^{\tilde{\mathcal{Q}}}$, the
following diagram commutes:

$\textstyle{I^{\tilde{\mathcal{Q}}}_{xz}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\scriptstyle{\psi^{-1}}$$\textstyle{\mathbb{R}\partial_{s}\oplus\mathbb{R}\tau_{z}\oplus
T\mathcal{W}_{xy}\oplus
T\mathcal{V}_{yz}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\psi}}$$\textstyle{I^{\mathcal{W}}_{xy}\oplus
I^{\mathcal{V}}_{yz}}$

Now let $y,y^{\prime}\in\mathcal{G}$, and consider the maps

$\alpha:\mathcal{W}_{xy}\times\mathcal{G}_{yy^{\prime}}\times\mathcal{V}_{y^{\prime}z}\times[0,\varepsilon)^{2}_{u,v}\to\mathcal{W}_{xy}\times\mathcal{V}_{yz}\times[0,\varepsilon)_{s}$

and

$\beta:\mathcal{W}_{xy}\times\mathcal{G}_{yy^{\prime}}\times\mathcal{V}_{y^{\prime}z}\times[0,\varepsilon)^{2}_{u,v}\to\mathcal{W}_{xy^{\prime}}\times\mathcal{V}_{y^{\prime}z}\times[0,\varepsilon)_{s}$

which we used as the gluing maps in the construction of $\tilde{\mathcal{Q}}$.
These were defined to send $(a,b,c,u,v)$ to $(a,\mathcal{C}(b,c;u),v)$ and
$(\mathcal{C}(a,b;v),c,u)$ respectively. The compatibility we need is that the
outside part of the following diagram commutes. Note that the top entry is
$V_{x}\oplus\textrm{(the tangent space of the
overlap)}\oplus\mathbb{R}\tau_{z}$.

$\textstyle{V_{x}\oplus\mathbb{R}\partial_{u}\oplus\mathbb{R}\partial_{v}\oplus
T\mathcal{W}_{xy}\oplus T\mathcal{G}_{yy^{\prime}}\oplus
T\mathcal{V}_{y^{\prime}z}\oplus\mathbb{R}\tau_{z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d\alpha}$$\scriptstyle{d\beta}$$\scriptstyle{=}$$\textstyle{V_{x}\oplus
T\mathcal{W}_{xy}\oplus\mathbb{R}\partial_{s}\oplus
T\mathcal{V}_{yz}\oplus\mathbb{R}\tau_{z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\psi}}$$\textstyle{V_{x}\oplus
I^{\tilde{\mathcal{Q}}}_{xz}}$$\textstyle{V_{x}\oplus
T\mathcal{W}_{xy^{\prime}}\oplus\mathbb{R}\partial_{s}\oplus
T\mathcal{V}_{y^{\prime}z}\oplus\mathbb{R}\tau_{z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\psi}}$$\textstyle{V_{x}\oplus
I^{\mathcal{W}}_{xy}\oplus
I^{\mathcal{V}}_{yz}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{st^{\mathcal{V}}\circ
st^{\mathcal{W}}}$$\scriptstyle{\psi}$$\textstyle{V_{x}\oplus
I^{\mathcal{W}}_{xy}\oplus I^{\mathcal{G}}_{yy^{\prime}}\oplus
I^{\mathcal{V}}_{y^{\prime}z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\scriptstyle{\psi}$$\scriptstyle{st^{\mathcal{V}}\circ
st^{\mathcal{G}}\circ st^{\mathcal{W}}}$$\textstyle{V_{x}\oplus
I^{\mathcal{W}}_{xy^{\prime}}\oplus
I^{\mathcal{V}}_{y^{\prime}z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\scriptstyle{st^{\mathcal{V}}\circ
st^{\mathcal{W}}}$$\textstyle{\mathbb{R}^{2}\oplus V_{z}}$

The two bottom triangles commute by the compatibility of the
$st^{\mathcal{W}}$, $st^{\mathcal{G}}$ and $\psi$. The middle diamond commutes
by associativity of the maps $\psi$. The top two squares together we can check
commutativity of directly: all three ways from the top entry to the entry in
the diagram below it send $\partial_{u}$ to $\nu_{y}$, $\partial_{v}$ to
$\nu_{y^{\prime}}$, $\tau_{z}$ to $\tau_{z}$, don’t do anything to $V_{x}$ and
are given by $dc$ on the other factors.

This implies that the $st^{\tilde{\mathcal{Q}}}$, which we defined on an oven
cover of $\tilde{\mathcal{Q}}_{xz}$, actually glue to give a well-defined
stable isomorphism over the entirety of $\tilde{\mathcal{Q}}_{xz}$.

We next must check that the $st^{\tilde{\mathcal{Q}}}$ form an actual framing
of $\tilde{\mathcal{Q}}$. Commutativity of $\star$ ‣ 65 over the boundary face
$\tilde{\mathcal{Q}}_{xx^{\prime};z}$ follows on from commutativity of the
following diagram; noting that going right and then down corresponds to
applying $st^{\mathcal{F}}$ then $st^{\tilde{\mathcal{Q}}}$ whereas going down
then right corresponds to applying $\psi$ then $st^{\tilde{\mathcal{Q}}}$:

$\textstyle{V_{x}\oplus I^{\mathcal{F}}_{xx^{\prime}}\oplus
I^{\tilde{\mathcal{Q}}}_{x^{\prime}z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{V_{x}\oplus
I^{\mathcal{F}}_{xx^{\prime}}\oplus I^{\mathcal{W}}_{x^{\prime}y}\oplus
I^{\mathcal{V}}_{yz}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\scriptstyle{\psi}$$\scriptstyle{st^{\mathcal{V}}\circ
st^{\mathcal{W}}\circ st^{\mathcal{F}}}$$\textstyle{V_{x}\oplus
I^{\tilde{\mathcal{Q}}}_{xz}}$$\textstyle{V_{x}\oplus
I^{\mathcal{W}}_{xy}\oplus
I^{\mathcal{V}}_{yz}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{st^{\mathcal{V}}\circ
st^{\mathcal{W}}}$$\scriptstyle{\psi}$$\textstyle{\mathbb{R}^{2}\oplus V_{z}}$

over the intersection of this face with each open subset of the form
$\mathcal{W}_{xy}\times\mathcal{V}_{yz}\times[0,\varepsilon)$. As usual, maps
of the form $st\circ st$, the first (i.e. rightmost) copy of $st$ applied
provides the first copy of $\mathbb{R}$ and the second (i.e. leftmost) $st$
provides the second copy of $\mathbb{R}$.

The outer square is $\star$ ‣ 65 over this face; the upper left triangle
commutes by compatibility of the maps $\psi$ and the lower right triangle
commutes by compatibility of the $\psi$ and $st$.

Commutativity of $\star$ ‣ 65 over the boundary face
$\tilde{\mathcal{Q}}_{x;z^{\prime}z}$ is essentially identical; commutativity
of $\star$ ‣ 65 over the boundary face
$\mathcal{W}_{xy}\times\mathcal{V}_{yz}\times\\{0\\}$ follows by construction
of $st^{\tilde{\mathcal{Q}}}$.

We frame $\tilde{\mathcal{S}}$ in exactly the same way: over each
$\mathcal{R}_{xy}\times\mathcal{V}_{yz}\times[0,\varepsilon)_{s}$, we define
$st^{\tilde{\mathcal{S}}}_{xz}$ to be the composition

$st^{\tilde{\mathcal{S}}}_{xz}:V_{x}\oplus
I^{\tilde{\mathcal{S}}}_{xz}\xrightarrow{\psi^{-1}}V_{x}\oplus
I^{\mathcal{R}}_{xy}\oplus
I^{\mathcal{V}}_{yz}\xrightarrow{st^{\mathcal{V}}\circ
st^{\mathcal{R}}}\mathbb{R}^{3}\oplus V_{z}$

where the first two entries of $\mathbb{R}^{3}$ are from $st^{\mathcal{R}}$
and the final one is from $st^{\mathcal{V}}$.

Similarly to with $\tilde{\mathcal{Q}}$, this glues together to define
$st^{\tilde{\mathcal{S}}}$ over the the entirety of
$\tilde{\mathcal{S}}_{xz}$. For this to be a framing, we have a compatibility
condition over each boundary face; over the boundary faces
$\tilde{\mathcal{S}}_{xx^{\prime};z}$, $\tilde{\mathcal{S}}_{x;z^{\prime}z}$
and $\mathcal{R}_{xy}\times\mathcal{V}_{yz}\times\\{0\\}$ this follows from
the same argument as for $\tilde{\mathcal{Q}}$, over the boundary faces
$\tilde{\mathcal{Q}}_{xz}$ and $\tilde{\mathcal{Q}}^{\prime}_{xz}$, 65
commutes by compatibility between the framings of $\mathcal{R}$ and
$\tilde{\mathcal{W}}$, $\tilde{\mathcal{W}}^{\prime}$.

######  Lemma 71.

Let $\mathcal{W}:\mathcal{F}\to\mathcal{G}$ and
$\mathcal{V}:\mathcal{G}\to\mathcal{H}$ be framed morphisms of framed flow
categories. Then their composition can be framed, and the equivalence class
$[\mathcal{V}\circ\mathcal{W}]$ defines in $[\mathcal{F},\mathcal{H}]^{fr}$ is
well-defined.

######  Proof 4.18.

Compatibly choose framed functions $(f_{xz},s_{xz})$ on each
$\tilde{\mathcal{Q}}_{xz}$, let $\tilde{\mathcal{Q}}^{\leq
1}=f_{xz}^{-1}(-\infty,1]$ and let $\mathcal{U}_{xz}=f_{xz}^{-1}$. Then
$\mathcal{U}$ is a representative of $\mathcal{V}\circ\mathcal{W}$ as an
unframed morphism; we use the framing we’ve constructed on
$\tilde{\mathcal{Q}}$ to frame $\mathcal{U}$.

We define stable isomorphisms
$st^{\mathcal{U}}_{xz}:I^{\mathcal{U}}_{xz}\oplus V_{z}\to\mathbb{R}\oplus
V_{x}$ with stabilising bundle $\mathbb{R}\xi$ (where $\xi$ is some formal
generator) to be the composition:

$st^{\mathcal{U}}_{xz}:I^{\mathcal{U}}_{xz}\oplus
V_{z}\oplus\mathbb{R}\xi\xrightarrow{di_{xz}+s_{xz}}I^{\tilde{\mathcal{Q}}}_{xz}\oplus
V_{z}\xrightarrow{st^{\tilde{\mathcal{Q}}}}\mathbb{R}^{2}\oplus
V_{x}=\mathbb{R}\oplus\mathbb{R}\xi\oplus V_{x}$

Here $i_{xz}:\mathcal{U}_{xz}\to\tilde{\mathcal{Q}}_{xz}$ is the natural
inclusion map, $s_{xz}$ takes $\mathbb{R}\xi$ as input, the first map sends
$\tau_{z}$ to $\tau_{z}$, and the final identification is to make it clear
that it is the second copy of $\mathbb{R}$ in $\mathbb{R}^{2}$ that is the
stabilising bundle.

Commutativity of $\star$ ‣ 65 over the appropriate boundary faces follows from
similar considerations to before, though now we must use the compatibility of
the framed functions with those on the boundary faces. This implies
$\mathcal{U}$ is a framed morphism.

Similarly we can frame $\tilde{\mathcal{Q}}^{\leq 1}$ in an appropriate sense;
note commutativity of 65 uses the fact that we identify
$\mathbb{R}^{2}=\mathbb{R}\oplus\mathbb{R}\xi$ rather than the other way
around. Proceeding as in the proof of Lemma 55 (but now with framings in hand)
shows that the class of $\mathcal{U}$ in $[\mathcal{F},\mathcal{H}]^{fr}$ is
independent of choices.

######  Lemma 72.

Composition of framed morphisms is compatible with the equivalence relation in
Definition 68, and induces a well-defined bilinear map

$\circ:[\mathcal{G},\mathcal{H}]^{fr}\otimes[\mathcal{F},\mathcal{G}]^{fr}\to[\mathcal{F},\mathcal{H}]^{fr}$

######  Proof 4.19.

Similarly to the proof of Lemma 71, we choose compatible framed functions on
each $\tilde{\mathcal{Q}}_{xz}$ and $\tilde{\mathcal{S}}_{xz}$, and proceed as
in the proof of Lemma 55, framing the bordism $\mathcal{T}$ similarly to
above, to conclude that composition induces the desired well-defined map; the
same argument as in the proof of Lemma 55 also shows it is a bilinear map.

Similarly to Lemma 57, we have:

######  Lemma 73.

The composition map is associative, i.e. the following diagram commutes:

$\textstyle{\left[\mathcal{H},\mathcal{I}\right]^{fr}\otimes\left[\mathcal{G},\mathcal{H}\right]^{fr}\otimes\left[\mathcal{F},\mathcal{G}\right]^{fr}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\circ}$$\scriptstyle{\circ}$$\textstyle{\left[\mathcal{G},\mathcal{I}\right]^{fr}\otimes\left[\mathcal{F},\mathcal{G}\right]^{fr}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\circ}$$\textstyle{\left[\mathcal{H},\mathcal{I}\right]^{fr}\otimes\left[\mathcal{F},\mathcal{H}\right]^{fr}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\circ}$$\textstyle{\left[\mathcal{F},\mathcal{I}\right]^{fr}}$

######  Proof 4.20.

We prove this by equipping all the manifolds encountered in the proof of Lemma
57 with (compatible) stable framings; we furthermore assume all of the same
notation as in Lemma 57.

Using the framings on $\mathcal{F}$, $\mathcal{G}$, $\mathcal{H}$,
$\mathcal{I}$, $\mathcal{W}$, $\mathcal{V}$ and $\mathcal{X}$, we equip
$\tilde{\mathcal{P}}$, $\tilde{\mathcal{P}}^{\prime}$, $\tilde{\mathcal{Q}}$,
$\tilde{\mathcal{Q}}^{\prime}$ and $\tilde{\mathcal{Z}}$ with stable framings,
just as in Lemma 71.

Using the given choices of framed functions, we equip $\mathcal{M}$,
$\mathcal{M}^{\prime}$, $\mathcal{U}$, $\mathcal{U}^{\prime}$, $\mathcal{Y}$,
$\mathcal{L}$ all with stable framings.

Unfortunately the embeddings from the collar neighbourhood theorem
$\mathcal{P}_{xw},\mathcal{P}^{\prime}_{xw}\hookrightarrow\mathcal{Y}_{xw}$
may not be compatible with the stable framings, so whilst $\mathcal{L}$ is a
bordism between $\mathcal{M}$ and $\mathcal{M}^{\prime}$, the framing we have
equipped $\mathcal{L}$ with might not be compatible with the framings on
$\mathcal{M}$ and $\mathcal{M}^{\prime}$, so it might not be a framed bordism
between them.

However note that the framings on each $\tilde{\mathcal{P}}_{xw}$ agree with
those on $\mathcal{Y}_{xw}$ when restricted to the carapace of
$\tilde{\mathcal{P}}_{xw}$. Since each $\tilde{\mathcal{P}}_{xw}$ deformation
retracts to its carapace (which also includes all its boundary), the framings
on $\tilde{\mathcal{P}}_{xw}$ pulled back from $\mathcal{Y}_{xw}$ are
homotopic to those constructed on the $\tilde{\mathcal{P}}_{xw}$ directly;
furthermore by inducting on the dimension and working relative to the boundary
we may assume that these homotopies are compatible with each other for all
$x\in\mathcal{F}$, $w\in\mathcal{I}$.

Since homotopic framings on $\tilde{\mathcal{P}}_{xw}$ induce homotopic
framings on $\mathcal{M}_{xw}$ (since the construction of this framing works
in 1-parameter families of framings on $\tilde{\mathcal{P}}$), and noting that
this discussion applies equally to $\tilde{\mathcal{P}}^{\prime}$ and
$\mathcal{M}^{\prime}$, it follows that $\mathcal{L}$ is a framed bordism
between $\mathcal{M}$ and $\mathcal{M}^{\prime}$, where $\mathcal{M}$ and
$\mathcal{M}^{\prime}$ are equipped with framings homotopic to the ones we
originally equipped them with.

Combined with Example 4.15, it follows that $\mathcal{M}$ and
$\mathcal{M}^{\prime}$ are framed bordant.

Similarly to Lemma 58 (and by the same proof, but incorporating the
‘deformation to the carapace’ argument from above), we have:

######  Lemma 74.

In the situation of Lemma 58, if $\mathcal{F}$, $\mathcal{G}$, $\mathcal{H}$,
$\mathcal{W}$, $\mathcal{V}$, $\mathcal{T}$ and $\mathcal{R}$ are all
compatibly framed, then $\mathcal{T}$ is a representative of
$\mathcal{V}\circ\mathcal{W}$ in $[\mathcal{F},\mathcal{H}]^{fr}$.

######  Definition 75.

We define the (homotopy) _category of framed flow categories_ , $\Flow^{fr}$,
to have objects framed flow categories, and morphisms framed morphisms of flow
categories. This is a (non-unital) category enriched in abelian groups.

######  Example 4.21.

Given a framed flow category $\mathcal{F}$, we can form $\mathcal{F}[n]$ as a
framed flow category for any $n\in\mathbb{Z}$, by setting
$V^{\mathcal{F}[n]}_{x}:=V^{\mathcal{F}}_{x}\oplus\mathbb{R}^{n}$ and
$st^{\mathcal{F}[n]}:=st^{\mathcal{F}}\oplus\id_{\mathbb{R}^{n}}$.

######  Example 4.22.

A morphism $*[i]\to*$ consists of a closed (stably) framed $i$-manifold; a
framed bordism between morphisms consists of a (stably) framed bordism. From
this, we see that

$[*[i],*]^{fr}=\Omega^{fr}_{i}(\ast)=\Omega^{fr}_{i}$

In particular, it follows that $\Flow^{fr}$ is enriched over graded modules
over $\Omega^{fr}_{*}$.

######  Conjecture 76 (Abouzaid-Blumberg).

$\Flow^{fr}$ is equivalent to the homotopy category of finite spectra.

Though not phrased like this, a functor from the category of framed flow
categories to the homotopy category of finite spectra is essentially
constructed in [Lar], following [CJS95].

