$\mathcal{N}^{n,n-1,\mathrm{DIII}}_{\mathrm{Gapless}}(H)=\operatorname{\mathrm{sf_{2}}}(H^{c})\in\mathbb{Z}_{2}$.

  * •

In class $\mathrm{AII}$, let
$\mathcal{N}^{n,n-1,\mathrm{AII}}_{\mathrm{Gapless}}(H)=\operatorname{\mathrm{sf_{2}}}(H^{c})\in\mathbb{Z}_{2}$.

  * •

In class $\mathrm{C}$, let
$\mathcal{N}^{n,n-1,\mathrm{C}}_{\mathrm{Gapless}}(H)=\operatorname{\mathrm{sf}}(H^{c})\in
2\mathbb{Z}$.

For each of the above classes $\spadesuit$, the numerical invariant
$\mathcal{N}^{n,n-1,\spadesuit}_{\mathrm{Gapless}}(H)$ is the image of the
gapless corner invariant
$\mathcal{I}^{n,n-1,\spadesuit}_{\mathrm{Gapless}}(H)$ through the map
$\operatorname{\mathrm{sf}}^{\spadesuit}$. These numerical invariants account
for strong invariants.

######  Remark 5.26.

In Definition 5.25, the numerical corner invariants for both class
$\mathrm{DIII}$ and class $\mathrm{AII}$ are defined by using
$\mathbb{Z}_{2}$-valued spectral flow, though these two $\mathbb{Z}_{2}$ are
different from the viewpoint of index theory in the sense that they sit in
different Bott clock. A similar remark holds for, e.g., cases of $n=k$ in
classes $\mathrm{BDI}$ and $\mathrm{CII}$, where both of these numerical
corner invariants are defined as Fredholm indices.

###  5.7. Product Formula

In Sect. $4$ of [32], a construction of the second-order topological
insulators of $3$-D class A systems is proposed, which is given by using the
Hamiltonians of $2$-D class A and $1$-D class AIII topological insulators. In
this subsection, we generalize this construction to other pairs in the
Altland–Zirnbauer classification. This provides a way to construct nontrivial
examples of each entry in Table 1 from the Hamiltonians of (first-order)
topological insulators131313For the case of $k=2$, the construction is
restricted to $\alpha=0$ and $\beta=\infty$ case.. For this purpose, we use an
exterior product of topological $KR$-groups [6].

For $j=1,2$, let $H_{j}$ be a bulk Hamiltonian of an $n_{j}$-D $k_{j}$-th
order topological insulator141414that in Sect. 5.1 satisfying Assumption 5.1
when $k_{j}=2$ or that in Sect. 5.5 satisfying Assumption 5.9 when $k_{j}\geq
3$. When $k_{j}=1$, the bulk Hamiltonian is assumed to be gapped. When
$k_{j}=2$, we consider the case of $\alpha=0$ and $\beta=\infty$. in real
Altland–Zirnbauer class $\spadesuit_{j}$ (AI, BDI, D, DIII, AII, CII, C or
CI). Let $n=n_{1}+n_{2}$, $k=k_{1}+k_{2}$ and $d_{j}=n_{j}-k_{j}$ for $j=1,2$,
and let $d=d_{1}+d_{2}$. Corresponding to the class in the Altland–Zirnbauer
classification (for which we write $\spadesuit_{j}$) to which the Hamiltonian
belongs, it preserves the symmetries as (even/odd) TRS, (even/odd) PHS or
chiral symmetry. We write $\Theta_{j}$, $\Xi_{j}$ and $\Pi_{j}$ for the
symmetry operator for $H_{j}$. As in Sect. 5, the models of corners
$H_{i}^{c}$ lead to a continuous family of self-adjoint or skew-adjoint
Fredholm operators and defines an element of the $KO$-group
$KO_{i^{\prime}(\spadesuit_{j})}(C(\mathbb{T}^{d_{j}}),\tau)$ where
$i^{\prime}(\spadesuit_{j})=i(\spadesuit_{j})-1$. As in Appendix A.1, we have
an exterior product of $KO$-groups