##  5 Algebra over flow categories

###  5.1 Bilinear maps of flow categories

######  Definition 77.

Let $\mathcal{F}$, $\mathcal{G}$ and $\mathcal{H}$ be flow categories. A
_bilinear map_ of flow categories
$\mathcal{W}:\mathcal{F}\times\mathcal{G}\rightarrow\mathcal{H}$ consists of
compact smooth manifolds with faces $\mathcal{W}_{xy;z}$ of dimension
$|x|+|y|-|z|$ for each $x$ in $\mathcal{F}$, $y$ in $\mathcal{G}$ and $z$ in
$\mathcal{H}$, along with maps

$c=c^{\mathcal{W}}:\mathcal{F}_{xu}\times\mathcal{W}_{uy;z}\rightarrow\mathcal{W}_{xy;z}$

and

$c=c^{\mathcal{W}}:\mathcal{G}_{yv}\times\mathcal{W}_{xv;z}\rightarrow\mathcal{H}_{xy;z}$

and

$c=c^{\mathcal{W}}:\mathcal{W}_{xy;w}\times\mathcal{H}_{wz}\rightarrow\mathcal{W}_{xy;z}$

which are compatible with each other and $c^{\mathcal{F}}$, $c^{\mathcal{G}}$
and $c^{\mathcal{H}}$, and which define a system of faces

$\mathcal{F}_{x_{0}\ldots x_{i}}\times\mathcal{G}_{y_{0}\ldots
y_{j}}\times\mathcal{W}_{x_{i}y_{j};z_{0}}\times\mathcal{H}_{z_{0}\ldots
z_{k}}$

for $x_{0}=x,\ldots,x_{i}$ in $\mathcal{F}$, $y_{0}=y,\ldots,y_{j}$ in
$\mathcal{G}$ and $z_{0},\ldots,z_{k}=z$ in $\mathcal{H}$. In particular,
$\mathcal{W}_{xy;z}$ has a system of boundary faces
$\mathcal{F}_{xx^{\prime}}\times\mathcal{W}_{x^{\prime}y;z}$,
$\mathcal{G}_{yy^{\prime}}\times\mathcal{W}_{xy^{\prime};z}$ and
$\mathcal{W}_{xy;z^{\prime}}\times\mathcal{H}_{z^{\prime}z}$ for
$x^{\prime}\in\mathcal{F}$, $y^{\prime}\in\mathcal{G}$ and
$z^{\prime}\in\mathcal{H}$.

A bilinear map $*[i]\times*[j]\rightarrow\mathcal{H}$ consists of the same
data as a morphism $*[i+j]\rightarrow\mathcal{H}$. A bilinear map
$*[i]\times\mathcal{G}\to\mathcal{H}$ consists of the same data as a morphism
$\mathcal{G}[i]\to\mathcal{H}$. Let $\mathcal{F}$, $\mathcal{G}$ and
$\mathcal{H}$ be flow categories,
$\mathcal{W}:\mathcal{F}\times\mathcal{G}\rightarrow\mathcal{H}$ a bilinear
map and $\mathcal{A}:*[i]\rightarrow\mathcal{F}$ and
$\mathcal{B}:*[j]\rightarrow\mathcal{G}$ morphisms. We define a morphism of
flow categories

$\mathcal{W}\circ\mathcal{A}:\mathcal{G}[i]\rightarrow\mathcal{H}$

as follows.

For $y$ in $\mathcal{G}$ and $z$ in $\mathcal{H}$, define

$\tilde{\mathcal{P}}_{yz}:=\tilde{\mathcal{P}}_{yz}(\mathcal{A},\mathcal{W})=\left(\bigsqcup_{x\in\mathcal{F}}\mathcal{A}_{*x}\times\mathcal{W}_{xy;z}\times[0,\varepsilon)\right)/\sim$

where $\sim$ is defined so that $\tilde{\mathcal{P}}_{yz}$ is the coequaliser
of

$\bigsqcup_{x,x^{\prime}\in\mathcal{F}}\mathcal{A}_{*x}\times\mathcal{F}_{xx^{\prime}}\times\mathcal{W}_{x^{\prime}y;z}\times[0,\varepsilon)^{2}_{u,v}\rightrightarrows\bigsqcup_{x\in\mathcal{F}}\mathcal{A}_{*x}\times\mathcal{W}_{xy;z}\times[0,\varepsilon)_{s}$

where the two maps send $(a,b,c,u,v)$ to $(a,\mathcal{C}(b,c,u),v)$ and
$(\mathcal{C}(a,b,v),c,u)$ respectively, as in the construction of the
composition of morphisms.

By Lemmas 37 and 38 (applying these lemmas with $I=ob\mathcal{F}$ and
$J=ob\mathcal{G}\sqcup ob\mathcal{H}$), this is a manifold with faces, with a
system of faces given by

$\mathcal{A}_{*;x_{0}\ldots x_{i}}\times\mathcal{G}_{y_{0}\ldots
y_{j}}\times\mathcal{W}_{x_{i}y_{j};z_{0}}\times\mathcal{H}_{z_{0}\ldots
z_{k}}\times\\{0\\}$

and

$\mathcal{G}_{y_{0}\ldots
y_{j}}\times\tilde{\mathcal{P}}_{y_{j}z_{0}}\times\mathcal{H}_{z_{0}\ldots
z_{k}}$

for $x,x_{0},\ldots,x_{i}$ in $\mathcal{F}$, $y_{0}=y,\ldots,y_{j}$ in
$\mathcal{G}$ and $z_{0},\ldots,z_{k}=z$ in $\mathcal{H}$. Pick framed
functions $(f_{yz},s_{yz})$ on each $\tilde{\mathcal{P}}_{yz}$ which are
compatible with each other on overlaps, and let
$\mathcal{V}_{yz}:=f^{-1}_{yz}\\{1\\}$ for all $y\in\mathcal{G}$,
$z\in\mathcal{H}$. These form a morphism of flow categories
$\mathcal{G}[i]\rightarrow\mathcal{H}$ which we define
$\mathcal{W}\circ\mathcal{A}$ to be. By the same argument as in the proof of
Lemma 55, this defines a well-defined morphism in $\Flow$.

Given instead $\mathcal{B}:*[j]\rightarrow\mathcal{G}$, we can similarly
define
$\tilde{\mathcal{P}}_{xz}=\tilde{\mathcal{P}}_{xz}(\mathcal{B},\mathcal{W})$
for $x\in\mathcal{F}$ and $z\in\mathcal{H}$, and define a morphism of flow
categories

$\mathcal{W}\circ\mathcal{B}:\mathcal{F}[j]\rightarrow\mathcal{H}$

######  Lemma 78.

For all morphisms of flow categories $\mathcal{A}:*[i]\to\mathcal{F}$ and
$\mathcal{B}:*[j]\to\mathcal{G}$, we have that

$[(\mathcal{W}\circ\mathcal{A})\circ\mathcal{B}]=[(\mathcal{W}\circ\mathcal{B})\circ\mathcal{A}]\in[*[i+j],\mathcal{H}]$

######  Proof 5.1.

For $z\in\mathcal{H}$, define
$\tilde{\mathcal{Q}}_{*z}(\mathcal{B},\tilde{\mathcal{P}}(\mathcal{A},\mathcal{W}))$
to be the coequaliser of the following diagram:

$\bigsqcup\limits_{y,y^{\prime}\in\mathcal{G}}\mathcal{B}_{*y}\times\mathcal{G}_{yy^{\prime}}\times\tilde{\mathcal{P}}_{y^{\prime}z}\times[0,\varepsilon)^{2}_{u,v}\rightrightarrows\bigsqcup\limits_{y\in\mathcal{G}}\mathcal{B}_{*y}\times\tilde{\mathcal{P}}_{yz}\times[0,\varepsilon)_{s}$

where the two maps send $(a,b,c,u,v)$ to $(a,\mathcal{C}(b,c,u),v)$ and
$(\mathcal{C}(a,b,v),c,u)$ respectively, as before.

By Lemmas 37 and 38,
$\tilde{\mathcal{Q}}_{*z}(\mathcal{B},\tilde{\mathcal{P}}(\mathcal{A},\mathcal{W}))$
has a system of boundary faces given by
$\tilde{\mathcal{Q}}_{*z^{\prime}}\times\mathcal{H}_{z^{\prime}z}$,
$\mathcal{A}_{*x}\times\\{0\\}\times\tilde{\mathcal{P}}_{xz}$ and
${\mathcal{B}}_{*y}\times\tilde{\mathcal{P}}_{yz}\times\\{0\\}$ for
$x\in\mathcal{F}$, $y\in\mathcal{G}$ and $z^{\prime}\in\mathcal{H}$; we note
that the final two types of boundary faces overlap in codimension two faces of
the form
$\mathcal{A}_{*x}\times\mathcal{B}_{*y}\times\mathcal{W}_{xy;z}\times\\{(0,0)\\}$,
which is a compact face.

Choose framed functions $(f_{xy},s_{xy})$ and $(f_{yz},s_{yz})$ on each
$\tilde{\mathcal{P}}_{xz}$ and $\tilde{\mathcal{P}}_{yz}$ respectively, and
let $\mathcal{U}_{xz}=f_{xz}^{-1}\\{1\\}$ and
$\mathcal{U}_{yz}=f_{yz}^{-1}\\{1\\}$ respectively; these are representatives
of $\mathcal{W}\circ\mathcal{B}$ and $\mathcal{W}\circ\mathcal{A}$
respectively.

Now choose framed functions $(g_{*z},t_{*z})$ on each
$\tilde{\mathcal{Q}}_{*z}$, extending the $(f_{xy},s_{xy})$ and
$(f_{yz},s_{yz})$ on appropriate boundary faces. Then considering
$g_{*z}^{-1}\\{1\\}$ and using the argument in Lemma 58, the result follows.

######  Definition 79.

We define a map

$\mathcal{W}_{*}:[*[i],\mathcal{F}]\otimes[*[j],\mathcal{G}]\rightarrow[*[i+j],\mathcal{H}]$

sending $[\mathcal{A}]\otimes[\mathcal{B}]$ to
$[(\mathcal{W}\circ\mathcal{A})\circ\mathcal{B}]=[(\mathcal{W}\circ\mathcal{B})\circ\mathcal{A}]\in[*[i+j],\mathcal{H}]$.
This is a well-defined map, by Lemmas 78 and 56.

######  Lemma 80.

$\mathcal{W}_{*}$ is a bilinear map.

######  Proof 5.2.

Since composition is bilinear,
$[(\mathcal{W}\circ\mathcal{A})\circ\mathcal{B}]$ is linear in
$[\mathcal{B}]$. Similarly $[(\mathcal{W}\circ\mathcal{B})\circ\mathcal{A}]$
is linear in $[\mathcal{A}]$.

######  Definition 81.

Let $\mathcal{W},\mathcal{V}:\mathcal{F}\times\mathcal{G}\to\mathcal{H}$ be
bilinear maps. A _bordism_ between $\mathcal{W}$ and $\mathcal{V}$ consists of
compact manifolds with faces $\mathcal{R}_{xy;z}$ of dimension $|x|+|y|-|z|+1$
for each $x\in\mathcal{F}$, $y\in\mathcal{G}$ and $z\in\mathcal{H}$, with
systems of boundary faces given by
$\mathcal{F}_{xx^{\prime}}\times\mathcal{R}_{x^{\prime}y;z}$,
$\mathcal{G}_{yy^{\prime}}\times\mathcal{R}_{xy^{\prime};z}$,
$\mathcal{R}_{xz;z^{\prime}}\times\mathcal{H}_{z^{\prime}z}$,
$\mathcal{W}_{xy;z}$ and $\mathcal{V}_{xy;z}$. Note all of these are broken
except the last two which are unbroken; we label them as incoming and outgoing
respectively.

In both cases, as usual we that the inclusions $c$ of each boundary face
satisfy appropriate associativity relations.

If $\mathcal{W}:\mathcal{F}\times\mathcal{G}\to\mathcal{H}$ is a bilinear map
and $\mathcal{A}:\mathcal{F}^{\prime}\to\mathcal{F}$ is a morphism, we can
form another bilinear map
$\mathcal{W}\circ\mathcal{A}:\mathcal{F}^{\prime}\times\mathcal{G}\to\mathcal{H}$
in the same way as in the previously treated case $\mathcal{F}=*[i]$: for
$w\in\mathcal{F}^{\prime}$, $y\in\mathcal{G}$ and $z\in\mathcal{H}$, we let
$\tilde{\mathcal{P}}_{wy;z}=\tilde{\mathcal{P}}_{wy;z}(\mathcal{A},\mathcal{W})$
be the coequaliser of

$\bigsqcup_{x,x^{\prime}\in\mathcal{F}}\mathcal{A}_{wx}\times\mathcal{F}_{xx^{\prime}}\times\mathcal{W}_{x^{\prime}y;z}\times[0,\varepsilon)^{2}_{u,v}\rightrightarrows\bigsqcup_{x\in\mathcal{F}}\mathcal{A}_{wx}\times\mathcal{W}_{xy;z}\times[0,\varepsilon)_{s}$

Then choosing appropriate compatible framed functions and taking the preimage
of 1 gives a bilinear map
$\mathcal{F}^{\prime}\times\mathcal{G}\to\mathcal{H}$; the usual arguments
show this is well-defined up to bordism. Note this generalises the previous
construction by Example 5.1.

If instead we are given a morphism
$\mathcal{B}:\mathcal{G}^{\prime}\to\mathcal{G}$, we can similarly form a
bilinear map
$\mathcal{W}\circ\mathcal{B}:\mathcal{F}\times\mathcal{G}^{\prime}\to\mathcal{H}$.

######  Lemma 82.

Let $\mathcal{W},\mathcal{V}:\mathcal{F}\times\mathcal{G}\to\mathcal{H}$ be
bilinear maps, $\mathcal{A}:\mathcal{F}^{\prime}\to\mathcal{F}$ a morphism and
$\mathcal{R}$ a bordism from $\mathcal{W}$ to $\mathcal{V}$. Then
$\mathcal{W}\circ\mathcal{A}$ and $\mathcal{V}\circ\mathcal{A}$ are bordant
bilinear maps $\mathcal{F}^{\prime}\times\mathcal{G}\to\mathcal{H}$.

In particular, $\mathcal{W}_{*}=\mathcal{V}_{*}$.

######  Proof 5.3.

For $w\in\mathcal{F}^{\prime}$, $y\in\mathcal{G}$ and $z\in\mathcal{H}$, let
$\tilde{\mathcal{P}}_{wy;z}=\tilde{\mathcal{P}}_{wy;z}(\mathcal{A},\mathcal{R})$
be the coequaliser of

$\bigsqcup_{x,x^{\prime}\in\mathcal{F}}\mathcal{A}_{wx}\times\mathcal{F}_{xx^{\prime}}\times\mathcal{R}_{x^{\prime}y;z}\times[0,\varepsilon)^{2}_{u,v}\rightrightarrows\bigsqcup_{x\in\mathcal{F}}\mathcal{A}_{wx}\times\mathcal{R}_{xy;z}\times[0,\varepsilon)_{s}$

Then choosing suitable compatible framed functions $(f_{wy;z},s_{wy;z})$ on
each $\tilde{\mathcal{P}}_{wy;z}$ extending compatible framed functions on the
boundary faces $\tilde{\mathcal{P}}_{wy;z}(\mathcal{A},\mathcal{W})$ and
$\tilde{\mathcal{P}}_{wy;z}(\mathcal{A},\mathcal{V})$ and taking the preimage
of 1 provides the desired bordism.

Now let $\mathcal{F}$, $\mathcal{G}$, $\mathcal{H}$, $\mathcal{I}$,
$\mathcal{J}$ and $\mathcal{K}$ be flow categories, and let
$\mathcal{W}:\mathcal{F}\times\mathcal{G}\rightarrow\mathcal{I}$,
$\mathcal{V}:\mathcal{G}\times\mathcal{H}\rightarrow\mathcal{J}$,
$\mathcal{X}:\mathcal{I}\times\mathcal{H}\rightarrow\mathcal{K}$ and
$\mathcal{Y}:\mathcal{F}\times\mathcal{J}\rightarrow\mathcal{K}$ be bilinear
maps. These induce trilinear maps of abelian groups

$\mathcal{X}_{*}\circ\mathcal{W}_{*},\mathcal{Y}_{*}\circ\mathcal{V}_{*}:[*[i],\mathcal{F}]\otimes[*[j],\mathcal{G}]\otimes[*[k],\mathcal{H}]\rightarrow[*[i+j+k],\mathcal{K}]$

We will give conditions for these two maps to agree.

######  Definition 83.

An _associator_ $\mathcal{Z}$ for the decuplet
$(\mathcal{F},\mathcal{G},\mathcal{H},\mathcal{I},\mathcal{J},\mathcal{K},\mathcal{W},\mathcal{V},\mathcal{X},\mathcal{Y})$
consists of compact manifolds with faces $\mathcal{Z}_{xyz;w}$ for $x$ in
$\mathcal{F}$, $y$ in $\mathcal{G}$, $z$ in $\mathcal{H}$ and $w$ in
$\mathcal{I}$, of dimension $|x|+|y|+|z|-|w|+1$, with a system of faces given
by

$\mathcal{F}_{x_{0}\ldots x_{i}}\times\mathcal{G}_{y_{0}\ldots
y_{j}}\times\mathcal{H}_{z_{0}\ldots
z_{k}}\times\mathcal{Z}_{x_{i}y_{j}z_{k};w_{0}}\times\mathcal{K}_{w_{0}\ldots
w_{l}}$

and

$\mathcal{F}_{x_{0}\ldots x_{i}}\times\mathcal{G}_{y_{0}\ldots
y_{j}}\times\mathcal{H}_{z_{0}\ldots
z_{k}}\times\mathcal{W}_{x_{i}y_{j};u_{0}}\times\mathcal{I}_{u_{0}\ldots
u_{r}}\times\mathcal{X}_{u_{r}z_{k};w_{0}}\times\mathcal{K}_{w_{0}\ldots
w_{l}}$

and

$\mathcal{F}_{x_{0}\ldots x_{i}}\times\mathcal{G}_{y_{0}\ldots
y_{j}}\times\mathcal{H}_{z_{0}\ldots
z_{k}}\times\mathcal{V}_{y_{j}z_{k};v_{0}}\times\mathcal{J}_{v_{0}\ldots
v_{s}}\times\mathcal{Y}_{x_{i}v_{j};w_{0}}\times\mathcal{K}_{w_{0}\ldots
w_{l}}$

for $x_{0}=x,\ldots,x_{i}$ in $\mathcal{F}$, $y_{0}=y,\ldots,y_{j}$ in
$\mathcal{G}$, $z_{0}=z,\ldots,z_{k}$ in $\mathcal{H}$, $u_{0},\ldots,u_{r}$
in $\mathcal{I}$, $v_{0},\ldots,v_{s}$ in $\mathcal{J}$ and
$w_{0},\ldots,w_{l}=w$ in $\mathcal{K}$.

Note that in particular for an associator $\mathcal{Z}$, each
$\mathcal{Z}_{xyz;w}$ has a system of boundary faces

$\mathcal{F}_{xx^{\prime}}\times\mathcal{Z}_{x^{\prime}yz;w}$

and

$\mathcal{G}_{yy^{\prime}}\times\mathcal{Z}_{xy^{\prime}z;w}$

and

$\mathcal{H}_{zz^{\prime}}\times\mathcal{Z}_{xyz^{\prime};w}$

and

$\mathcal{W}_{xy;u}\times\mathcal{X}_{uz;w}$

and

$\mathcal{V}_{yz;v}\times\mathcal{Y}_{xv;w}$

and

$\mathcal{Z}_{xyz;w^{\prime}}\times\mathcal{K}_{w^{\prime}w}$

for $x,x^{\prime}$ in $\mathcal{F}$, $y,y^{\prime}$ in $\mathcal{G}$,
$z,z^{\prime}$ in $\mathcal{H}$, $u$ in $\mathcal{I}$, $v$ in $\mathcal{J}$
and $w,w^{\prime}$ in $\mathcal{K}$.

######  Lemma 84.

If there exists an associator $\mathcal{Z}$ for
$(\mathcal{F},\mathcal{G},\mathcal{H},\mathcal{I},\mathcal{J},\mathcal{K},\mathcal{W},\mathcal{V},\mathcal{X},\mathcal{Y})$,
then the two induced trilinear maps are equal:

$\mathcal{X}_{*}\circ\mathcal{W}_{*}=\mathcal{Y}_{*}\circ\mathcal{V}_{*}:[*[i],\mathcal{F}]\otimes[*[j],\mathcal{G}]\otimes[*[k],\mathcal{H}]\rightarrow[*[i+j+k],\mathcal{K}]$

######  Proof 5.4.

$\mathcal{B}\circ{\mathcal{Z}}$ along with the proof of Lemma 58 provides a
bordism between the bilinear maps
$\mathcal{X}\circ(\mathcal{W}\circ\mathcal{B})$ and
$\mathcal{Y}\circ(\mathcal{V}\circ\mathcal{B})$; the result then follows from
Lemma 82.

###  5.2 Bilinear maps: framings

We incorporate stable framings into Section 5.1 in the same way as in Section
4.3. Similarly to Definition 66, we have:

######  Definition 85.

Let $\mathcal{F}$, $\mathcal{G}$ and $\mathcal{H}$ be framed flow categories.
A _stable framing_ on a bilinear map
$\mathcal{W}:\mathcal{F}\times\mathcal{G}\to\mathcal{H}$ consists of stable
isomorphisms

$st=st^{\mathcal{W}}_{xyz}:V_{x}\oplus V_{y}\oplus
I^{\mathcal{W}}_{xy;z}\to\mathbb{R}\oplus V_{z}$

where $I^{\mathcal{W}}_{xy;z}\tau_{z}$ is defined to be
$T\mathcal{W}_{xy;z}\oplus\mathbb{R}\tau_{z}$ (the same as in Definition 62),
with abstract gluing isomorphisms $\psi$ defined similarly (noting that all
boundary faces are broken), such that these, combined with those for
$\mathcal{F}$, $\mathcal{G}$, $\mathcal{H}$ and each lower-dimensional
$\mathcal{W}_{x^{\prime}y^{\prime};z^{\prime}}$, form systems of framings for
each $\mathcal{W}_{xy;z}$.

The analogues of Examples 5.1 and 5.1 hold for framed bilinear maps and framed
morphisms.

Assume $\mathcal{W}:\mathcal{F}\times\mathcal{G}\to\mathcal{H}$ be a framed
bilinear map, and let $\mathcal{A}:\mathcal{F}^{\prime}\to\mathcal{F}$ and
$\mathcal{B}:\mathcal{G}^{\prime}\to\mathcal{G}$ be framed morphisms. We equip
each $\tilde{\mathcal{P}}(\mathcal{A},\mathcal{W})$ and
$\tilde{\mathcal{P}}(\mathcal{B},\mathcal{W})$ (constructed as in Section 5.1)
with framings, in exactly the same way we framed each
$\tilde{\mathcal{Q}}(\cdot,\cdot)$ in Lemma 70, and from this we frame
$\mathcal{W}\circ\mathcal{A}$ and $\mathcal{W}\circ\mathcal{B}$ in exactly the
same way as in Lemma 71. From this plus (the argument in) Lemma 72, we have:

######  Lemma 86.

The bilinear maps
$\mathcal{W}\circ\mathcal{A}:\mathcal{F}^{\prime}\times\mathcal{G}\to\mathcal{H}$
and
$\mathcal{W}\circ\mathcal{B}:\mathcal{F}\times\mathcal{G}^{\prime}\to\mathcal{H}$
admit framings, and define well-defined framed morphisms up to framed bordism.

Similar to Lemma 80 (and by the same proof, with the usual incorporation of
framings), we have:

######  Lemma 87.

For all framed morphisms of flow categories $\mathcal{A}:*[i]\to\mathcal{F}$
and $\mathcal{B}:*[j]\to\mathcal{G}$, we have that

$[(\mathcal{W}\circ\mathcal{A})\circ\mathcal{B}]=[(\mathcal{W}\circ\mathcal{B})\circ\mathcal{A}]\in[*[i+j],\mathcal{H}]^{fr}$

The same arguments as in Lemmas 72 and 80 prove

######  Lemma 88.

This construction defines a well-defined bilinear map induced by
$\mathcal{W}$:

$\mathcal{W}_{*}:[*[i],\mathcal{F}]^{fr}\otimes[*[j],\mathcal{G}]^{fr}\to[*[i+j],\mathcal{H}]^{fr}$

Next is bordisms of bilinear maps. Let $\mathcal{R}$ be a bordism from
$\mathcal{W}$ to $\mathcal{V}$, where
$\mathcal{W},\mathcal{V}:\mathcal{F}\times\mathcal{G}\to\mathcal{H}$ are
bilinear maps, and assume
$\mathcal{W},\mathcal{V},\mathcal{F},\mathcal{G},\mathcal{H}$ are all
compatibly framed. By labelling the boundary faces of each
$\mathcal{R}_{xy;z}$ as broken or incoming/outgoing unbroken as in Definition
81, from Section 4.2 we obtain abstract index bundles $I^{\mathcal{R}}_{xy;z}$
along with coherent abstract gluing isomorphisms over each boundary face.

######  Definition 89.

A _framing_ on the bordism $\mathcal{R}$ consists of stable isomorphisms

$st_{xy;z}:V_{x}\oplus V_{y}\oplus
I^{\mathcal{R}}_{xy;z}\to\mathbb{R}^{2}\oplus V_{z}$

over each $\mathcal{R}_{xy;z}$, such that these, combined with the framings
$\mathcal{W},\mathcal{V},\mathcal{F},\mathcal{G},\mathcal{H}$ and the framings
for each lower-dimensional $\mathcal{R}_{x^{\prime}y^{\prime};z^{\prime}}$,
form systems of framings for each $\mathcal{R}_{xy;z}$.

Incorporating framings into the proof of Lemma 82 just as in Lemma 71, we have
that:

######  Lemma 90.

Let $\mathcal{W},\mathcal{V}:\mathcal{F}\times\mathcal{G}\to\mathcal{H}$ be
framed bilinear maps, $\mathcal{A}:\mathcal{F}^{\prime}\to\mathcal{F}$ a
framed morphism and $\mathcal{R}$ a framed bordism from $\mathcal{W}$ to
$\mathcal{V}$. Then $\mathcal{W}\circ\mathcal{A}$ and
$\mathcal{V}\circ\mathcal{A}$ are framed bordant bilinear maps
$\mathcal{F}^{\prime}\times\mathcal{G}\to\mathcal{H}$.

In particular, $\mathcal{W}_{*}=\mathcal{V}_{*}$.

We now move onto associators. Let $\mathcal{Z}$ be an associator for the
decuplet
$(\mathcal{F},\mathcal{G},\mathcal{H},\mathcal{I},\mathcal{J},\mathcal{K},\mathcal{W},\mathcal{V},\mathcal{X},\mathcal{Y})$.
Note all boundary faces of each $\mathcal{Z}_{xyz;w}$ are broken.

######  Definition 91.

A _framing_ on the associator $\mathcal{Z}$ consists of stable isomorphisms

$st_{xyz;w}:V_{x}\oplus V_{y}\oplus V_{z}\oplus
I^{\mathcal{Z}}_{xyz;w}\to\mathbb{R}^{3}\oplus V_{w}$

over each $\mathcal{Z}_{xyz;w}$, such that these, combined with the framings
for all items in the decuplet and the framings for each lower-dimensional
$\mathcal{Z}_{x^{\prime}y^{\prime}z^{\prime};w^{\prime}}$, form systems of
framings for each $\mathcal{Z}_{xyz;w}$.

Similarly to Lemma 84, by incorporating framings in the usual way, we have the
following lemma:

######  Lemma 92.

If there exists an framed associator $\mathcal{Z}$ for a framed decuplet
$(\mathcal{F},\mathcal{G},\mathcal{H},\mathcal{I},\mathcal{J},\mathcal{K},\mathcal{W},\mathcal{V},\mathcal{X},\mathcal{Y})$
as above, then the two induced trilinear maps are equal:

$\mathcal{X}_{*}\circ\mathcal{W}_{*}=\mathcal{Y}_{*}\circ\mathcal{V}_{*}:[*[i],\mathcal{F}]^{fr}\otimes[*[j],\mathcal{G}]^{fr}\otimes[*[k],\mathcal{H}]^{fr}\to[*[i+j+k],\mathcal{K}]^{fr}$

###  5.3 Morse complex of a flow category

Let $\mathcal{M}$ be a flow category (at this point not necessarily framed).
We define

$CM_{i}(\mathcal{M})=\oplus_{|x|=i}\mathbb{Z}/2\langle x\rangle.$

We define a differential $\partial:CM_{i}(\mathcal{M})\to
CM_{i-1}(\mathcal{M})$ by setting

$\partial(x)=\sum_{y:|y|=|x|-1}|\mathcal{M}_{xy}|\cdot y$

where $\mathcal{M}_{xy}$ is a manifold of dimension $|x|-|y|-1=0$ and
$|\mathcal{M}_{xy}|\in\mathbb{Z}/2$ denotes its cardinality.

######  Lemma 93.

$\partial$ is a differential.

The proof is a simplification of the case of the Morse complex of a framed
flow category given below in Lemma 95.

There is a canonical isomorphism of rings
$\Gamma:\Omega^{fr}_{0}\xrightarrow{\simeq}\mathbb{Z}$ between the
0-dimensional framed bordism group and $\mathbb{Z}$.

######  Definition 94.

Let $\mathcal{M}$ be a framed flow category. For each object $x$, there are
two homotopy classes of stable isomorphism $V_{x}\simeq\mathbb{R}^{-|x|}$. We
let $F(x)$ be the set of such homotopy classes of isomorphisms.

We define the _Morse chain complex_ $CM_{*}(\mathcal{M})$ to be the chain
complex with underlying graded abelian groups given by

$CM_{i}(\mathcal{M}):=\left(\bigoplus_{|x|=i,\phi\in
F(x)}\mathbb{Z}\phi\right)/A_{i}$

where $A_{i}$ is the subgroup generated by expressions

$\sum_{\phi\in F(x)}\phi$

for all $x$. This has differential $\partial$, which is defined as follows. We
first note that each $\mathcal{M}_{xy}$ is not naturally a stably framed
manifold, but does acquire a natural stable framing if we fix stable
isomorphisms between $V_{x}$ and $V_{y}$ and $\mathbb{R}^{N}$ for appropriate
$N$.

Let $x$ be an object in $\mathcal{M}$, with index $i=\mu(x)$. For a generator
$\phi$ in $F(x)$,

$\partial\phi:=\sum_{\mu(y)=i-1}\Gamma(\mathcal{M}_{xy})\psi_{y}$

where we choose some $\psi_{y}$ in $F(y)$ for each $y$, and $\mathcal{M}_{xy}$
is equipped with the stable framing it acquires from $\phi$ and $\psi_{y}$.
Then $\partial$ is a linear map which sends $A_{i}$ to $A_{i-1}$ and so
descends to a well-defined linear map $CM_{i}(\mathcal{M})\rightarrow
CM_{i-1}(\mathcal{M})$, independent of the choices of $\psi_{y}$.

Note that an evidently equivalent definition to this is to initially pick a
(stable) isomorphism $\phi_{x}:V_{x}\simeq\mathbb{R}^{N_{x}+\mu(x)}$ for each
$x$, and only take one generator for each $x$, and then to not quotient by the
subgroups $A_{*}$, observing that the resulting complex is independent of the
choices of isomorphisms $\phi_{x}$. This is in some sense a nicer definition
but requires us to keep track of more choices along the way.

######  Lemma 95.

$\partial$ is a differential. Equivalently,

$\partial\circ\partial=0.$

######  Proof 5.5.

Let $x$ be an object of $\mathcal{M}$, of index $i=\mu(x)$. Choose $\phi$ in
$F(x)$. Then

$\partial\circ\partial(x)=\sum\limits_{\mu(y)=i-1}\sum\limits_{\mu(z)=i-2}\Gamma(\mathcal{M}_{xy})\Gamma(\mathcal{M}_{yz})\,\theta_{z}$

where we choose $\psi_{y}$ in $F(y)$ and $\theta_{z}$ in $F(z)$ for each $y$,
$z$, which induces framings on $\mathcal{M}_{xy}$ and $\mathcal{M}_{yz}$. We
will show that the coefficient of each $\theta_{z}$ is 0, which implies the
whole sum is 0.

The coefficient of each $\theta_{z}$ is

$\sum\limits_{\mu(y)=i-1}\Gamma(\mathcal{M}_{xy})\Gamma(\mathcal{M}_{yz})$

which, since $\Gamma$ is a ring map, is equal to

$\Gamma\left(\bigsqcup\limits_{\mu(y)=i-1}\mathcal{M}_{xy}\times\mathcal{M}_{yz}\right)$

which is equal to

$\Gamma\left(\partial\mathcal{M}_{xz}\right)$

and hence vanishes.

######  Definition 96.

The _Morse homology_ $HM_{*}(\mathcal{M})$ of $\mathcal{M}$ is defined to be
the homology of the chain complex $CM_{*}(\mathcal{M})$.

######  Definition 97.

Given a morphism $\mathcal{W}$ of framed flow categories from
$\mathcal{M}^{0}$ to $\mathcal{M}^{1}$, there is an induced map

$CM_{*}\left(\mathcal{W}\right):CM_{*}\left(\mathcal{M}^{0}\right)\rightarrow
CM_{*}\left(\mathcal{M}^{1}\right)$

defined as follows. Let $x$ be an object in $\mathcal{M}^{0}$, with index
$i=|x|$. For a generator $\phi$ in $F(x)$,

$CM_{*}\left(\mathcal{W}\right)\phi:=\sum\limits_{|y|=i}\Gamma\left(\mathcal{W}_{xy}\right)\,\psi_{y}$

where we choose some $\psi_{y}$ in $F(y)$ for each $y$, and $\mathcal{W}_{xy}$
is equipped with the stable framing it acquires from $\phi$ and $\psi$. Then
$CM_{*}\left(\mathcal{W}\right)$ is a linear map which sends subgroups
$A_{i}^{0}$ to $A_{i}^{1}$ and so descends to a well-defined linear map
$CM_{*}\left(\mathcal{M}^{0}\right)$ to $CM_{*}\left(\mathcal{M}^{1}\right)$,
independent of the choices of $\psi_{y}$.

######  Lemma 98.

$CM_{*}\left(\mathcal{W}\right)$ is a chain map. Equivalently,

$\partial^{1}\circ
CM_{*}\left(\mathcal{W}\right)=CM_{*}\left(\mathcal{W}\right)\circ\partial^{0}.$

######  Proof 5.6.

Let $x$ be an object of $\mathcal{M}^{0}$, of index $i=|x|$. Choose $\phi$ in
$F(x)$. Then

$\partial^{1}\circ
CM_{*}\left(\mathcal{W}\right)(x)=\sum\limits_{\begin{subarray}{c}y\in\mathcal{M}^{1}\\\
|y|=i\end{subarray}}\sum\limits_{\begin{subarray}{c}z\in\mathcal{M}^{1}\\\
|z|=i-1\end{subarray}}\Gamma\left(\mathcal{W}_{xy}\right)\Gamma\left(\mathcal{M}^{1}_{yz}(y,z)\right)\theta_{z}$

where we choose $\psi_{y}$ in $F(y)$ and $\theta_{z}$ in $F(z)$ for each
$y,z$, similarly to the proof of Lemma 95. Similarly,

$CM_{*}\left(\mathcal{W}\right)\circ\partial^{0}(x)=\sum\limits_{\begin{subarray}{c}y\in\mathcal{M}^{0}\\\
|y|=i-1\end{subarray}}\sum\limits_{\begin{subarray}{c}z\in\mathcal{M}^{1}\\\
|z|=i-1\end{subarray}}\Gamma\left(\mathcal{M}^{0}_{xy}\right)\Gamma\left(\mathcal{W}_{yz}\right)\theta_{z}.$

The coefficient of each $\theta_{z}$ in the difference between these
expressions is

$\sum\limits_{\begin{subarray}{c}y\in\mathcal{M}^{1}\\\
|y|=i\end{subarray}}\Gamma\left(\mathcal{W}_{xy}\right)\Gamma\left(\mathcal{M}^{1}_{yz}\right)-\sum\limits_{\begin{subarray}{c}y\in\mathcal{M}^{0}\\\
|y|=i-1\end{subarray}}\Gamma\left(\mathcal{M}^{0}_{xy}\right)\Gamma\left(\mathcal{M}^{01}_{yz}\right)$

which, since $\Gamma$ is a ring map, is

$\Gamma\left(\bigsqcup\limits_{\begin{subarray}{c}y\in\mathcal{M}^{1}\\\
|y|=i\end{subarray}}\mathcal{W}_{xy}\times\mathcal{M}^{1}_{yz}\sqcup\bigsqcup\limits_{\begin{subarray}{c}y\in\mathcal{M}^{0}\\\
|y|=i-1\end{subarray}}\overline{\mathcal{M}^{0}_{xy}\times\mathcal{W}_{yz}}\right)$

which is equal to

$\Gamma\left(\partial\mathcal{W}_{xy}\right)$

and hence vanishes. Note that
$\overline{\mathcal{M}^{0}_{xy}\times\mathcal{W}_{yz}}$ appears with the
framing reversed; see Remark 3.6.

Now let $\mathcal{M}^{i}$ be framed flow categories, for $i\in\\{0,1,2\\}$,
and $\mathcal{R}:\mathcal{M}^{0}\times\mathcal{M}^{1}\to\mathcal{M}^{2}$ be a
framed bilinear map.

######  Lemma 99.

The framed bilinear map $\mathcal{R}$ induces a degree zero product

$\mu_{\mathcal{R}}:CM_{*}(\mathcal{M}^{0})\otimes
CM_{*}(\mathcal{M}^{1})\longrightarrow CM_{*}(\mathcal{M}^{2})$

which descends to homology. If $\mathcal{R}$ admits a framed associator, then
the product on homology is associative.