$KO_{i^{\prime}(\spadesuit_{1})}(C(\mathbb{T}^{d_{1}}),\tau{T})\times
KO_{i^{\prime}(\spadesuit_{2})}(C(\mathbb{T}^{d_{2}}),\tau{T})\to
KO_{i^{\prime}(\spadesuit_{1})+i^{\prime}(\spadesuit_{2})}(C(\mathbb{T}^{d}),\tau{T}),$

described through these Fredholm operators. By using this form of the product,
we obtain an explicit form of the product of the gapless invariants of $H_{1}$
and $H_{2}$. As a result, we can write down a bulk Hamiltonian $H$ of an $n$-D
$k$-th order topological insulator of class $\spadesuit$. The lattice on which
we consider $H^{c}$ as a model of the codimension-$k$ corner is introduced as
the product of that151515When $k_{j}=1$, the lattice is $\mathbb{Z}_{\geq
0}\times\mathbb{Z}^{d_{j}}$, where $H^{c}_{j}$ is the compression of the bulk
Hamiltonian onto this half-space. Topological invariants for them are the one
discussed in topological insulators. To clarify our sign choices, we mention
that they are obtained by applying the discussion in Sect. 5 to the Toeplitz
extension (4.1) in place of (2.2) or (4.2). of $H^{c}_{1}$ and that of
$H^{c}_{2}$. By this construction, we have the following relation between
gapless invariants.

######  Theorem 5.27.

For the Hamiltonian $H$ indicated in Table LABEL:producttalbe, we have

$\mathcal{I}_{\mathrm{Gapless}}^{n_{1},k_{1},\spadesuit_{1}}(H_{1})\cdot\mathcal{I}_{\mathrm{Gapless}}^{n_{2},k_{2},\spadesuit_{2}}(H_{2})=\mathcal{I}_{\mathrm{Gapless}}^{n,k,\spadesuit}(H),$

where $\cdot$ denotes the exterior product of elements of the $KO$-groups.

Note that Theorem 5.27 is the product formula at the level of $KO$-group
elements and accounts for both strong and weak invariants. In order to show
this theorem, we need to write down the explicit form of $H$. In the
following, we discuss them for some classes.

Let us consider the case where $\spadesuit_{1}=\mathrm{BDI}$ and
$\spadesuit_{2}=\mathrm{BDI}$. In this case, each $H_{j}$ has even TRS, even
PHS and chiral symmetry. We now consider the following $n$-dimensional
Hamiltonian:

(5.7) $H=H_{1}\otimes 1+\Pi_{1}\otimes H_{2},$

which satisfies even TRS given by $\Theta=\Theta_{1}\otimes\Theta_{2}$, even
PHS given by $\Xi=\Xi_{1}\otimes\Xi_{2}$ and the chiral symmetry given by
$\Pi=\Pi_{1}\otimes\Pi_{2}$. Thus, the Hamiltonian $H$ belongs to the class
$\spadesuit=\mathrm{BDI}$. The model of the codimension-$k$ corner $H^{c}$ of
$H$ is written by using the model $H^{c}_{j}$ of the codimension-$k_{j}$
corner as follows:

$H^{c}({\bm{t}}_{1},{\bm{t}}_{2})=H^{c}_{1}({\bm{t}}_{1})\otimes
1+\Pi_{1}\otimes H^{c}_{2}({\bm{t}}_{2}),$

where ${\bm{t}}_{j}$ is an element of the $d_{j}$-dimensional torus (momentum
space) corresponding to a direction parallel to the corner of $H^{c}_{j}$ for
$j=1,2$. Note that $({\bm{t}}_{1},{\bm{t}}_{2})$ constitute the parameter of
the $d$-dimensional momentum space in a direction parallel to the corner of
$H^{c}$. By our assumption, $H^{c}_{j}({\bm{t}}_{j})$ is an element of the
space $\mathrm{Fred}^{(0,\underline{2})}_{*}$ and gives a $\mathbb{Z}_{2}$-map
$(\mathbb{T}^{d_{j}},\zeta)\to(\mathrm{Fred}^{(0,\underline{2})}_{*},\mathfrak{r}_{\Theta_{j}})$.
The operator $H^{c}({\bm{t}}_{1},{\bm{t}}_{2})$ is the image of the pair
$(H^{c}_{1},H^{c}_{2})$ through the map,