######  Proof 5.7.

Fix objects $x$ and $y$ of $\mathcal{M}^{0}$ and $\mathcal{M}^{1}$ of degrees
$|x|=i$ and $|y|=j$. For each $z\in\mathcal{M}^{2}$ with $|z|=i+j$ we have a
zero-dimensional manifold $\mathcal{W}_{xy;z}$ and we set

$\mu_{\mathcal{R}}(\phi_{x}\otimes\phi_{y})=\sum_{|z|=i+j}\Gamma(\mathcal{R}_{xy;z})\,\phi_{z}$

where $\phi_{\bullet}\in F(\bullet)$ for $\bullet=x,y,z$ and the choices of
$\phi_{\bullet}$ equip $\mathcal{W}_{xy;z}$ with a stable framing. As in the
construction of the differential on $CM_{i}(\mathcal{M})$ above, the map does
not depend on the choices of the $\phi_{\bullet}$. The fact that
$\mathcal{W}_{xy;z}$, with $|z|=|x|+|y|-1$, has a system of boundary faces
indexed by

$\mathcal{M}^{0}_{xx^{\prime}}\times\mathcal{R}_{x^{\prime}y;z},\,\mathcal{M}^{1}_{yy^{\prime}}\times\mathcal{R}_{xy^{\prime};z},\,\mathcal{R}_{xy;z^{\prime}}\times\mathcal{M}^{2}_{z^{\prime}z}$

for appropriate $x^{\prime},y^{\prime},z^{\prime}$ shows that
$\mu_{\mathcal{R}}$ satisfies the Leibniz rule and hence descends to a map on
homology. The proof of the final claim is analogous.

###  5.4 Truncations

######  Definition 100.

Let $\mathcal{F}$ and $\mathcal{G}$ be flow categories. A _pre- $\tau_{\leq
n}$-morphism_ $\mathcal{W}$ from $\mathcal{F}$ to $\mathcal{G}$ consists of
compact smooth manifolds with faces $\mathcal{W}_{xy}$ (of dimension
$|x|-|y|$) for all $x$ in $\mathcal{F}$ and $y$ in $\mathcal{G}$ such that
$|x|-|y|\leq n$, along with maps

$c=c^{\mathcal{W}}:\mathcal{F}_{xz}\times\mathcal{W}_{zy}\rightarrow\mathcal{W}_{xy}$

and

$c=c^{\mathcal{W}}:\mathcal{W}_{xz}\times\mathcal{G}_{zy}\rightarrow\mathcal{W}_{xy}$

which are compatible with $c^{\mathcal{F}}$ and $c^{\mathcal{G}}$, which
define a system of faces

$\mathcal{W}_{x_{0}\ldots x_{i};y_{0}\ldots y_{j}}$

for $x_{0}=x,\ldots x_{i}$ in $\mathcal{F}$ and $y_{0},\ldots,y_{j}=y$ in
$\mathcal{G}$.

A _pre- $\tau_{\leq n}$-bordism_ $\mathcal{R}$ between two pre-$\tau_{\leq
n-1}$-morphisms $\mathcal{W}$ and $\mathcal{V}$ consists of compact smooth
manifolds with faces $\mathcal{R}_{xy}$ (of dimension $|x|-|y|$ \+ 1), for all
$x$ in $\mathcal{F}$ and $y$ in $\mathcal{G}$ such that $|x|-|y|+1\leq n$,
along with maps

$c=c^{\mathcal{R}}:\mathcal{W}_{xy},\mathcal{V}_{xy}\rightarrow\mathcal{R}_{xy}$

and

$c=c^{\mathcal{R}}:\mathcal{F}_{xz}\times\mathcal{R}_{zy}\rightarrow\mathcal{R}_{xy}$

and

$c=c^{\mathcal{R}}:\mathcal{R}_{xz}\times\mathcal{G}_{zy}\rightarrow\mathcal{R}_{xy}$

compatible with $c^{\mathcal{F}}$, $c^{\mathcal{G}}$, $c^{\mathcal{W}}$,
$c^{\mathcal{V}}$, such that the images of the faces in the domains defines a
system of faces for $\mathcal{R}_{xy}$ (after deleting repeats). More
precisely, there is a system of faces for $\mathcal{R}_{xy}$ given by the
following:

$\mathcal{W}_{x_{0}\ldots x_{i};y_{0}\ldots y_{j}}$

and

$\mathcal{V}_{x_{0}\ldots x_{i};y_{0}\ldots y_{j}}$

and

$\mathcal{R}_{x_{0}\ldots x_{i};y_{0}\ldots y_{j}}$

for $x_{0}=x,\ldots,x_{i}$ in $\mathcal{F}$ and $y_{0},\ldots,y_{j}$ in
$\mathcal{G}$.

A more intuitive description is the following. A pre-$\tau_{\leq n}$-morphism
$\mathcal{W}$ is like a morphism of flow categories $\mathcal{W}$ except we
only require the manifolds with faces $\mathcal{W}_{xy}$ to exist when their
dimension is $\leq n$, but when they do exist they have the same concatenation
maps, defining a system of faces in the same way as before. A similar
description applies to pre-bordisms. Similarly to with morphisms and bordisms,
we equip any pre-$\tau_{\leq n}$-morphism or pre-$\tau_{\leq n}$-bordism we
encounter with a system of collars, compatible with those on $\mathcal{F}$ and
$\mathcal{G}$ (and $\mathcal{W}$ and $\mathcal{V}$ in the case of a
pre-$\tau_{\leq n}$-bordism). Any morphism of flow categories
$\mathcal{W}:\mathcal{F}\rightarrow\mathcal{G}$ determines a pre-$\tau_{\leq
n}$-morphism $\tau_{\leq n}\mathcal{W}:\mathcal{F}\rightarrow\mathcal{G}$ by
forgetting the manifolds $\mathcal{W}_{xy}$ whenever $|x|-|y|>n$.

Similarly, whenever $m\geq n$, any pre-$\tau_{\leq m}$-morphism $\mathcal{W}$
determines a pre-$\tau_{\leq n}$-morphism $\tau_{\leq n}\mathcal{W}$ in the
same way.

Clearly composing these truncation operations is associative, i.e. $\tau_{\leq
n}\tau_{\leq m}\mathcal{W}=\tau_{\leq n}\mathcal{W}$, for $n\leq m$.

Note that these operations are not necessarily surjective: not every
pre-$\tau_{\leq n}$-morphism $\mathcal{F}\rightarrow\mathcal{G}$ is
$\tau_{\leq n}\mathcal{W}$ for $\mathcal{W}$ some pre-$\tau_{\leq
n+1}$-morphism.

There are similar truncation operations for pre-$\tau_{\leq n}$-bordisms.

######  Definition 101.

Let $\mathcal{F}$ and $\mathcal{G}$ be flow categories. A _$\tau_{\leq n}$
-morphism_ $\mathcal{W}$ from $\mathcal{F}$ to $\mathcal{G}$ is a
pre-$\tau_{\leq n}$-morphism $\mathcal{W}:\mathcal{F}\rightarrow\mathcal{G}$
such that $\mathcal{W}=\tau_{\leq n}\mathcal{W}^{\prime}$ for some
pre-$\tau_{\leq n+1}$-morphism $\mathcal{W}^{\prime}$. We call
$\mathcal{W}^{\prime}$ an _extension_ of $\mathcal{F}$.

A _$\tau_{\leq n}$ -bordism_ between two $\tau_{\leq n}$-morphisms
$\mathcal{W}$ and $\mathcal{V}$ is a pre-$\tau_{\leq n}$-bordism $\mathcal{R}$
between $\tau_{\leq n-1}\mathcal{W}$ and $\tau_{\leq n-1}\mathcal{V}$ such
that $\mathcal{R}=\tau_{\leq n}\mathcal{R}^{\prime}$ for
$\mathcal{R}^{\prime}$ some pre-$\tau_{\leq n+1}$-bordism between
$\mathcal{W}$ and $\mathcal{V}$. We call $\mathcal{R}^{\prime}$ an _extension_
of $\mathcal{R}$.

We emphasise that in these definitions, the extensions $\mathcal{W}^{\prime}$
and $\mathcal{R}^{\prime}$ are assumed to exist, but are not kept track of as
part of the data.

Similarly to Remark 5.4, we think of these as morphisms (or bordisms of
morphisms) of flow categories, where we only specify the manifolds with faces
$\mathcal{W}_{xy}$ when their dimension is $\leq n$ (again with the
appropriate system of faces), and such that there exist manifolds with faces
$\mathcal{W}_{xy}$ when the dimension is $n+1$ with the appropriate system of
faces, but without keeping track of these $(n+1)$-dimensional manifolds with
faces as part of the data. We may think of a morphism of flow categories as a
$\tau_{\leq\infty}$-morphism, or equivalently as a
pre-$\tau_{\leq\infty}$-morphism. Similarly to Example 5.4, any morphism of
flow categories $\mathcal{W}:\mathcal{F}\rightarrow\mathcal{G}$ determines a
$\tau_{\leq n}$-morphism $\tau_{\leq n}\mathcal{W}$ by forgetting all
manifolds of dimension $\geq n$. A pre-$\tau_{\leq 0}$-morphism
$\mathcal{W}:\mathcal{F}\rightarrow\mathcal{G}$ determines a linear map
$CM_{*}(\mathcal{W}):CM_{*}(\mathcal{F};\mathbb{Z}/2)\rightarrow
CM_{*}(\mathcal{G};\mathbb{Z}/2)$, defined by

$CM_{*}(\mathcal{W})(x):=\sum\limits_{y\in\mathcal{G}\,,\,|y|=|x|}|\mathcal{W}_{xy}|\cdot
y$

for $x$ in $\mathcal{F}$. Moreover, $\mathcal{W}$ is a $\tau_{\leq
0}$-morphism if and only if $CM_{*}(\mathcal{W})$ is a chain map.

If $\mathcal{W}=\tau_{\leq 0}\mathcal{V}$ for some morphism
$\mathcal{V}:\mathcal{F}\to\mathcal{G}$, then this is the same as the
construction in Section 5.3. Given a morphism
$\mathcal{W}:\mathcal{F}\rightarrow\mathcal{G}$, there is a natural
isomorphism

$CM_{*}(\Cone(\mathcal{W}))\simeq\Cone(CM_{*}(\mathcal{W}):CM_{*}(\mathcal{F})\to
CM_{*}(\mathcal{G}))$

Example 5.4 is one of the main reasons we will define a category built out of
$\tau_{\leq n}$-morphisms, rather that pre-$\tau_{\leq n}$-morphisms.
Similarly to Definition 51, we give the following definition.

######  Definition 102.

Let $\mathcal{F}$ and $\mathcal{G}$ be flow categories. Define
$[\mathcal{F},\mathcal{G}]_{\tau_{\leq n}}$ to be the set of $\tau_{\leq
n}$-morphisms $\mathcal{F}\rightarrow\mathcal{G}$, modulo the equivalence
relation generated by

  1. 1.

$\mathcal{W}\sim\mathcal{V}$ if $\mathcal{W}$ and $\mathcal{V}$ are
$\tau_{\leq n}$-bordant.

  2. 2.

$\mathcal{W}\sim\mathcal{V}$ if $\mathcal{W}$ and $\mathcal{V}$ are
diffeomorphic.

This is an abelian group under disjoint union, with unit given by the empty
morphism.

Truncation as in Example 5.4 determines morphisms of abelian groups

$[\mathcal{F},\mathcal{G}]\xrightarrow{\tau_{\leq
m}}[\mathcal{F},\mathcal{G}]_{\tau_{\leq m}}\xrightarrow{\tau_{\leq
n}}[\mathcal{F},\mathcal{G}]_{\tau_{\leq n}}$

for $m\geq n$.

######  Lemma 103.

There is a natural isomorphism $\Xi:[\mathcal{F},\mathcal{G}]_{\tau_{\leq
0}}\rightarrow\Hom(CM_{*}(\mathcal{F}),CM_{*}(\mathcal{G}))$ (where the target
is the set of chain maps $CM_{*}(\mathcal{F})\rightarrow CM_{*}(\mathcal{G})$
up to homotopy).

######  Proof 5.8.

We first show that $[\mathcal{F},\mathcal{G}]_{\tau_{\leq
0}}\to\Hom(CM_{*}(\mathcal{F}),CM_{*}(\mathcal{G}))$ is injective. Suppose
$\mathcal{W}$ and $\mathcal{V}$ are flow $\tau_{\leq 0}$-morphisms which are
bordant, via a bordism $\mathcal{R}$. Recall that the bordism relation means
that we have compact smooth manifolds with faces $\mathcal{R}_{xy}$ of
dimensions $|x|-|y|+1$ for $x\in\mathcal{F}$ and $y\in\mathcal{G}$ (with
appropriate faces). In particular we can define a linear map
$CM_{*}(\mathcal{F})\to CM_{*+1}(\mathcal{G})$ via

$x\mapsto\sum_{|y|=|x|+1}\mathcal{R}_{xy}\,y$

The bordism relation implies this defines a chain homotopy between the chain
maps $x\mapsto\sum_{|y|=|x|}\mathcal{W}_{xy}\,y$ and
$x\mapsto\sum_{|y|=|x|}\mathcal{V}_{xy}\,y$. A corresponding simpler argument
when $\mathcal{W}$ and $\mathcal{V}$ are related by diffeomorphism then shows
injectivity of $\Xi$. Analogously, any linear map $CM_{*}(\mathcal{F})\to
CM_{*}(\mathcal{G})$, say $x\mapsto\sum_{|y|=|x|}n_{xy}y$ for integers
$n_{xy}$, can be viewed as a collection of framed $0$-manifolds $n_{xy}$ which
define a pre-$\tau_{\leq 0}$-morphism. The condition that the linear map is a
chain map exactly says this lifts to a $\tau_{\leq 0}$-morphism, which gives
surjectivity of $\Xi$.

Given pre-$\tau_{\leq n}$-morphisms
$\mathcal{W}:\mathcal{F}\rightarrow\mathcal{G}$ and
$\mathcal{V}:\mathcal{G}\rightarrow\mathcal{H}$, we can form their
composition, $\mathcal{V}\circ\mathcal{W}$, identically to the case of
morphisms of flow categories.

We first define
$\tilde{\mathcal{Q}}_{xy}=\tilde{\mathcal{Q}}_{xy}(\mathcal{W},\mathcal{V})$
as in Equation 5 whenever $|x|-|y|\leq n$. The $\tilde{\mathcal{Q}}_{xy}$ are
manifolds with faces by the same argument as before, with a system of faces
given by the same description. Choosing compatible framed functions
$(f_{xy},s_{xy})$ as before and defining $\mathcal{U}_{xy}=f_{xy}^{-1}\\{p\\}$
whenever $|x|-|y|\leq n$. defines a pre-$\tau_{\leq n}$ morphism
$\mathcal{U}:\mathcal{F}\to\mathcal{H}$, which we define
$\mathcal{V}\circ\mathcal{G}$ to be. Composition of this form is compatible
with further truncation: if $n>k$ and we apply the above construction to
$\tau_{\leq k}\mathcal{W}$ and $\tau_{\leq k}\mathcal{V}$ and take the
restrictions of the same framed functions functions, the output will be
exactly $\tau_{\leq k}\mathcal{U}$.

######  Definition 104.

If $\mathcal{W}$ and $\mathcal{V}$ are $\tau_{\leq n}$-morphisms, we choose
extensions $\mathcal{W}^{\prime}$ and $\mathcal{V}^{\prime}$ and compose these
as above to get a pre-$\tau_{\leq n+1}$-morphism $\mathcal{U}^{\prime}$. Then
we define the _composition_ of $\mathcal{W}$ and $\mathcal{V}$ to be the
$\tau_{\leq n}$-morphism $\tau_{\leq n}\mathcal{U}^{\prime}$.

As a pre-$\tau_{\leq n}$-morphism, $\tau_{\leq n}\mathcal{U}^{\prime}$ is the
composition of $\mathcal{W}$ and $\mathcal{V}$. However if we composed
$\mathcal{W}$ and $\mathcal{V}$ as pre-$\tau_{\leq n}$-morphisms and used
different choices of framed functions, we may produce a pre-$\tau_{\leq
n}$-morphism that does not admit an extension, which is why we instead take
the truncation of the composition of the extensions $\mathcal{W}^{\prime}$ and
$\mathcal{V}^{\prime}$.

######  Lemma 105.

Composition of $\tau_{\leq n}$-morphisms is compatible with the equivalence
relation defined in Definition 101, and induces a well-defined bilinear map

$\circ:[\mathcal{G},\mathcal{H}]_{\tau_{\leq
n}}\otimes[\mathcal{F},\mathcal{G}]_{\tau_{\leq
n}}\rightarrow[\mathcal{F},\mathcal{H}]_{\tau_{\leq n}}.$

######  Proof 5.9.

Identical to that of Lemma 56 but only considering manifolds of appropriate
dimensions.

######  Lemma 106.

Composition of $\tau_{\leq n}$-morphisms is associative.

######  Proof 5.10.

Identical to that of Lemma 57 but only considering manifolds of appropriate
dimensions.

Similarly we may define $\tau_{\leq n}$ bilinear maps, bordisms of bilinear
maps and associators, satisfying the same properties as we saw before; in
particular they can be associatively composed.

We have thus shown that there is a category, $\Flow_{\tau_{\leq n}}$, with
objects flow categories, and morphisms $[\cdot,\cdot]_{\tau_{\leq n}}$.
Compatibility of truncations with composition defines functors

(8) $\Flow\rightarrow\Flow_{\tau_{\leq m}}\rightarrow\Flow_{\tau_{\leq n}}$

whenever $m\geq n$ which are suitably associative.

Generalising Lemma 103, we conjecture the following.

######  Conjecture 107.

The category $\Flow_{\tau_{\leq n}}$ is equivalent to the category with
objects perfect $MO$-modules, and morphisms $M\rightarrow N$ given by homotopy
classes of $\tau_{\leq n}MO$-linear maps $M\otimes_{MO}\tau_{\leq
n}MO\rightarrow N\otimes_{MO}\tau_{\leq n}MO$, where $\tau_{\leq n}$ denotes
the truncation of a spectrum with respect to the standard $t$-structure.

Furthermore this equivalence should be compatible (in an appropriate sense)
with the equivalence of Conjecture 60.

Suppose $\mathcal{F}$ and $\mathcal{G}$ are framed flow categories, with
framing stable vector spaces $V_{x}$ and $V_{y}$ for $x\in\mathcal{F}$ and
$y\in\mathcal{G}$. A pre- $\tau_{\leq n}$-framed morphism is a pre-$\tau_{\leq
n}$ morphism for which the stable isomorphisms $st_{xy}$ from Definition 66
are defined and satisfy the same compatibility whenever they can i.e. whenever
$|x|-|y|\leq n$. This is a $\tau_{\leq n}$-framed morphism if it arises by
truncation from a pre-$\tau_{\leq(n+1)}$-framed morphism as usual. All of the
definitions and properties of $\Flow^{fr}$ are identical in this truncated
setting, except that the various diagrams now only make sense when the degrees
of the objects are constrained. In particular, we have the analogue

(9) ${\Flow}^{fr}\rightarrow{\Flow}^{fr}_{\tau_{\leq
m}}\rightarrow{\Flow}^{fr}_{\tau_{\leq n}}$

for functors of truncations of framed flow categories, again associative.
Entirely analogously to Conjecture 107, we have:

######  Conjecture 108.

The category $\Flow^{fr}_{\tau_{\leq n}}$ is equivalent to the category with
objects perfect $\mathbb{S}$-modules, and morphisms $M\rightarrow N$ given by
homotopy classes of $\tau_{\leq n}\mathbb{S}$-linear maps
$M\otimes_{\mathbb{S}}\tau_{\leq n}\mathbb{S}\rightarrow
N\otimes_{\mathbb{S}}\tau_{\leq n}\mathbb{S}$.