$(\mathrm{Fred}^{(0,\underline{2})}_{*},\mathfrak{r}_{\Theta_{1}})\times(\mathrm{Fred}^{(0,\underline{2})}_{*},\mathfrak{r}_{\Theta_{2}})\to(\mathrm{Fred}^{(0,\underline{2})}_{*},\mathfrak{r}_{\Theta}),$

in (A.2), where the action of $\mathit{Cl}_{0,1}$ to define the left-hand side
is given by $\epsilon_{j}=\Pi_{j}$ and that for the right-hand side is given
by $\epsilon=\epsilon_{1}\otimes\epsilon_{2}=\Pi$. Since this map induces the
exterior product of $KO$-groups (Appendix A.1),

$KO_{0}(C(\mathbb{T}^{d_{1}}),\tau{T})\times
KO_{0}(C(\mathbb{T}^{d_{2}}),\tau{T})\to KO_{0}(C(\mathbb{T}^{d}),\tau{T}),$

we obtain Theorem 5.27 in this case.

We next consider the case where $\spadesuit_{1}=\mathrm{DIII}$ and
$\spadesuit_{2}=\mathrm{D}$. In this case, $H_{1}$ has odd TRS, even PHS and
the chiral symmetry, and $H_{2}$ has even PHS. As in Sect. 5.3,
$H^{c}_{1}({\bm{t}}_{1})$ belongs to
$(\mathrm{Fred}^{(1,\underline{1})}_{*},\mathfrak{q}_{\Theta_{1}})$, and
$iH^{c}_{2}({\bm{t}}_{2})$ belongs to
$(\mathrm{Fred}^{(\underline{1},0)}_{*},\mathfrak{r}_{\Xi_{2}})$. By using
Proposition A.4, we identify
$(\mathrm{Fred}^{(1,\underline{1})}_{*},\mathfrak{q}_{\Theta_{1}})$ with
$(\mathrm{Fred}^{(2,\underline{2})}_{*},\mathfrak{q}_{\Theta_{1}\oplus\Theta_{1}})$
and $(\mathrm{Fred}^{(\underline{1},0)}_{*},\mathfrak{r}_{\Xi_{2}})$ with
$(\mathrm{Fred}^{(0,\underline{3})}_{*},\mathfrak{r}_{\Xi_{2}\oplus\Xi_{2}})$.
We then use the map (A.2) of the form

$(\mathrm{Fred}^{(2,\underline{2})}_{*},\mathfrak{q}_{\Theta_{1}\oplus\Theta_{1}})\times(\mathrm{Fred}^{(0,\underline{3})}_{*},\mathfrak{r}_{\Xi_{2}\oplus\Xi_{2}})\to(\mathrm{Fred}^{(2,\underline{3})}_{*},\mathfrak{q}^{\prime}),$

where $\mathfrak{q}^{\prime}$ is the conjugation of the fourfold direct sum of
$\Theta_{1}\otimes\Xi_{2}$. By Proposition A.4, we have the
$\mathbb{Z}_{2}$-homeomorphism
$(\mathrm{Fred}^{(2,\underline{3})}_{*},\mathfrak{q}^{\prime})\cong(\mathrm{Fred}^{(0,\underline{1})}_{*},\mathfrak{q}_{\Theta_{1}\otimes\Xi_{2}})$.
Thus, we obtain a $\mathbb{Z}_{2}$-map
$H^{c}\colon(\mathbb{T}^{d},\zeta)\to(\mathrm{Fred}^{(0,\underline{1})}_{*},\mathfrak{q}_{\Theta_{1}\otimes\Xi_{2}})$
from $H_{1}^{c}$ and $H_{2}^{c}$ which is a model for the codimension-$k$
corner in class $\spadesuit=\mathrm{AII}$. Its bulk Hamiltonian $H$ and (odd)
TRS operator $\Theta$ is expressed as (5.7) and
$\Theta=\Theta_{1}\otimes\Xi_{2}$, respectively. Note that
$\Pi_{1}=i\Theta_{1}\Xi_{1}$ in class DIII and $\Theta$ commutes with $H$.
Since the map (A.2) induces the exterior product of $KO$-groups, Theorem 5.27
holds for this class AII Hamiltonian $H$.