###  5.5 Obstruction theory

In this section, we perform some more involved homological algebra in the
category of flow categories. In each case, we provide the analogous result
from (ordinary) homological algebra as motivation; we expect that these should
be equivalent under Conjectures 107 and 108.

######  Theorem 109.

Let $\mathcal{F}$ and $\mathcal{G}$ be flow categories such that either
$\mathcal{F}=*[i]$ or $\mathcal{G}=*[i]$ for some $i$. Let
$\mathcal{W}:\mathcal{F}\rightarrow\mathcal{G}$ be a $\tau_{\leq n}$-morphism.
Then there is a class

$[\mathcal{O}]=[\mathcal{O}(\mathcal{W})]\in\Hom\left(CM_{*+n+2}(\mathcal{F};\mathbb{Z}/2),CM_{*}(\mathcal{G};\Omega_{n+1})\right)$

such that $[\mathcal{O}]$ vanishes if and only if $\mathcal{W}=\tau_{\leq
n}\mathcal{V}$, for $\mathcal{V}:\mathcal{F}\rightarrow\mathcal{G}$ some
$\tau_{\leq n+1}$-morphism.

Similarly if $\mathcal{F},\mathcal{G}$ and $\mathcal{W}$ are all framed, there
is a class

$[\mathcal{O}^{fr}]=[\mathcal{O}^{fr}(\mathcal{W})]\in\Hom\left(CM_{*+n+2}(\mathcal{F};\mathbb{Z}),CM_{*}(\mathcal{G};\Omega_{n+1}^{fr})\right)$

such that $[\mathcal{O}^{fr}]$ vanishes if and only if $\mathcal{W}=\tau_{\leq
n}\mathcal{V}$, for $\mathcal{V}:\mathcal{F}\rightarrow\mathcal{G}$ some
framed $\tau_{\leq n+1}$-morphism.

We conjecture that the condition that either $\mathcal{F}$ or $\mathcal{G}$ is
$*[i]$ is unnecessary.

We first prove the unframed case, then incorporate framings into the same
proof.

######  Proof 5.11 (Proof without framings).

We assume that $\mathcal{F}=*[i]$ here, the case where $\mathcal{G}=*[i]$ is
similar.

Choose $\mathcal{W}^{\prime}$ an extension of $\mathcal{W}$. We first
construct a linear map

$\mathcal{O}=\mathcal{O}(\mathcal{W}^{\prime}):CM_{*+n+2}(\mathcal{F};\mathbb{Z}/2)\rightarrow
CM_{*}(\mathcal{G},\Omega_{n+1})$

We define

$\tilde{\mathcal{Y}}_{*y}=\tilde{\mathcal{Y}}_{*y}(\mathcal{W}^{\prime}):=\left(\bigsqcup\limits_{w\in\mathcal{G}}\mathcal{W}^{\prime}_{*w}\times\mathcal{G}_{wy}\times[0,\varepsilon)\right)/\sim$

for all $y$ in $\mathcal{G}$ such that $i-|y|=n+2$, where $\sim$ is the
equivalence relation defined so that as a topological space,
$\tilde{\mathcal{Y}}_{*y}$ is the coequaliser of the diagram

$\bigsqcup\limits_{w,v\in\mathcal{G}}\mathcal{W}^{\prime}_{*w}\times\mathcal{G}_{wv}\times\mathcal{G}_{vy}\times[0,\varepsilon)^{2}_{u,v}\rightrightarrows\bigsqcup\limits_{w\in\mathcal{G}}\mathcal{W}^{\prime}_{*w}\times\mathcal{G}_{wy}\times[0,\varepsilon)_{s}$

where the two maps send $(a,b,c,u,v)$ to $(a,\mathcal{C}(b,c,u),v)$ and
$(\mathcal{C}(a,b,v),c,u)$ respectively.

$\tilde{\mathcal{Y}}_{*y}$ is a manifold with faces, with a system of faces
given by

$\mathcal{W}^{\prime}_{*;y_{0}\ldots y_{j}}$

for $y_{0},\ldots,y_{j}=y$ in $\mathcal{G}$, and in particular a system of
boundary faces given by
$\mathcal{W}^{\prime}_{*w}\times\mathcal{G}_{wy}\times\\{0\\}$ for
$w\in\mathcal{G}$. Note that all these faces are compact.

We choose framed functions $(f_{*y},s_{*y})$ on each
$\tilde{\mathcal{Y}}_{*y}$, noting that on all proper faces each $f_{*y}$ is
negative.

Then define

$\mathcal{Z}_{*y}=\mathcal{Z}_{*y}(\mathcal{W}^{\prime}):=f_{*y}^{-1}\\{1\\}$

This is a closed smooth manifold of dimension $n+1$, and its bordism class
$[\mathcal{Z}_{*y}]$ in the bordism group $\Omega_{n+1}$ is independent of the
choice of $f_{*y}$, though it does depend on the choice of extension
$\mathcal{W}^{\prime}$ (we will use this dependence later).

We define the linear map $\mathcal{O}$ by

$\mathcal{O}(*):=\sum\limits_{i-|y|=n+2}[\mathcal{Z}_{*y}]\cdot y$

We next show this is a chain map. This is equivalent to showing that the
closed manifold

$\bigsqcup\limits_{s\in\mathcal{G},|s|-|y|=1}\mathcal{Z}_{*s}\times\mathcal{G}_{sy}$

represents the zero class in the bordism group $\Omega_{n+1}$ (i.e. it bounds
a compact smooth manifold with boundary), for all $y$ in $\mathcal{G}$ such
that $i-|y|=n+3$. We note here that since a closed smooth manifold which
bounds a topological manifold also bounds a smooth manifold, it would be
sufficient (and easier) to show that this class is zero in the topological
bordism group, but this would not adapt so well to the framed setting.

We define manifolds with faces

$\tilde{\mathcal{X}}_{*y}:=\left(\bigsqcup\limits_{w\in\mathcal{G},|w|-|y|>1}\mathcal{W}^{\prime}_{xw}\times\mathcal{G}_{wy}\times[0,\varepsilon)\right)/\sim$

for all $y$ in $\mathcal{G}$ such that $i-|y|=n+3$, where $\sim$ is defined so
that as a topological space, $\tilde{\mathcal{X}}_{*y}$ is the coequaliser of
the diagram

$\bigsqcup_{v,w\in\mathcal{G},|w|-|y|>1}\mathcal{W}^{\prime}_{*v}\times\mathcal{G}_{vw}\times\mathcal{G}_{wy}\times[0,\varepsilon)^{2}_{u,v}\rightrightarrows\bigsqcup_{w\in\mathcal{G},|w|-|y|>1}\mathcal{W}^{\prime}_{xw}\times\mathcal{G}_{wy}\times[0,\varepsilon)$

where the two maps send $(a,b,c,u,v)$ to $(a,\mathcal{C}(b,c,u),v)$ and
$(\mathcal{C}(a,b,v),c,u)$ respectively. Note that there are natural
embeddings

$\tilde{\mathcal{Y}}_{*s}\times\mathcal{G}_{sy}\hookrightarrow\tilde{\mathcal{X}}_{*y}$

for all $s$ in $\mathcal{G}$ such that $|s|-|y|=1$, and that these have
disjoint images.

$\tilde{\mathcal{X}}_{*y}$ has a system of faces given by

$\mathcal{W}^{\prime}_{*;y_{0}\ldots y_{j}}$

for $y_{0},\ldots,y_{j}=y$ in $\mathcal{G}$ such that $j>1$ or
$|y_{j-1}|-|y_{j}|>1$, along with the codimension 1 faces

$\tilde{\mathcal{Y}}_{*s}\times\mathcal{G}_{sy}$

for $s$ in $\mathcal{G}$ such that $|s|-|y|=1$. Note that the only non-compact
faces are the ones of the form
$\tilde{\mathcal{Y}}_{*s}\times\mathcal{G}_{sy}$.

Now choose framed functions $(g_{*y},t_{*y})$ on each
$\tilde{\mathcal{X}}_{*y}$, compatible with the $(f_{*s},s_{*s})$ on each
$\tilde{\mathcal{Y}}_{*s}\times\mathcal{G}_{sy}$. Let
$\mathcal{L}_{*y}=g_{*y}^{-1}\\{1\\}$. Then $\mathcal{L}_{*y}$ is a compact
smooth manifold with boundary

$\bigsqcup\limits_{s\in\mathcal{G},|s|-|y|=1}\mathcal{Z}_{*s}\times\mathcal{G}_{sy}$

which implies that $\mathcal{O}$ is a chain map.

We now assume that $\mathcal{O}$ is a null-homotopic chain map, i.e. its class
in

$\Hom\left(CM_{*+n+2}(\mathcal{F};\mathbb{Z}/2),CM_{*}(\mathcal{G},\Omega_{n+1})\right)$

is 0. We show that there is some other choice of extension
$\mathcal{W}^{\prime\prime}$ such that
$\mathcal{O}(\mathcal{W}^{\prime\prime})=0$.

This assumption says that there is some linear map

$\mathcal{P}:CM_{*+n+1}(\mathcal{F};\mathbb{Z}/2)\rightarrow
CM_{*}(\mathcal{G},\Omega_{n+1})$

such that $\mathcal{O}(*)=d\mathcal{P}(*)$. More explicitly, $\mathcal{P}$
sends $*$ to

$\sum\limits_{y\in\mathcal{G},i-|y|=n+1}[\mathcal{P}_{*y}]\cdot y$

for some closed manifolds $\mathcal{P}_{*y}$, where $[\mathcal{P}_{*y}]$
represents the bordism classes in the bordism group $\Omega_{n+1}$ of each
$\mathcal{P}_{*y}$, and such that

$\bigsqcup\limits_{s\in\mathcal{G},|s|-|y|=1}\mathcal{P}_{*s}\times\mathcal{G}_{sy}$

is cobordant to $\mathcal{Z}_{*y}$ for all $y$ in $\mathcal{G}$ such that
$i-|y|=n+2$.

We define $\mathcal{W}^{\prime\prime}_{*y}:=\mathcal{W}^{\prime}_{*y}$ for all
$y$ in $\mathcal{G}$ such that $i-|y|\leq n$, and
$\mathcal{W}^{\prime\prime}_{*y}:=\mathcal{W}^{\prime}_{*y}\sqcup\mathcal{P}_{*y}$
for all $y$ in $\mathcal{G}$ such that $i-|y|=n+1$. Then we see that

$\tilde{\mathcal{Y}}_{*y}(\mathcal{W}^{\prime\prime})=\tilde{\mathcal{Y}}_{*y}(\mathcal{W}^{\prime})\bigsqcup\limits_{s\in\mathcal{G},|s|-|y|=1}\mathcal{P}_{*s}\times\mathcal{G}_{sy}\times[0,\varepsilon)$

so for appropriate choices of framed functions $(f_{*x},s_{*x})$, we see that

$\mathcal{Z}_{*y}(\mathcal{W}^{\prime\prime})=\mathcal{Z}_{*y}(\mathcal{W}^{\prime})\bigsqcup\limits_{s\in\mathcal{G},|s|-|y|=1}\mathcal{P}_{*s}\times\mathcal{G}_{sy}$

Each of these manifolds is null-cobordant, so
$\mathcal{O}(\mathcal{W}^{\prime\prime})=0$.

Replacing $\mathcal{W}^{\prime}$ with $\mathcal{W}^{\prime\prime}$, we may now
assume that $\mathcal{O}=0$. We build a pre-$\tau_{\leq n+2}$-morphism
$\mathcal{U}$ such that $\tau_{\leq n+1}\mathcal{U}=\mathcal{W}^{\prime}$,
which implies that $\mathcal{W}^{\prime}$ is the desired $\tau_{\leq
n+1}$-morphism.

We define $\mathcal{U}_{*y}=\mathcal{W}^{\prime}_{*y}$ for all $y$ in
$\mathcal{G}$ such that $i-|y|\leq n+1$. If we were to set $\mathcal{U}_{*y}$
to be $\tilde{\mathcal{Y}}_{*y}$ for $y$ in $\mathcal{G}$ such that
$i-|y|=n+2$, it has the right system of faces for $\mathcal{U}$ to be a
pre-$\tau_{\leq n+2}$-morphism, but is non-compact. Instead consider the map
$f_{*y}:\tilde{\mathcal{Y}}_{*y}\rightarrow\mathbb{R}$.
$f_{*y}^{-1}(-\infty,1]$ is compact, and by assumption
$\mathcal{Z}_{*y}=f_{*y}^{-1}\\{1\\}$ is a closed smooth manifold which is
nullcobordant, so we can glue such a nullcobordism onto
$f_{*y}^{-1}(-\infty,1]$, and set $\mathcal{U}_{*y}$ to be this new manifold.
Then by construction $\mathcal{U}$ is a pre-$\tau_{\leq n+2}$-morphism such
that $\tau_{n+1}\mathcal{U}=\mathcal{W}^{\prime}$.

We now assume that $\mathcal{W}=\tau_{\leq n}\mathcal{V}$ for
$\mathcal{V}:\mathcal{F}\rightarrow\mathcal{G}$ some $\tau_{\leq
n+1}$-morphism. Choose an extension $\mathcal{V}^{\prime}$ of $\mathcal{V}$.
Then when $*-|y|\leq n+2$, $\mathcal{V}^{\prime}_{*y}$ is a compact manifold
with faces with a system of faces given by

$\mathcal{V}^{\prime}_{*;y_{0}\ldots y_{j}}$

for $y_{0},\ldots,y_{j}=y$ in $\mathcal{G}$. When $*-|y|=n+2$, the collar
neighbourhood theorem for manifolds with faces implies that there is a
neigbourhood of all the faces in $\mathcal{V}^{\prime}_{*y}$ diffeomorphic to
$\tilde{\mathcal{Y}}_{*y}$. Choosing $f_{*y}$ as in the definition of
$\mathcal{Z}_{*y}$,

$\mathcal{V}^{\prime}_{*y}\setminus f_{*y}^{-1}(-\infty,1)$

is a compact manifold with boundary $\mathcal{Z}_{*y}$. Therefore the linear
map $\mathcal{O}$ is 0.

######  Proof 5.12 (Incorporating framings).

We incorporate framings into the unframed proof, assuming
$\mathcal{F},\mathcal{G},\mathcal{W},\mathcal{W}^{\prime}$ are all framed.

We first note that all boundary faces of each $\tilde{\mathcal{Y}}_{*y}$ are
broken. We frame each of these manifolds compatibly.

For each $y\in\mathcal{G}$, we define

$st_{*y}:V_{*[i]}\oplus I^{\tilde{\mathcal{R}}}_{*y}\to\mathbb{R}\oplus V_{y}$

to be given by, on each
$\mathcal{W}^{\prime}_{*w}\times\mathcal{G}_{wy}\times[0,\varepsilon)$, the
composition

$V_{*[i]}\oplus
I^{\tilde{\mathcal{R}}}_{*y}\xrightarrow{\psi^{-1}}V_{*[i]}\oplus
I^{\mathcal{W}^{\prime}}_{*w}\oplus
I^{\mathcal{G}}_{wy}\xrightarrow{st^{\mathcal{G}}\circ
st^{\mathcal{W}^{\prime}}}\mathbb{R}\oplus V_{y}$

These glue on overlaps just as in Lemma 70, and similarly are compatible on
boundary faces in the sense that they (combined with the other framings) form
systems of framings on each $\tilde{\mathcal{Y}}_{*y}$.

As in Lemma 70, we use the sections $s_{*y}$ to then obtain stable framings

$st^{\mathcal{Z}}:V_{*[i]}\oplus I^{\mathcal{Z}}_{*y}\to V_{y}$

on each closed manifold $\mathcal{Z}_{*y}$. We choose stable isomorphisms
$V_{q}\simeq\mathbb{R}^{|q|}$ for each $q\in\mathcal{G}$ or $q=*[i]$, then
this framing on each $\mathcal{Z}_{*y}$ determines a class
$[\mathcal{Z}_{*y}]$ in $\Omega^{fr}_{*}$.

We then define $\mathcal{O}^{fr}$ in the same way as the unframed case, but
now using the fact that the manifolds in question are framed, so
$\mathcal{O}^{fr}$ is now a linear map

$\mathcal{O}^{fr}=\mathcal{O}^{fr}(\mathcal{W}^{\prime}):CM_{*+n+2}(\mathcal{F};\mathbb{Z})\rightarrow
CM_{*}(\mathcal{G},\Omega^{fr}_{n+1})$

Note all boundary faces of each $\tilde{\mathcal{X}}_{*y}$ are broken. We
frame them in exactly the same way as the $\tilde{\mathcal{Y}}_{*y}$, to give
stable framings

$st^{\tilde{\mathcal{Y}}}_{*y}:V_{*[i]}\oplus
I^{\tilde{\mathcal{X}}}_{*y}\to\mathbb{R}\oplus V_{y}$

which are compatible with the framings on the boundary faces in the sense that
they form systems of faces over each $\tilde{\mathcal{Y}}_{*s}$; these
framings induce framings on each $\mathcal{L}_{*y}$, which implies that
$\mathcal{O}^{fr}$ is a chain map.

The assumption says that there is some linear map

$\mathcal{P}^{fr}:CM_{*+n+1}(\mathcal{F};\mathbb{Z})\rightarrow
CM_{*}(\mathcal{G},\Omega^{fr}_{n+1})$

such that $\mathcal{O}^{fr}(*)=d\mathcal{P}^{fr}(*)$. By replacing
$\mathcal{W}^{\prime}$ with a different extension $\mathcal{W}^{\prime\prime}$
of $\mathcal{W}$ by attaching disjoint copies of $\overline{\mathcal{P}}_{*y}$
to each $\mathcal{W}^{\prime}_{*y}$ when $i-|y|=n+1$ as in the unframed case,
we may assume that $\mathcal{O}^{fr}=0$.

We take the same (unframed) pre-$\tau_{\leq n+2}$ morphism $\mathcal{U}$ as in
the unframed case, and by using the framings on the $\tilde{\mathcal{Y}}_{*y}$
and the framing of the nullbordism of each $\tilde{\mathcal{Z}}_{*y}$, and
gluing these framings together on the overlap, we obtain framings on
$\mathcal{U}$.

The claim that if $\mathcal{W}=\tau_{\leq n}\mathcal{V}$ for some $\tau_{\leq
n+1}$ morphism $\mathcal{V}:\mathcal{F}\to\mathcal{G}$ then the associated
linear map $\mathcal{O}^{fr}$ is trivial is essentially the same as in the
unframed case.

Theorem 109 should be considered an analogue in $\Flow$ (respectively
$\Flow^{fr}$) of (a special case of) the following fact in homological
algebra, applied to the ring spectra $MO$ (respectively $\mathbb{S}$).

######  Proposition 110.

Let $R$ be a $(-1)$-connected dga or ring spectrum, and let $M$ and $N$ be
objects in $R\modules$, the category of dg modules or module spectra over $R$.
All tensor products we take are assumed to be derived. Let

$[\phi]\in\Hom_{\tau_{\leq n}R\modules}\left(M\otimes_{R}\tau_{\leq
n}R,N\otimes_{R}\tau_{\leq n}R\right)$

be a morphism in the derived category of $\tau_{\leq n}R$ (here $\tau$ refers
to truncation with respect to the standard $t$-structure), with some
representative $\phi$ in $[\phi]$.

Then there is a class

$[\mathcal{O}]\in\Hom_{HR_{0}\modules}\left(\Sigma^{-n-2}M\otimes_{R}HR_{0},N\otimes_{R}HR_{n+1}\right)$

(here $HR_{i}$ is the appropriate Eilenberg-Maclane chain complex or spectrum)
which vanishes if and only if $[\phi]$ is in the image of the natural map

$\tau_{\leq n}:\Hom_{\tau_{\leq n+1}R\modules}\left(M\otimes_{R}\tau_{\leq
n+1}R,N\otimes_{R}\tau_{\leq n+1}R\right)\rightarrow\Hom_{\tau_{\leq
n}R\modules}\left(M\otimes_{R}\tau_{\leq n}R,N\otimes_{R}\tau_{\leq
n}R\right)$

Formulating this at the level of $HR_{0}$-modules is analogous to working with
chain complexes in Theorem 109.

######  Proof 5.13 (Sketch proof).

There is a fibration sequence in $\tau_{\leq n+1}R\modules$ (using the fact
that $R$ is $(-1)$-connected here):

$N\otimes_{R}\tau_{\leq n+1}R\rightarrow N\otimes_{R}\tau_{\leq n}R\rightarrow
N\otimes_{R}\Sigma^{n+2}HR_{n+1}$

and therefore an exact sequence

$\textstyle{\Hom_{\tau_{\leq n+1}R\modules}\left(M\otimes_{R}\tau_{\leq
n+1}R,N\otimes_{R}\tau_{\leq
n+1}R\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Hom_{\tau_{\leq
n+1}R\modules}\left(M\otimes_{R}\tau_{\leq n+1}R,N\otimes_{R}\tau_{\leq
n}R\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\Hom_{\tau_{\leq
n+1}R\modules}\left(M\otimes_{R}\tau_{\leq
n+1}R,N\otimes_{R}\Sigma^{n+2}HR_{n+1}\right)}$

We identify the middle term with

$\Hom_{\tau_{\leq n}R\modules}\left(M\otimes_{R}\tau_{\leq
n}R,N\otimes_{R}\tau_{\leq n}R\right)$

the final term with

$\Hom_{HR_{0}\modules}\left(\Sigma^{-n-2}M\otimes_{R}HR_{0},N\otimes_{R}HR_{n+1}\right)$

and take $[\mathcal{O}]$ to be $f([\phi])$.

######  Theorem 111.

Let $\mathcal{W}:\mathcal{F}\to\mathcal{G}$ be a morphism of flow categories
such that the induced map
$CM_{*}(\mathcal{W};\mathbb{Z}/2):CM_{*}(\mathcal{F};\mathbb{Z}/2)\to
CM_{*}(\mathcal{G};\mathbb{Z}/2)$ is a quasiisomorphism over $\mathbb{Z}/2$.

Then the induced map
$\mathcal{W}_{*}:\left[*[u],\mathcal{F}\right]\to\left[*[u],\mathcal{G}\right]$
is surjective for all $u$.

Similarly if $\mathcal{F},\mathcal{G},\mathcal{W}$ are all framed and
$CM_{*}(\mathcal{W};\mathbb{Z}):CM_{*}(\mathcal{F};\mathbb{Z})\to
CM_{*}(\mathcal{G},\mathbb{Z})$ is a quasiisomorphism over $\mathbb{Z}$, then
the induced map
$\mathcal{W}_{*}:[*[u],\mathcal{F}]^{fr}\to[*[u],\mathcal{G}]^{fr}$ is
surjective for all $u$.

We conjecture (but do not require for our purposes) that the same statement
holds for injectivity as well as surjectivity.

######  Proof 5.14 (Proof without framings).

We show that $\mathcal{W}_{*}:\left[*[u],\mathcal{F}\right]_{\tau_{\leq
v}}\to\left[*[u],\mathcal{G}\right]_{\tau_{\leq v}}$ is surjective for each
$v$ inductively; this holds for $v=0$ by assumption. Assume this holds for
$v$; we will show it holds for $v+1$.

Let $\mathcal{B}:*[u]\to\mathcal{G}$ be a $\tau_{\leq v+1}$-morphism. By
assumption, $\tau_{\leq v}\mathcal{B}$ lifts to a $\tau_{\leq v}$-morphism
$\mathcal{A}:*[u]\to\mathcal{F}$, and there is a bordism $\mathcal{R}$ from
$\mathcal{W}\circ\mathcal{A}$ to $\tau_{\leq v}\mathcal{B}$.

Much more explicitly, we first choose an extension $\mathcal{B}^{\prime}$ of
$\mathcal{B}$, and assume that there are compact manifolds with faces
$\mathcal{A}^{\prime}_{*x}$ for each $x$ in $\mathcal{F}$ with $u-|x|\leq v+1$
of dimension $u-|x|$, each of which has a system of faces given by

$\mathcal{A}^{\prime}_{*;x_{0}\ldots x_{i}}$

for $x_{0},\ldots,x_{i}=x$ in $\mathcal{F}$ (and in particular a system of
boundary faces given by $\mathcal{A}^{\prime}_{*;x^{\prime}x}$ for
$x^{\prime}$ in $\mathcal{F}$), as well as compact manifolds with faces
$\mathcal{R}^{\prime}_{*y}$ for each $y$ in $\mathcal{G}$ with $u-|y|\leq v$
of dimension $u-|y|$, with a system of faces given by

$\mathcal{B}_{*;y_{0}\ldots y_{j}}$

for $y_{0},\ldots,y_{j}=y$ in $\mathcal{G}$,

$\mathcal{A}^{\prime}_{*;x_{0}\ldots
x_{i}}\times\mathcal{W}_{x_{i};y_{0}\ldots y_{j}}$

for $x_{0},\ldots,x_{i}$ in $\mathcal{F}$ and $y_{0},\ldots,y_{j}$ in
$\mathcal{G}$, and

$\mathcal{R}^{\prime}_{*;y_{0}\ldots y_{j}}$

for $y_{0},\ldots,y_{j}=y$ in $\mathcal{G}$. In particular,
$\mathcal{R}^{\prime}_{*y}$ has a system of boundary faces given by
$\mathcal{B}_{*y}$, $\mathcal{A}_{*x}\times\mathcal{W}_{xy}$ for $x$ in
$\mathcal{F}$, and $\mathcal{R}^{\prime}_{*;y^{\prime}y}$ for $y^{\prime}$ in
$\mathcal{G}$.

We next define an element $(a,r)$ in

$CM_{u-v-2}\left(\mathcal{F};\Omega_{v+1}\right)\oplus
CM_{u-v-1}\left(\mathcal{G};\Omega_{v+1}\right)$

For $x$ in $\mathcal{F}$ with $u-|x|=v+2$, we define
$\tilde{\mathcal{D}}_{*x}$ to be the coequaliser of the following.

$\bigsqcup\limits_{x^{\prime},x^{\prime\prime}\in\mathcal{F}}\mathcal{A}^{\prime}_{*;x^{\prime\prime}x^{\prime}x}\times[0,\varepsilon)^{2}\rightrightarrows\bigsqcup\limits_{x^{\prime}\in\mathcal{F}}\mathcal{A}^{\prime}_{*;x^{\prime}x}\times[0,\varepsilon)$

By Lemma 38, this is a manifold with faces, with a system of faces given by

$\mathcal{A}^{\prime}_{*;x_{0}\ldots x_{i}}\times\\{0\\}$

for $x_{0},\ldots,x_{i}=x$ in $\mathcal{F}$. In particular, all these faces
are compact.

Choose framed functions $(f_{*x},s_{*x})$ on each $\tilde{\mathcal{D}}_{*x}$,
and let $\mathcal{S}_{*x}:=f_{*x}^{-1}\\{1\\}$. Now define $a$ by

$a:=\sum\limits_{\begin{subarray}{c}x\in\mathcal{F}\\\
u-|x|=v+2\end{subarray}}[\mathcal{S}_{*x}]\cdot x\in
CM_{u-v-2}\left(\mathcal{F};\Omega_{v+1}\right)$

Note that $a$ is a specific instance of the obstruction chain (written there
as $\mathcal{O}$) constructed in (the proof of) Theorem 109, and therefore has
already been shown to be a closed chain.

For $y$ in $\mathcal{G}$ with $u-|y|=v+1$, we define
$\tilde{\mathcal{E}}_{*y}$ to be the coequaliser of the following diagram.

---
$\textstyle{\bigsqcup\limits_{y^{\prime}\in\mathcal{G}}\mathcal{B}_{*y^{\prime}}\times\mathcal{G}_{y^{\prime}y}\times[0,\varepsilon)\times[0,-\varepsilon)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{B}_{*y}\times[0,-\varepsilon)}$$\textstyle{\bigsqcup\limits_{y^{\prime},y^{\prime\prime}\in\mathcal{G}}\mathcal{R}^{\prime}_{*;y^{\prime\prime}y^{\prime}y}\times[0,\varepsilon)^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigsqcup\limits_{y^{\prime}\in\mathcal{G}}\mathcal{R}^{\prime}_{*;y^{\prime}y}\times[0,\varepsilon)}$$\textstyle{\bigsqcup\limits_{\begin{subarray}{c}x\in\mathcal{F}\\\
y^{\prime}\in\mathcal{G}\end{subarray}}\mathcal{A}^{\prime}_{*x}\times\mathcal{W}_{x;y^{\prime}y}\times[0,\varepsilon)^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigsqcup\limits_{x\in\mathcal{F}}\mathcal{A}^{\prime}_{*x}\times\mathcal{W}_{xy}\times[0,\varepsilon)}$$\textstyle{\bigsqcup\limits_{x,x^{\prime}\in\mathcal{F}}\mathcal{A}_{*;x^{\prime}x}\times\mathcal{W}_{xy}\times[0,\varepsilon)^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Note the copies of $[0,-\varepsilon)$ in some places instead of
$[0,\varepsilon)$; this will be relevant when we work in the framed setting.
Here the two maps out of each
$\mathcal{B}_{*y^{\prime}}\times\mathcal{G}_{y^{\prime}y}\times[0,\varepsilon)\times[0,-\varepsilon)$
send $(a,b,u,v)$ to $\alpha(a,b,u,v):=(\mathcal{C}(a,b;u),v)$ and
$\beta(a,b,u,v):=(\mathcal{C}(a;-v),b,u)$ respectively.

Each $\tilde{\mathcal{E}}_{*y}$ is a manifold with corners by Lemma 37 (noting
$[0,-\varepsilon)$ is naturally diffeomorphic to $[0,\varepsilon)$), and by
applying Lemma 38 (using the indexing sets
$I=\\{*\\}\cup\mathcal{G}\cup\mathcal{F}$ and $J=\emptyset$),
$\tilde{\mathcal{E}}$ is a manifold with faces, with a system of faces given
by

$\mathcal{B}_{*;y_{0}\ldots y_{j}}\times\\{0\\}$

for $y_{0},\ldots,y_{j}=y$ in $\mathcal{G}$,

$\mathcal{R}^{\prime}_{*;y_{0}\ldots y_{j}}\times\\{0\\}$

for $j\geq 1$ and $y_{0},\ldots,y_{j}=y\in\mathcal{G}$, and

$\mathcal{A}^{\prime}_{*;x_{0}\ldots
x_{i}}\times\mathcal{W}_{x_{i};y_{0}\ldots y_{j}}\times\\{0\\}$

for $x_{0},\ldots,x_{i}$ in $\mathcal{F}$ and $y_{0},\ldots,y_{j}=y$ in
$\mathcal{G}$. In particular, all these faces are compact.

Choose framed functions $(f_{*y},s_{*y})$ on each $\tilde{\mathcal{E}}_{*y}$,
and let $\mathcal{T}_{*y}=f_{*y}^{-1}\\{1\\}$. Now define $r$ by

$r:=\sum\limits_{\begin{subarray}{c}y\in\mathcal{G}\\\
u-|y|=v+1\end{subarray}}\left[\mathcal{T}_{*y}\right]\cdot y\in
CM_{u-v-1}\left(\mathcal{G};\Omega_{v+1}\right)$

We next show that $(a,r)$ is closed with respect to the mapping cone
differential: explicitly, that $da=0$ and $CM_{*}(\mathcal{W})(a)+dr=0$. We
have already seen that $da=0$.

We next define several auxiliary manifolds. For $y$ in $\mathcal{G}$ with
$u-|y|\leq v+1$, we define $\tilde{\mathcal{K}}_{*y}$ to be the coequaliser of
the following diagram.

$\textstyle{\bigsqcup\limits_{x,x^{\prime}\in\mathcal{F}}\mathcal{A}^{\prime}_{*;x^{\prime}x}\times\mathcal{W}_{xy^{\prime}}\times[0,\varepsilon)^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigsqcup\limits_{x\in\mathcal{F}}\mathcal{A}^{\prime}_{*x}\times\mathcal{W}_{xy^{\prime}}\times[0,\varepsilon)}$

By Lemma 38 (with $I=\mathcal{F}$ and $J=\mathcal{G}\setminus\\{y\\}$),
$\tilde{\mathcal{K}}_{*y}$ is a manifold with faces, with a system of faces
given by

$\mathcal{A}_{*;x_{0}\ldots x_{i}}\times\mathcal{W}_{x_{i};y_{0}\ldots
y_{j}}\times\\{0\\}$

for $x_{0},\ldots,x_{i}$ in $\mathcal{F}$ and $y_{0},\ldots,y_{j}=y$ in
$\mathcal{G}$, and

$\tilde{\mathcal{K}}_{*;y_{0}\ldots y_{j}}$

for $y_{0},\ldots,y_{j}=y$ in $\mathcal{G}$.

For $y$ in $\mathcal{G}$ with $u-|y|=v+2$, we define
$\tilde{\mathcal{J}}_{*y}$ to be the coequaliser of the following diagram.

$\textstyle{\bigsqcup\limits_{y^{\prime},y^{\prime\prime}\in\mathcal{G}}\tilde{\mathcal{K}}_{*;y^{\prime\prime}y^{\prime}y}\times[0,\varepsilon)^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigsqcup\limits_{y^{\prime}\in\mathcal{G}}\tilde{\mathcal{K}}_{*;y^{\prime}y}\times[0,\varepsilon)}$

By Lemma 37, this is a manifold with corners.

For $y$ in $\mathcal{G}$ with $u-|y|=v+2$, we define
$\tilde{\mathcal{L}}_{*y}$ to be the coequaliser of the following diagram.

$\textstyle{\bigsqcup\limits_{\begin{subarray}{c}x,x^{\prime}\in\mathcal{F}\\\
|x|-|y|>0\end{subarray}}\mathcal{A}^{\prime}_{*;x^{\prime}x}\times\mathcal{W}_{xy}\times[0,\varepsilon)^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigsqcup\limits_{\begin{subarray}{c}x\in\mathcal{F}\\\
|x|-|y|>0\end{subarray}}\mathcal{A}^{\prime}_{*x}\times\mathcal{W}_{xy}\times[0,\varepsilon)}$

By Lemma 38 (applied with $I=\left\\{x\in\mathcal{F}\,|\,|x|-|y|>0\right\\}$
and
$J=\left\\{x\in\mathcal{F}\,|\,|x|-|y|=0\right\\}\cup\left(\mathcal{G}\setminus\\{y\\}\right)$),
$\tilde{\mathcal{L}}_{*y}$ is a manifold with faces, with a system of faces
given by

$\mathcal{A}_{*;x_{0}\ldots x_{i}}\times\mathcal{W}_{x_{i};y_{0}\ldots
y_{j}}\times\\{0\\}$

when for $x_{0},\ldots,x_{i}$ in $\mathcal{F}$ and $y_{0},\ldots,y_{j}$ in
$\mathcal{G}$, when either $i+j>0$ or $i=j=0$ and $|x_{0}|-|y_{0}|>0$, and

$\tilde{\mathcal{D}}_{*x}\times\mathcal{W}_{xy}$

for $x$ in $\mathcal{F}$ with $|x|-|y|=0$, and

$\tilde{\mathcal{K}}_{*;y_{0}\ldots y_{j}}$

for $y_{0},\ldots,y_{j}=y$ in $\mathcal{G}$ when $j\geq 1$.

For $y$ in $\mathcal{G}$ with $u-|y|=v+2$, we define
$\tilde{\mathcal{M}}_{*y}$ to be the coequaliser of the following diagram.