The other cases are computed in a similar way, and the results are summarized
in Table LABEL:producttalbe, where we write

$H_{\bigstar}=\left(\begin{array}[]{cc}0&H_{1}\otimes 1-i\otimes H_{2}\\\
H_{1}\otimes 1+i\otimes H_{2}&0\end{array}\right),\
\Theta_{\clubsuit}=\left(\begin{array}[]{cc}\Theta_{1}\otimes\Xi_{2}&0\\\
0&\Theta_{1}\otimes\Xi_{2}\end{array}\right),$
$H_{\Box}=\left(\begin{array}[]{cc}0&-H_{1}\otimes i-1\otimes H_{2}\\\
H_{1}\otimes i-1\otimes
H_{2}&0\end{array}\right),\Theta_{\triangle}=\left(\begin{array}[]{cc}\Xi_{1}\otimes\Theta_{2}&0\\\
0&\Xi_{1}\otimes\Theta_{2}\end{array}\right),$
$\Theta_{\diamondsuit}=\left(\begin{array}[]{cc}0&-\Xi_{1}\otimes\Xi_{2}\\\
\Xi_{1}\otimes\Xi_{2}&0\end{array}\right)\ \ \text{and}\ \
\Theta_{\heartsuit}=\left(\begin{array}[]{cc}0&\Theta_{1}\otimes\Theta_{2}\\\
\Theta_{1}\otimes\Theta_{2}&0\end{array}\right).$

Table 12. The forms of the Hamiltonians and symmetry operators in class $\spadesuit$ constructed from two pairs of Hamiltonians and symmetry operators in classes $\spadesuit_{1}$ and $\spadesuit_{2}$. Complex cases are also included [32, 33]. $\spadesuit_{1}$ | $\spadesuit_{2}$ | $\spadesuit$ | Hamiltonian ($H$) | TRS ($\Theta$) | PHS ($\Xi$) | Chiral ($\Pi$)
---|---|---|---|---|---|---
$\mathrm{AI}$ | $\mathrm{AI}$ | $\mathrm{CI}$ | $H_{\bigstar}$ | $\Theta_{\heartsuit}$ | $i\Theta\Pi$ | $\mathrm{diag}(1,-1)$
$\mathrm{AI}$ | $\mathrm{BDI}$ | $\mathrm{AI}$ | $H_{1}\otimes\Pi_{2}+1\otimes H_{2}$ | $\Theta_{1}\otimes\Theta_{2}$ | — | —
$\mathrm{AI}$ | $\mathrm{D}$ | $\mathrm{BDI}$ | $H_{\bigstar}$ | $\Theta_{\clubsuit}$ | $\Xi=\Theta\Pi$ | $\mathrm{diag}(1,-1)$
$\mathrm{AI}$ | $\mathrm{DIII}$ | $\mathrm{D}$ | $H_{1}\otimes\Pi_{2}+1\otimes H_{2}$ | — | $\Theta_{1}\otimes\Theta_{2}\Pi_{2}$ | —
$\mathrm{AI}$ | $\mathrm{AII}$ | $\mathrm{DIII}$ | $H_{\bigstar}$ | $\Theta_{\heartsuit}$ | $i\Pi\Theta$ | $\mathrm{diag}(1,-1)$
$\mathrm{AI}$ | $\mathrm{CII}$ | $\mathrm{AII}$ | $H_{1}\otimes\Pi_{2}+1\otimes H_{2}$ | $\Theta_{1}\otimes\Theta_{2}$ | — | —
$\mathrm{AI}$ | $\mathrm{C}$ | $\mathrm{CII}$ | $H_{\bigstar}$ | $\Theta_{\clubsuit}$ | $\Xi=-\Theta\Pi$ | $\mathrm{diag}(1,-1)$
$\mathrm{AI}$ | $\mathrm{CI}$ | $\mathrm{C}$ | $H_{1}\otimes\Pi_{2}+1\otimes H_{2}$ | — | $\Theta_{1}\otimes\Theta_{2}\Pi_{2}$ | —
$\mathrm{BDI}$ | $\mathrm{AI}$ | $\mathrm{AI}$ | $H_{1}\otimes 1+\Pi_{1}\otimes H_{2}$ | $\Theta_{1}\otimes\Theta_{2}$ | — | —
$\mathrm{BDI}$ | $\mathrm{BDI}$ | $\mathrm{BDI}$ | $H_{1}\otimes 1+\Pi_{1}\otimes H_{2}$ | $\Theta_{1}\otimes\Theta_{2}$ | $\Xi_{1}\otimes\Xi_{2}$ | $\Pi_{1}\otimes\Pi_{2}$
$\mathrm{BDI}$ | $\mathrm{D}$ | $\mathrm{D}$ | $H_{1}\otimes 1+\Pi_{1}\otimes H_{2}$ | — | $\Xi_{1}\otimes\Xi_{2}$ | —
$\mathrm{BDI}$ | $\mathrm{DIII}$ | $\mathrm{DIII}$ | $H_{1}\otimes 1+\Pi_{1}\otimes H_{2}$ | $\Theta_{1}\otimes\Theta_{2}$ | $\Xi_{1}\otimes\Xi_{2}$ | $\Pi_{1}\otimes\Pi_{2}$
$\mathrm{BDI}$ | $\mathrm{AII}$ | $\mathrm{AII}$ | $H_{1}\otimes 1+\Pi_{1}\otimes H_{2}$ | $\Theta_{1}\otimes\Theta_{2}$ | — | —
$\mathrm{BDI}$ | $\mathrm{CII}$ | $\mathrm{CII}$ | $H_{1}\otimes 1+\Pi_{1}\otimes H_{2}$ | $\Theta_{1}\otimes\Theta_{2}$ | $\Xi_{1}\otimes\Xi_{2}$ | $\Pi_{1}\otimes\Pi_{2}$
$\mathrm{BDI}$ | $\mathrm{C}$ | $\mathrm{C}$ | $H_{1}\otimes 1+\Pi_{1}\otimes H_{2}$ | — | $\Xi_{1}\otimes\Xi_{2}$ | —
$\mathrm{BDI}$ | $\mathrm{CI}$ | $\mathrm{CI}$ | $H_{1}\otimes 1+\Pi_{1}\otimes H_{2}$ | $\Theta_{1}\otimes\Theta_{2}$ | $\Xi_{1}\otimes\Xi_{2}$ | $\Pi_{1}\otimes\Pi_{2}$
$\mathrm{D}$ | $\mathrm{AI}$ | $\mathrm{BDI}$ | $H_{\Box}$ | $\Theta_{\triangle}$ | $\Xi=\Theta\Pi$ | $\mathrm{diag}(1,-1)$