$\textstyle{\bigsqcup\limits_{\begin{subarray}{c}y^{\prime}\in\mathcal{G}\\\
|y^{\prime}|-|y|>1\end{subarray}}\mathcal{B}_{*;y^{\prime}y}\times[0,\varepsilon)\times[0,-\varepsilon)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{B}^{\prime}_{*y}\times[0,-\varepsilon)}$$\textstyle{\bigsqcup\limits_{\begin{subarray}{c}y^{\prime},y^{\prime\prime}\in\mathcal{G}\\\
|y^{\prime}|-|y|>1\end{subarray}}\mathcal{R}^{\prime}_{*;y^{\prime\prime}y^{\prime}y}\times[0,\varepsilon)^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigsqcup\limits_{\begin{subarray}{c}y^{\prime}\in\mathcal{G}\\\
|y^{\prime}|-|y|>1\end{subarray}}\mathcal{R}^{\prime}_{*;y^{\prime}y}\times[0,\varepsilon)}$$\textstyle{\bigsqcup\limits_{\begin{subarray}{c}x\in\mathcal{F};y^{\prime}\in\mathcal{G}\\\
|y^{\prime}|-|y|>1\end{subarray}}\mathcal{A}^{\prime}_{*x}\times\mathcal{W}_{x;y^{\prime}y}\times[0,\varepsilon)^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigsqcup\limits_{\begin{subarray}{c}x\in\mathcal{F}\\\
|x|-|y|>0\end{subarray}}\mathcal{A}^{\prime}_{*x}\times\mathcal{W}_{xy}\times[0,\varepsilon)}$$\textstyle{\bigsqcup\limits_{\begin{subarray}{c}x,x^{\prime}\in\mathcal{F}\\\
|x|-|y|>0\end{subarray}}\mathcal{A}_{*;x^{\prime}x}\times\mathcal{W}_{xy}\times[0,\varepsilon)^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

This is a manifold with corners by Lemma 37; we see (either by iteratively
applying Lemma 38 or by direct inspection of the corner strata) that
$\tilde{\mathcal{M}}_{*y}$ is a manifold with faces, with a system of faces
given by

$\mathcal{B}_{*;y_{0}\ldots y_{j}}\times\\{0\\}$

for $y_{0},\ldots,y_{j}=y$ in $\mathcal{G}$,

$\mathcal{R}^{\prime}_{*;y_{0}\ldots y_{j}}\times\\{0\\}$

for $y_{0},\ldots,y_{j}=y$ in $\mathcal{G}$ when either $j>1$ or $j=0$ and
$|y_{0}|-|y_{1}|>1$, and

$\mathcal{A}^{\prime}_{*;x_{0}\ldots
x_{i}}\times\mathcal{W}_{x_{i};y_{0}\ldots y_{j}}\times\\{0\\}$

for $x_{0},\ldots,x_{i}$ in $\mathcal{F}$ and $y_{0},\ldots,y_{j}$ in
$\mathcal{G}$, when either $i+j>0$ or $i=j=0$ and $|x_{0}|-|y_{0}|>0$, and

$\tilde{\mathcal{K}}_{*;y_{0}\ldots y_{j}}$

for $y_{0},\ldots,y_{j}$ in $\mathcal{G}$ with $j\geq 1$, and

$\tilde{\mathcal{E}}_{*;y^{\prime}y}$

for $y^{\prime}$ in $\mathcal{G}$ with $|y^{\prime}|-|y|=1$.

Both $\tilde{\mathcal{L}}_{*y}$ and $\tilde{\mathcal{M}}_{*y}$ contain faces
of the form $\tilde{\mathcal{K}}_{*;y_{0}\ldots y_{k}}$ (for
$y_{0},\ldots,y_{j}$ in $\mathcal{G}$ with $j\geq 1$), and in both cases these
faces are non-compact but their intersections with the other boundary faces
not of this form are always compact. By the collar neighbourhood theorem for
manifolds with faces, there are embeddings
$\tilde{\mathcal{J}}_{*y}\hookrightarrow\tilde{\mathcal{L}}_{*y},\tilde{\mathcal{M}}_{*y}$
onto neighbourhoods of all the faces of the form
$\tilde{\mathcal{K}}_{*;y_{0}\ldots y_{j}}$.

Pick a framed function $(h_{*y},w_{*y})$ on $\tilde{\mathcal{J}}_{*y}$, and
remove $h_{*y}^{-1}(-\infty,1)$ from $\tilde{\mathcal{L}}_{*y}$ and
$\tilde{\mathcal{M}}_{*y}$ to obtain $\tilde{\mathcal{L}}^{r}_{*y}$ and
$\tilde{\mathcal{M}}^{r}_{*y}$ respectively. These are still manifolds with
faces, but now they both have a boundary face identified with
$h_{*y}^{-1}\\{1\\}$, which is a compact smooth manifold with faces.

Now glue $\tilde{\mathcal{L}}^{r}_{*y}$ to $\tilde{\mathcal{M}}^{r}_{*y}$
along this common boundary face to obtain a new manifold with faces
$\mathcal{V}_{*y}$. Its non-compact boundary faces are given by
$\tilde{\mathcal{D}}_{*x}\times\mathcal{W}_{xy}$ for $x$ in $\mathcal{F}$ with
$|x|-|y|=1$ (coming from $\tilde{\mathcal{L}}_{*y}^{r}$), and also other ones
which are diffeomorphic to $\tilde{\mathcal{E}}_{*;y^{\prime}y}$ away from a
neighbourhood of the compact faces of this face, for $y^{\prime}$ in
$\mathcal{G}$ with $|y^{\prime}|-|y|=1$ (coming from
$\tilde{\mathcal{M}}^{r}_{*y}$).

For each $y$ in $\mathcal{G}$ with $u-|y|=v+2$, pick framed functions
$(g_{*y},t_{*y})$ on $\mathcal{V}_{*y}$, agreeing with the framed functions
$(f_{*x},s_{*x})$ and $(f_{*y},s_{*y})$ on appropriate faces.

Then $g_{*y}^{-1}\\{1\\}$ is a nullbordism of the manifold representing the
$y$-coefficient of $CM_{*}(\mathcal{W})(a)+dr$, so this chain must vanish,
completing the proof that $(a,r)$ is closed.

We proceed similarly to the proof of Theorem 109. Since $CM_{*}(\mathcal{W})$
is a quasiisomorphism (over $\mathbb{Z}/2\mathbb{Z}$, which implies it is also
one over $\Omega_{v+1}$), its cone is acyclic, so there is some
$(a^{\prime},r^{\prime})$ in

$CM_{u-v-1}\left(\mathcal{F};\Omega_{v+1}\right)\oplus
CM_{u-v}\left(\mathcal{G};\Omega_{v+1}\right)$

such that $da^{\prime}=a$ and $CM_{*}(\mathcal{W})(a^{\prime})+dr^{\prime}=r$.

Replacing each $\mathcal{A}^{\prime}_{*x}$ (for $u-|x|=v+1$) with
$\mathcal{A}^{\prime}_{*x}\sqcup a^{\prime}_{x}$ (where $a^{\prime}_{x}$ is a
choice of some representative of the $x$-coefficient of $a$) (so the new
$\mathcal{A}^{\prime}$ is another extension of $\mathcal{A}$) and
$\mathcal{R}^{\prime}_{*y}$ (for $u-|y|=v$) with
$\mathcal{R}^{\prime}_{*y}\sqcup r^{\prime}_{y}$, we may assume that the chain
$(a,r)$ is zero, i.e. that each $\mathcal{S}_{*x}$ and each $\mathcal{T}_{*y}$
is nullbordant.

We now define extensions of $\mathcal{A}^{\prime}$ and $\mathcal{R}^{\prime}$:
explicitly, we define compact manifolds with faces $\mathcal{A}^{\prime}_{*x}$
for $x$ in $\mathcal{F}$ with $u-|x|=v+2$ and $\mathcal{R}^{\prime}_{*y}$ for
$y$ in $\mathcal{G}$ with $u-|y|=v+1$, with systems of faces as described
earlier.

For $x$ in $\mathcal{F}$ with $u-|x|=v+2$, let
$\mathcal{A}^{\prime\prime}_{*x}=f_{*x}^{-1}(-\infty,1]$. Since
$\mathcal{S}_{*x}=f_{*x}^{-1}\\{1\\}$ is closed and nullbordant, we glue such
a nullbordism onto this boundary face to obtain $\mathcal{A}^{\prime}_{*x}$.

For $y$ in $\mathcal{G}$ with $u-|y|=v+1$, we let
$\mathcal{R}^{\prime\prime}_{*y}=f_{*y}^{-1}(-\infty,1]$. Since
$\mathcal{T}_{*y}=f_{*y}^{-1}\\{1\\}$ is closed and nullbordant, we glue such
a nullbordism onto this boundary face to obtain $\mathcal{R}^{\prime}_{*y}$,
completing the inductive step.

######  Proof 5.15 (Incorporating framings).

Similarly to the proof of Theorem 111, we coherently frame all manifolds
arising in the unframed case to prove the framed case. We assume
$\mathcal{A}^{\prime},\mathcal{B}^{\prime},\mathcal{R}^{\prime}$ are all
framed, and we recall that each $\mathcal{B}^{\prime}_{*y}$ is an _outgoing_
boundary face of each $\mathcal{R}^{\prime}_{*y}$. We first equip each
$\tilde{\mathcal{D}}_{*x}$, $\tilde{\mathcal{K}}_{*y}$,
$\tilde{\mathcal{J}}_{*y}$, $\tilde{\mathcal{L}}_{*y}$ with compatible systems
of framings in the same way as we did with $\tilde{\mathcal{Q}}(\ldots)$ in
the proof of Lemma 70; these then induce compatible systems of framings on
each $\mathcal{S}_{*x}$ and $\tilde{\mathcal{L}}^{r}_{*y}$ in the usual way.

We next frame each $\tilde{\mathcal{E}}_{*y}$. The only unbroken boundary face
of $\tilde{\mathcal{E}}_{*y}$ is $\mathcal{B}_{*y}\times\\{0\\}$; we declare
this to be outgoing. We define $st_{*y}^{\tilde{\mathcal{E}}}:V_{*[u]}\oplus
I^{\tilde{\mathcal{E}}}_{*y}\to\mathbb{R}^{2}\oplus V_{y}$ as follows. Over
each $\mathcal{B}^{\prime}_{*y}\times[0,-\varepsilon)$, we define
$st^{\tilde{\mathcal{E}}}$ to be the composition

$V_{*[u]}\oplus
I^{\tilde{\mathcal{E}}}_{*y}\xrightarrow{\psi^{-1}}V_{*[u]}\oplus
I^{\mathcal{B}^{\prime}}_{*y}\oplus\mathbb{R}\sigma\xrightarrow{st^{\mathcal{B}^{\prime}}}\mathbb{R}\oplus\mathbb{R}\sigma\oplus
V_{y}$

where the first copy of $\mathbb{R}$ comes from $st^{\mathcal{B}}$. On
$\mathcal{R}^{\prime}_{*y^{\prime}}\times\mathcal{G}_{y^{\prime}y}\times[0,\varepsilon)$,
we define $st^{\tilde{\mathcal{E}}}$ to be the composition

$V_{*[u]}\oplus I^{\tilde{\mathcal{E}}}\xrightarrow{\psi^{-1}}V_{*[u]}\oplus
I^{\mathcal{R}^{\prime}}_{*y^{\prime}}\oplus
I^{\mathcal{G}}_{y^{\prime}y}\xrightarrow{st^{\mathcal{G}}\circ
st^{\mathcal{R}^{\prime}}}\mathbb{R}^{2}\oplus V_{y}$

and on $\mathcal{A}^{\prime}_{*x}\times\mathcal{W}_{xy}\times[0,\varepsilon)$,
we define $st^{\tilde{\mathcal{E}}}$ to be the composition

$V_{*[u]}\oplus
I^{\tilde{\mathcal{E}}}_{*y}\xrightarrow{\psi^{-1}}V_{*[u]}\oplus
I^{\mathcal{A}^{\prime}}_{*x}\oplus
I^{\mathcal{W}}_{xy}\xrightarrow{st^{\mathcal{W}}\circ
st^{\mathcal{A}^{\prime}}}\mathbb{R}\oplus\mathbb{R}\oplus V_{y}$

where the first copy of $\mathbb{R}$ in the final term comes from
$st^{\mathcal{W}}$ and the second from $st^{\mathcal{A}^{\prime}}$.

We need these to agree on the regions where they overlap. Over all regions
except those of the form
$\mathcal{B}_{*y^{\prime}}\times\mathcal{G}_{y^{\prime}y}\times[0,\varepsilon)_{u}\times[0,-\varepsilon)_{v}$,
this is identical to checking the gluing on overlaps in Lemma 70; on the
region
$\mathcal{B}_{*y^{\prime}}\times\mathcal{G}_{y^{\prime}y}\times[0,\varepsilon)_{u}\times[0,-\varepsilon)_{v}$,
it follows from a similar argument, using the following diagram.

$\textstyle{V_{*[u]}\oplus
T\mathcal{B}^{\prime}_{*y}\oplus\mathbb{R}\partial_{s}\oplus\mathbb{R}\tau_{y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\psi}}$$\textstyle{V_{*[u]}\oplus\mathbb{R}\partial_{u}\oplus\mathbb{R}\partial_{v}\oplus
T\mathcal{B}^{\prime}_{*y^{\prime}}\oplus
T\mathcal{G}_{y^{\prime}y}\oplus\mathbb{R}\tau_{y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d\alpha}$$\scriptstyle{d\beta}$$\scriptstyle{=}$$\textstyle{V_{*[u]}\oplus
T\mathcal{R}^{\prime}_{*y^{\prime}}\oplus\partial_{s}\oplus
T\mathcal{G}_{y^{\prime}y}\oplus\mathbb{R}\tau_{y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\psi}}$$\textstyle{V_{*[u]}\oplus
I^{\mathcal{B}^{\prime}}_{*y}\oplus\mathbb{R}\sigma\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{V_{*[u]}\oplus
I^{\tilde{\mathcal{E}}}_{*y}}$$\textstyle{V_{*[u]}\oplus
I^{\mathcal{R}^{\prime}}_{*y^{\prime}}\oplus
I^{\mathcal{G}}_{y^{\prime}y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$

Here the $\bar{\psi}$ on the left is defined to send $\partial_{s}$ to
$\sigma$ and is the identity on all other factors, and the $\bar{\psi}$ on the
right is defined to send $\partial_{s}$ to $\tau_{y^{\prime}}$ and is the
identity on all other factors. Then we can see directly that this diagram
commutes since all three ways of going from the top middle entry to the bottom
middle entry send $\tau_{y}$ to $\tau_{y}$, $\partial_{u}$ to
$\nu_{y^{\prime}}$ and $\partial_{v}$ to $-\nu^{\mathcal{B}}$ (and are the
identity on all other factors).

The $st^{\tilde{\mathcal{E}}}$ (along with the other framings) assemble to
form systems of framings for all $\tilde{\mathcal{E}}_{*y}$, and similarly we
may frame all $\tilde{\mathcal{M}}_{*y}$ in exactly the same way. These
framings then induce framings on each $\mathcal{T}_{*y}$ and each
$\tilde{\mathcal{M}}^{r}_{*y}$ in the usual way.

The framings on $\mathcal{S}_{*x}$ and $\mathcal{T}_{*y}$ gives us a well-
defined element

$(a^{fr},r^{fr})\in CM_{u-v-2}\left(\mathcal{F};\Omega_{v+1}^{fr}\right)\oplus
CM_{u-v-1}\left(\mathcal{G};\Omega^{fr}_{v+1}\right)$

We have already seen that $da^{fr}=0$. We would like to show that the pair
$(a^{fr},r^{fr})$ is closed; in this case, an identical argument to before
allows us to complete the induction step.

It therefore remains to show that $\mathcal{W}_{*}(a^{fr})-d(r^{fr})=0$. To do
this we next equip $\mathcal{V}_{*y}$ with appropriate framings. Note that
both $\tilde{\mathcal{L}}^{r}_{*y}$ and $\tilde{\mathcal{M}}^{r}_{*y}$ have
$h^{-1}_{*y}\\{1\\}$ as an incoming face, so we can frame $\mathcal{V}_{*y}$
by equipping it with the framing of $\tilde{\mathcal{L}}^{r}_{*y}$ over
$\tilde{\mathcal{L}}^{r}_{*y}$ and with the opposite framing of
$\tilde{\mathcal{M}}^{r}_{*y}$ over $\tilde{\mathcal{M}}^{r}_{*y}$. Then
$g^{-1}$ is a framed nullbordism of

$\bigsqcup\limits_{x\in\mathcal{F}}\mathcal{S}_{*x}\times\mathcal{W}_{xy}\bigsqcup_{y^{\prime}\in\mathcal{G}}\overline{\mathcal{T}}_{*;y^{\prime}y}$

from which the required closedness follows.

The analogous algebraic result is the following corollary of Hurewicz’ and
Whitehead’s theorems:

######  Proposition 112.

Let $R$ be a $(-1)$-connected dga or ring spectrum, and $f:M\to N$ a morphism
in $R\modules$. If the induced map

$f:M\otimes_{R}HR_{0}\to N\otimes_{R}HR_{0}$

is an equivalence of $HR_{0}$-modules, then $f$ is an equivalence of
$R$-modules.

##  6 Baas-Sullivan singularities

For each $k>0$, we fix a set $\mathbb{M}^{k}$ of closed $k$-manifolds $P$ with
stable framings $st^{P}:TP\to\mathbb{R}^{k}$ containing a representative of
each isomorphism (i.e. diffeomorphism respecting stable framings) class of
stably framed $k$-manifolds.

###  6.1 Singularities

######  Definition 113.

A _manifold with faces with $k$-Baas-Sullivan singularities_ (abbreviated to
_with $\kBS$-singularities_, or just _$\kBS$ -manifold (with faces)_)
$(M,N_{P},\iota)$ (often just abbreviated to $M$) consists of a manifold with
faces $M$, along with manifolds with faces $N_{P}$ for each
$P\in\mathbb{M}^{k}$, such that only finitely many $N_{P}$ are non-empty, and
$\iota$ is a proper embedding of a boundary face

$\iota:\bigsqcup\limits_{P\in\mathbb{M}^{k}}N_{P}\times P\hookrightarrow M$

We call faces of $M$ which have no components lying in the image of $\iota$
_essential faces_ and faces of $M$ which lie in the image of $\iota$ _singular
faces_. We write $D_{M}$ for the set of essential faces of $M$, and if
$F\subseteq M$ is an essential face, we write $D_{F\subseteq M}$ for the set
of essential faces containing $F$. Note that any essential face $F$ of $M$
naturally inherits the structure of a $\kBS$-manifold with faces. In
particular, if $(M^{\prime},N^{\prime}_{P},\iota^{\prime})$ is another
$\kBS$-manifold with faces, for $M^{\prime}\hookrightarrow M$ to be an
inclusion of a face, there are inclusions of faces of non-singular manifolds
$N_{P}^{\prime}\hookrightarrow N_{P}$ such that the following diagram
commutes:

$\textstyle{\bigsqcup\limits_{P}N_{P}^{\prime}\times
P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{\prime}}$$\textstyle{\bigsqcup\limits_{P}N_{P}\times
P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{M^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M}$

Note that in particular, even if $P,P^{\prime}\in\mathbb{M}^{k}$ are
diffeomorphic, we do not treat them as the same singularity type.

A _closed $\kBS$-manifold_ is a $\kBS$-manifold with faces which has no
essential faces.

A _non-singular_ $\kBS$-manifold is one where all $N_{P}$ are empty; this is
just an ordinary manifold.

A _system of essential faces_ for $M$ is a collection of faces $F_{i}^{j}$ is
a collection of essential faces $F_{i}^{j}$ of $M$ such that each $F_{i}^{j}$
has codimension $j$, all $F_{i}^{j}$ have disjoint smooth loci, and any
(positive codimension) essential face of $M$ is a disjoint union of components
of the $F_{i}^{j}$.

Let $X$ be some other space. We say a continuous function $f:M\to X$ is
_compatible with the singularities_ if, on each $\iota(N_{P}\times P)$, $f$ is
constant in the $P$ direction, i.e. it only depends on the $N_{P}$-co-
ordinate.

One should think of such a manifold as a singular manifold with singular loci
locally modelled on cones over $k$-manifolds. Indeed taking $(M,N_{P},\iota)$
a $\kBS$-manifold with faces and attaching cells $N_{P}\times C_{P}$ to each
$N_{P}\times P$ (where $C_{P}$ is the cone over $P$) gives such a singular
manifold. The definition above can be thought of as removing neighbourhoods of
the singular loci but keeping track of how to glue them back in; this
definition is easier to work with in practice.