$\mathrm{D}$ | $\mathrm{BDI}$ | $\mathrm{D}$ | $H_{1}\otimes\Pi_{2}+1\otimes H_{2}$ | — | $\Xi_{1}\otimes\Xi_{2}$ | —
$\mathrm{D}$ | $\mathrm{D}$ | $\mathrm{DIII}$ | $H_{\bigstar}$ | $\Theta_{\diamondsuit}$ | $i\Pi\Theta$ | $\mathrm{diag}(1,-1)$
$\mathrm{D}$ | $\mathrm{DIII}$ | $\mathrm{AII}$ | $H_{1}\otimes\Pi_{2}+1\otimes H_{2}$ | $\Xi_{1}\otimes\Theta_{2}$ | — | —
$\mathrm{D}$ | $\mathrm{AII}$ | $\mathrm{CII}$ | $H_{\Box}$ | $\Theta_{\triangle}$ | $\Xi=-\Theta\Pi$ | $\mathrm{diag}(1,-1)$
$\mathrm{D}$ | $\mathrm{CII}$ | $\mathrm{C}$ | $H_{1}\otimes\Pi_{2}+1\otimes H_{2}$ | — | $\Xi_{1}\otimes\Xi_{2}$ | —
$\mathrm{D}$ | $\mathrm{C}$ | $\mathrm{CI}$ | $H_{\bigstar}$ | $\Theta_{\diamondsuit}$ | $i\Theta\Pi$ | $\mathrm{diag}(1,-1)$
$\mathrm{D}$ | $\mathrm{CI}$ | $\mathrm{AI}$ | $H_{1}\otimes\Pi_{2}+1\otimes H_{2}$ | $\Xi_{1}\otimes\Theta_{2}$ | — | —
$\mathrm{DIII}$ | $\mathrm{AI}$ | $\mathrm{D}$ | $H_{1}\otimes 1+\Pi_{1}\otimes H_{2}$ | — | $\Theta_{1}\Pi_{1}\otimes\Theta_{2}$ | —
$\mathrm{DIII}$ | $\mathrm{BDI}$ | $\mathrm{DIII}$ | $H_{1}\otimes\Pi_{2}+1\otimes H_{2}$ | $\Theta_{1}\otimes\Theta_{2}$ | $\Xi_{1}\otimes\Xi_{2}$ | $\Pi_{1}\otimes\Pi_{2}$
$\mathrm{DIII}$ | $\mathrm{D}$ | $\mathrm{AII}$ | $H_{1}\otimes 1+\Pi_{1}\otimes H_{2}$ | $\Theta_{1}\otimes\Xi_{2}$ | — | —
$\mathrm{DIII}$ | $\mathrm{DIII}$ | $\mathrm{CII}$ | $H_{1}\otimes 1+\Pi_{1}\otimes H_{2}$ | $\Theta_{1}\otimes\Theta_{2}\Pi_{2}$ | $\Pi_{1}\Theta_{1}\otimes\Theta_{2}$ | $\Pi_{1}\otimes\Pi_{2}$
$\mathrm{DIII}$ | $\mathrm{AII}$ | $\mathrm{C}$ | $H_{1}\otimes 1+\Pi_{1}\otimes H_{2}$ | — | $\Theta_{1}\Pi_{1}\otimes\Theta_{2}$ | —
$\mathrm{DIII}$ | $\mathrm{CII}$ | $\mathrm{CI}$ | $H_{1}\otimes\Pi_{2}+1\otimes H_{2}$ | $\Theta_{1}\otimes\Theta_{2}$ | $\Xi_{1}\otimes\Xi_{2}$ | $\Pi_{1}\otimes\Pi_{2}$
$\mathrm{DIII}$ | $\mathrm{C}$ | $\mathrm{AI}$ | $H_{1}\otimes 1+\Pi_{1}\otimes H_{2}$ | $\Theta_{1}\otimes\Xi_{2}$ | — | —
$\mathrm{DIII}$ | $\mathrm{CI}$ | $\mathrm{BDI}$ | $H_{1}\otimes 1+\Pi_{1}\otimes H_{2}$ | $\Theta_{1}\otimes\Theta_{2}\Pi_{2}$ | $\Theta_{1}\Pi_{1}\otimes\Theta_{2}$ | $\Pi_{1}\otimes\Pi_{2}$
$\mathrm{AII}$ | $\mathrm{AI}$ | $\mathrm{DIII}$ | $H_{\bigstar}$ | $\Theta_{\heartsuit}$ | $i\Pi\Theta$ | $\mathrm{diag}(1,-1)$
$\mathrm{AII}$ | $\mathrm{BDI}$ | $\mathrm{AII}$ | $H_{1}\otimes\Pi_{2}+1\otimes H_{2}$ | $\Theta_{1}\otimes\Theta_{2}$ | — | —
$\mathrm{AII}$ | $\mathrm{D}$ | $\mathrm{CII}$ | $H_{\bigstar}$ | $\Theta_{\clubsuit}$ | $\Xi=-\Theta\Pi$ | $\mathrm{diag}(1,-1)$
$\mathrm{AII}$ | $\mathrm{DIII}$ | $\mathrm{C}$ | $H_{1}\otimes\Pi_{2}+1\otimes H_{2}$ | — | $\Theta_{1}\otimes\Theta_{2}\Pi_{2}$ | —