In particular, we think of the singular faces as part of a “singular locus”
rather than actual faces of the manifold, whose role the essential faces play.

######  Lemma 114.

Let $M$ be a $\kBS$-manifold with faces, $G$ a face of $M$, and $F$ a face of
$G$. Then if $G$ is a singular face, $F$ is too.

It follows from this that if $F$ is an essential face, $G$ is too.

######  Proof 6.1.

If $G$ lies in the image of $\iota$ and $F\subset G$ then $F$ lies in the
image of $\iota$.

A $\kBS$-manifold with faces of dimension $\leq k$ is the same as a manifold
with faces.

######  Definition 115.

A framed function $(f,s)$ on a $\kBS$-manifold $(M,N_{P},\iota)$ is a framed
function on the manifold with corners $M$ such that over each $N_{P}\times P$,
$f$ is constant in the $P$-direction (so $f$ is compatible with the
singularities as a continuous function), and $s$ is also constant in the
$P$-direction (noting that since $f$ is constant in the $P$-direction,
$f^{-1}\\{1\\}\cap(N_{P}\times P)$ naturally splits into a product of
something with $P$).

The same proofs show that Lemma 43 (respectively Lemma 44) holds for framed
functions on $\kBS$-manifolds, where we restrict to (respectively extend from)
only essential faces.

From the preimage theorem for manifolds with faces, we see that

######  Proposition 116.

Let $(M,N_{P},\iota)$ be a $\kBS$-manifold with face with a system of
essential faces $\\{F_{i}\\}$, and $(f,s)$ a framed function on $M$. Then
$f^{-1}\\{1\\}$ and $f^{-1}(-\infty,1]$ are naturally $\kBS$-manifolds with
faces.

There is a system of essential faces for $f^{-1}\\{1\\}$ given by the
$F_{i}\cap f^{-1}\\{1\\}$, and one for the $f^{-1}(-\infty,1]$ given by the
$F_{i}\cap f^{-1}\\{1\\}$ and the $F_{i}\cap f^{-1}(-\infty,1]$.

######  Lemma 117.

Let $M$ be a $\kBS$-manifold with faces, and $N$ a manifold with faces. Then
$M\times N$ and $N\times M$ each canonically admit the structure of a manifold
with faces with $\kBS$ singularities.

######  Proof 6.2.

$M\times N$ is a manifold with faces, and admits a canonical proper embedding
of a codimension 1 face

$\iota\times\id_{N}:\bigsqcup\limits_{P}(N_{P}\times N)\times P\hookrightarrow
M\times N$

where the $N_{P}$, $\iota$ are as in Definition 113. The case of $N\times M$
is the same.

######  Corollary 118.

Let $M$ and $N$ be $\kBS$-manifolds with faces, of dimensions $m$ and $n$
respectively. If either $m\leq k$ or $n\leq k$, in particular if $2k>m+n$,
then $M\times N$ is naturally a $\kBS$-manifold with faces.

######  Proof 6.3.

By Remark 6.1 one of $M$ and $N$ is non-singular, and so by Lemma 117, their
product is a $\kBS$-manifold with faces.

We have the following abstract gluing lemma, similarly to Lemma 37 (but now
with essential faces playing the role of faces).

Let $(M_{i},Q_{i,P},\iota_{i})$ be $\kBS$-manifolds (so each
$\iota_{i}:\sqcup_{P}Q_{i,P}\times P\hookrightarrow M_{i}$ is the inclusion of
a boundary face). Let $\alpha_{w}:N_{w}\hookrightarrow M_{i_{w}}$ and
$\beta_{w}:N_{w}\hookrightarrow M_{j_{w}}$ be inclusions of essential boundary
faces; each of these induces the structure of a manifold with
$\kBS$-singularities on $N_{w}$, and we assume these structures agree. We
write $(N_{w},R_{w,P},\iota_{w})$ for this structure.

In particular, we have diffeomorphisms $R_{w,P}\simeq\partial
Q_{i_{w},P}\simeq\partial Q_{j_{w},P}$.

We now have inclusions of boundary faces $\alpha_{w}:R_{w,P}\hookrightarrow
Q_{i,P}$ and $\beta_{w}:R_{w,P}\hookrightarrow Q_{j,P}$ as in the hypothesis
of Lemma 37; for each $P$ we let $Y_{P}$ be the coequaliser of the diagram

$\bigsqcup\limits_{w}R_{w,P}\times[0,\varepsilon)^{2}\rightrightarrows\bigsqcup\limits_{i}Q_{i,P}\times[0,\varepsilon)$

which is a manifold with corners by Lemma 37.

Let $Z$ be constructed as in Lemma 37. There are natural disjoint embeddings
$\iota_{j}:Y_{P}\times P\hookrightarrow Z$ given by gluing together the
$\iota_{i}$.

######  Lemma 119.

$(Z,Y_{P},\iota:=\sqcup_{P}\iota_{P})$ is a manifold with corners with
$\kBS$-singularities.

######  Proof 6.4.

Lemma 37 shows that $Z$ is a manifold with corners. Equipping this with the
map $\iota$ defines a system of singular faces which makes this a manifold
with corners with $\kBS$-singularities.

Furthermore if we can index the $M_{i}$ and $N_{w}$ as in Lemma 38, the
statement of Lemma 38 still holds for a system of essential faces, with the
same proof.

######  Definition 120.

Let $\mathcal{F}$ and $\mathcal{G}$ be flow categories. A _$\kBS$
-pre-$\tau_{\leq n}$-morphism_ $\mathcal{W}$ from $\mathcal{F}$ to
$\mathcal{G}$ is the same as a $pre$-$\tau_{\leq n}$-morphism except all
manifolds are allowed to have $\kBS$-singularities, and all systems of faces
are essential systems of faces.

More explicitly, this consists of compact $\kBS$-manifolds with faces, say
$\mathcal{W}_{xy}=(\mathcal{W}_{xy},N_{xy,P},\iota_{xy})$ (of dimension
$|x|-|y|$) for all $x$ in $\mathcal{F}$ and $y$ in $\mathcal{G}$ such that
$|x|-|y|\leq n$, along with maps

$c=c^{\mathcal{W}}:\mathcal{F}_{xz}\times\mathcal{W}_{zy}\rightarrow\mathcal{W}_{xy}$

and

$c=c^{\mathcal{W}}:\mathcal{W}_{xz}\times\mathcal{G}_{zy}\rightarrow\mathcal{W}_{xy}$

which are compatible with $c^{\mathcal{F}}$ and $c^{\mathcal{G}}$, which
define a system of essential faces

$\mathcal{W}_{x_{0}\ldots x_{i};y_{0}\ldots y_{j}}$

for $x_{0}=x,\ldots,x_{i}$ in $\mathcal{F}$ and $y_{0},\ldots,y_{j}=y$ in
$\mathcal{G}$.

We define _$\kBS$ -$\tau_{\leq n}$-pre-bordisms_ and _$\kBS$ -$\tau_{\leq
n}$-bordisms_ analogously.

There are similar definitions of flow categories with $\kBS$-singularities,
bilinear maps, associators etc., but we will not use them.

Note that since the flow categories $\mathcal{F}$ and $\mathcal{G}$ don’t have
any singularities, the products $\mathcal{W}_{x_{0}\ldots x_{i};y_{0}\ldots
y_{j}}$ admit the structure of manifolds with $\kBS$ singularities, by Lemma
118. In practice we will only use these definitions when $2k-1>n$, so that we
can compose these: if we did not have this inequality, then when one attempts
to compose such things, eventually one will have to take a product of two
manifolds with non-empty singular faces, which does not have $\kBS$
singularities in the sense of the definition we’ve given. Compare to Remark
1.3. Composition of pre-$\tau_{\leq n}$-morphisms with $\kBS$-singularities
can be defined the same as in the non-singular case, with exactly the same
proofs of associativity and well-definedness, except all framed functions
which are chosen must be chosen to be compatible with the singularities.
Composing pre-$\tau_{\leq n}$-morphisms with $\kBS$-singularities with
bilinear maps is entirely analogous. In order to avoid the above truncation
issue, we could amend our definition to allow boundary faces of the form
$N\times P_{1}\times\ldots\times P_{k}$ where each $P_{i}\in\mathbb{M}^{k}$,
in which case products of manifolds with this structure still have this
structure; this however is not necessary for the current application.

One can make an analogous construction for Baas-Sullivan singularities
modelled after any set of closed manifolds $\mathbb{M}$; we conjecture that
the resulting category is equivalent to (the homotopy category of) the
category of perfect modules over $R$, where $R$ is the
$\mathbb{E}_{1}$-quotient of $MO$ by each $[M]\in\Omega_{*}\simeq\pi_{*}MO$
for $M\in\mathbb{M}$.

###  6.2 Bordism with singularities

Let $X$ be a space.

######  Definition 121.

Let $M$ and $N$ be two closed manifolds with $\kBS$-singularities. A _$\kBS$
-bordism_ from $M$ to $N$ is a compact manifold $W$ with faces with
$\kBS$-singularities, with a system of essential faces given by $\\{M,N\\}$.

The _$i$ th $\kBS$-bordism group of $X$_, $\Omega^{\kBS}_{i}(X)$, is the set
of pairs $(M,f)$, where $M$ is a closed manifold with $\kBS$-singularities and
$f:M\rightarrow X$ is a continuous map which is compatible with the
singularities, modulo the equivalence relation $\sim$, where we say
$(M,f)\sim(M^{\prime},f^{\prime})$ if there is a $\kBS$-bordism $W$ from $M$
to $M^{\prime}$, along with a map $F:W\rightarrow X$ extending $f$ and
$f^{\prime}$ which is again compatible with the singularities.

We write $\Omega_{i}^{\kBS}$ and $\Omega_{i}$ for $\Omega_{i}^{\kBS}(*)$ and
$\Omega_{i}(*)$ respectively. Recall that $\Omega_{*}(X)$ is naturally an
$\Omega_{*}$-module, though $\Omega^{\kBS}_{*}$ is not necessarily a ring.

######  Proposition 122.

Let $X$ be any space. The group $\Omega_{k}^{\kBS}(X)$ is zero.

######  Proof 6.5.

Let $f:M\to X$ represent a class in $\Omega_{k}^{\kBS(X)}$. $M$ is a closed
$k$-dimensional $\kBS$-manifold, and so is non-singular.

Let $W=M\times[0,1]$, viewed as a $\kBS$-manifold with singular boundary face
$M\times\\{1\\}$, and $g:W\to X$ given by $f$ on the first factor. This is a
bounding $\kBS$-manifold witnessing the vanishing of $[(f,M)]$.

######  Proposition 123.

Let $X$ be any space. The natural map
$\theta:\Omega_{k+1}(X)\rightarrow\Omega_{k+1}^{\kBS}(X)$ which sends
$(M,f:M\to X)$ to itself, has kernel $\Omega_{k}\cdot\Omega_{1}(X)$.

######  Proof 6.6.

Choose a class in $\ker\theta$, represented by a pair $(M,f:M\to X)$, where
$M$ is a closed $k$-manifold and $f$ is a continuous map. Then there is some
$\kBS$-manifold $(W,N_{P},\iota)$ with faces, such that its only essential
face is $M$ (and itself), and a map $F:W\to X$ which is compatible with the
singularities. This means that, viewed as a smooth manifold (i.e. without
$\kBS$-singularities), $(W,F)$ is a bordism from $(M,f)$ to

$\left(\bigsqcup\limits_{P}N_{P}\times P,F\right)$

and so we must have that

$[(M,f)]=\sum\limits_{P}[(N_{P},F)]\cdot[P]$

in $\Omega_{k+1}(X)$, using the fact that on each $N_{P}\times P$, $F$ is
constant in the $P$ direction. Clearly the right hand side lives in
$\Omega_{k}\cdot\Omega_{1}(X)$.

Conversely, any product $N\times Q$ of a 1-manifold $N$ and some
$Q\in\mathbb{M}^{k}$, along with a map $f:N\times Q\to X$ which is constant in
the $Q$-direction, represents a class in $\ker\theta$, since it bounds the
$\kBS$-manifold with faces $N\times Q\times[0,1]$ where $\iota$ is the
inclusion $N\times Q\times\\{1\\}\hookrightarrow N\times Q\times[0,1]$, along
with the map to $X$ given by $f$.

###  6.3 Framings and singularities

We would like to equip our manifolds with singularities with framings which
are suitably compatible with the singularities. Roughly speaking, a stable
framing on a manifold with $\kBS$-singularities consists of a stable framing
on the underlying manifold with faces, which respects the product structure
over each singular face. Note however that “respecting the product structure”
will in fact consist of more data.

######  Definition 124.

Let $M=(M,N_{P},\iota)$ be a $\kBS$-manifold with faces. We treat each
singular boundary face $N_{P}\times P$ as an unbroken face.

Given a stable isomorphism $E\oplus TM\to F$, we say it’s _compatible with the
singularities_ if we are also given stable isomorphisms $st^{N}_{P}:E\oplus
TN_{P}\oplus\mathbb{R}\sigma_{P}\to F-\mathbb{R}^{k}$ such that the following
diagram commutes over each $N_{P}\times P$:

$\textstyle{E\oplus TN_{P}\oplus\mathbb{R}\sigma_{P}\oplus
TP\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\scriptstyle{st^{N}_{P}\oplus
st^{P}}$$\textstyle{F-\mathbb{R}^{k}\oplus\mathbb{R}^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\oplus
TM\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F}$

where the right vertical arrow uses the natural stable isomorphism
$-\mathbb{R}^{k}\oplus\mathbb{R}^{k}\to 0$. Note that this definition requires
the extra data of the $st^{N}_{P}$.

From this, we can define the framed bordism groups with $\kBS$-singularities,
which we denote by $\Omega^{fr;\kBS}_{*}(X)$ for any space $X$. As before,
there is a natural map $\Omega^{fr}_{*}(X)\to\Omega^{fr;\kBS}_{*}(X)$, though
$\Omega^{fr;\kBS}_{*}:=\Omega^{fr;\kBS}_{*}(*)$ is not a ring.

Similarly to Proposition 122 and by the same proof with framings incorporated,
we have

######  Proposition 125.

Let $X$ be any space. The group $\Omega_{k}^{fr;\kBS(X)}$ is zero.

######  Proposition 126.

The natural map $\Omega_{k+1}^{fr}(X)\to\Omega_{k+1}^{fr;\kBS}(X)$ has kernel
$\Omega_{k}^{fr}(\ast)\cdot\Omega_{1}^{fr}(X)$.

######  Proof 6.7.

The proof is a minor variation on that of Proposition 123. If $M$ is a framed
closed $\kBS$-manifold and a map $f:M\to X$, such that $(M,f)$ vanishes in
$\Omega_{k+1}^{fr;\kBS}(X)$, then there is some framed $\kBS$-manifold
$(W,N_{P},\iota)$ mapping to $X$, with $M$ as its only essential face, and
such that the framing on $W$ and the map $W\to X$ are both compatible with the
singularities. Thus, since the framing is product-like on each $N_{P}\times P$
and the map to $X$ is constant on $P$, it follows that $(M,f)$ lies in
$\Omega_{k}^{fr}(\ast)\cdot\Omega_{1}^{fr}(X)$. The other direction is exactly
as in _op.cit._

######  Definition 127.

Let $\mathcal{S}:\mathcal{F}\to\mathcal{G}$ be a $\kBS$-$\tau_{\leq
n}$-morphism (respectively bordism). A _framing_ on $\mathcal{S}$ consists of
the same data as a framing on a non-singular morphism (resp. bordism), along
with the extra data that each stable framing $V_{x}\oplus
I^{\mathcal{S}}_{xy}\to\mathbb{R}^{i}\oplus V_{y}$, where $i=1$ (resp. $i=2$),
is compatible with the singularities.

For concreteness, we spell out what this definition says explicitly.

Write
$\mathcal{S}_{xy}=(\mathcal{S}_{xy},N^{\mathcal{S}}_{P;xy},\iota^{\mathcal{S}}_{xy})$
for the structure of a $\kBS$-manifold on each $\mathcal{S}_{xy}$ and let
$I^{N^{\mathcal{S}}}_{P;xy}:=TN^{\mathcal{S}}_{P;xy}\oplus\mathbb{R}\tau_{y}$.

Then the new extra data consists of stable isomorphisms

$st^{N^{\mathcal{S}}}_{P;xy}:V_{x}\oplus
I^{N^{\mathcal{S}}}_{P;xy}\oplus\mathbb{R}\sigma_{P}\to\mathbb{R}^{i}\oplus
V_{y}-\mathbb{R}^{k}$

(where $i=1$ (resp. $i=2$)), such that the following diagram commutes over
each $N^{\mathcal{S}}_{P;xy}\times P$:

$\textstyle{V_{x}\oplus
I^{N^{\mathcal{S}}}_{P;xy}\oplus\mathbb{R}\sigma_{P}\oplus
TP\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{st^{N^{\mathcal{S}}}_{P;xy}\oplus
st^{P}}$$\scriptstyle{\psi}$$\textstyle{\mathbb{R}^{i}\oplus
V_{y}-\mathbb{R}^{k}\oplus\mathbb{R}^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{x}\oplus
I^{\mathcal{S}}_{xy}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{st^{\mathcal{S}}}$$\textstyle{\mathbb{R}^{i}\oplus
V_{y}}$

where $i=1$ (resp. $i=2$) and the right vertical arrow uses the natural stable
isomorphism $-\mathbb{R}^{k}\oplus\mathbb{R}^{k}\to 0$.

Note that this definition implies that the following diagrams commute. Over a
broken face $\mathcal{S}_{xx^{\prime};y}\subseteq\mathcal{S}_{xy}$:

$\textstyle{V_{x}\oplus I^{\mathcal{F}}_{xx^{\prime}}\oplus
I^{N^{\mathcal{S}}}_{P;x^{\prime}y}\oplus\mathbb{R}\sigma_{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{st^{\mathcal{F}}}$$\scriptstyle{\psi}$$\textstyle{V_{x^{\prime}}\oplus
I^{N^{\mathcal{S}}}_{P;x^{\prime}y}\oplus\mathbb{R}\sigma_{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{st^{N^{\mathcal{S}}}}$$\textstyle{V_{x}\oplus
I^{N^{\mathcal{S}}}_{P;xy}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{st^{N^{\mathcal{S}}}}$$\textstyle{\mathbb{R}^{i}\oplus
V_{y}-\mathbb{R}^{k}}$

(where $i=1$ (resp. $i=2$) as well as a similar diagram over each
$\mathcal{S}_{x;y^{\prime}y}$.

If $\mathcal{S}$ is a bordism, then over an unbroken face
$\mathcal{W}_{xy}\subseteq\mathcal{S}_{xy}$:

$\textstyle{V_{x}\oplus