$\mathrm{AII}$ | $\mathrm{AII}$ | $\mathrm{CI}$ | $H_{\bigstar}$ | $\Theta_{\heartsuit}$ | $i\Theta\Pi$ | $\mathrm{diag}(1,-1)$
$\mathrm{AII}$ | $\mathrm{CII}$ | $\mathrm{AI}$ | $H_{1}\otimes\Pi_{2}+1\otimes H_{2}$ | $\Theta_{1}\otimes\Theta_{2}$ | — | —
$\mathrm{AII}$ | $\mathrm{C}$ | $\mathrm{BDI}$ | $H_{\bigstar}$ | $\Theta_{\clubsuit}$ | $\Xi=\Theta\Pi$ | $\mathrm{diag}(1,-1)$
$\mathrm{AII}$ | $\mathrm{CI}$ | $\mathrm{D}$ | $H_{1}\otimes\Pi_{2}+1\otimes H_{2}$ | — | $\Theta_{1}\otimes\Theta_{2}\Pi_{2}$ | —
$\mathrm{CII}$ | $\mathrm{AI}$ | $\mathrm{AII}$ | $H_{1}\otimes 1+\Pi_{1}\otimes H_{2}$ | $\Theta_{1}\otimes\Theta_{2}$ | — | —
$\mathrm{CII}$ | $\mathrm{BDI}$ | $\mathrm{CII}$ | $H_{1}\otimes 1+\Pi_{1}\otimes H_{2}$ | $\Theta_{1}\otimes\Theta_{2}$ | $\Xi_{1}\otimes\Xi_{2}$ | $\Pi_{1}\otimes\Pi_{2}$
$\mathrm{CII}$ | $\mathrm{D}$ | $\mathrm{C}$ | $H_{1}\otimes 1+\Pi_{1}\otimes H_{2}$ | — | $\Xi_{1}\otimes\Xi_{2}$ | —
$\mathrm{CII}$ | $\mathrm{DIII}$ | $\mathrm{CI}$ | $H_{1}\otimes 1+\Pi_{1}\otimes H_{2}$ | $\Theta_{1}\otimes\Theta_{2}$ | $\Xi_{1}\otimes\Xi_{2}$ | $\Pi_{1}\otimes\Pi_{2}$
$\mathrm{CII}$ | $\mathrm{AII}$ | $\mathrm{AI}$ | $H_{1}\otimes 1+\Pi_{1}\otimes H_{2}$ | $\Theta_{1}\otimes\Theta_{2}$ | — | —
$\mathrm{CII}$ | $\mathrm{CII}$ | $\mathrm{BDI}$ | $H_{1}\otimes 1+\Pi_{1}\otimes H_{2}$ | $\Theta_{1}\otimes\Theta_{2}$ | $\Xi_{1}\otimes\Xi_{2}$ | $\Pi_{1}\otimes\Pi_{2}$
$\mathrm{CII}$ | $\mathrm{C}$ | $\mathrm{D}$ | $H_{1}\otimes 1+\Pi_{1}\otimes H_{2}$ | — | $\Xi_{1}\otimes\Xi_{2}$ | —
$\mathrm{CII}$ | $\mathrm{CI}$ | $\mathrm{DIII}$ | $H_{1}\otimes 1+\Pi_{1}\otimes H_{2}$ | $\Theta_{1}\otimes\Theta_{2}$ | $\Xi_{1}\otimes\Xi_{2}$ | $\Pi_{1}\otimes\Pi_{2}$
$\mathrm{C}$ | $\mathrm{AI}$ | $\mathrm{CII}$ | $H_{\Box}$ | $\Theta_{\triangle}$ | $\Xi=-\Theta\Pi$ | $\mathrm{diag}(1,-1)$
$\mathrm{C}$ | $\mathrm{BDI}$ | $\mathrm{C}$ | $H_{1}\otimes\Pi_{2}+1\otimes H_{2}$ | — | $\Xi_{1}\otimes\Xi_{2}$ | —
$\mathrm{C}$ | $\mathrm{D}$ | $\mathrm{CI}$ | $H_{\bigstar}$ | $\Theta_{\diamondsuit}$ | $i\Theta\Pi$ | $\mathrm{diag}(1,-1)$
$\mathrm{C}$ | $\mathrm{DIII}$ | $\mathrm{AI}$ | $H_{1}\otimes\Pi_{2}+1\otimes H_{2}$ | $\Xi_{1}\otimes\Theta_{2}$ | — | —
$\mathrm{C}$ | $\mathrm{AII}$ | $\mathrm{BDI}$ | $H_{\Box}$ | $\Theta_{\triangle}$ | $\Xi=\Theta\Pi$ | $\mathrm{diag}(1,-1)$
$\mathrm{C}$ | $\mathrm{CII}$ | $\mathrm{D}$ | $H_{1}\otimes\Pi_{2}+1\otimes H_{2}$ | — | $\Xi_{1}\otimes\Xi_{2}$ | —
$\mathrm{C}$ | $\mathrm{C}$ | $\mathrm{DIII}$ | $H_{\bigstar}$ | $\Theta_{\diamondsuit}$ | $i\Pi\Theta$ | $\mathrm{diag}(1,-1)$
$\mathrm{C}$ | $\mathrm{CI}$ | $\mathrm{AII}$ | $H_{1}\otimes\Pi_{2}+1\otimes H_{2}$ | $\Xi_{1}\otimes\Theta_{2}$ | — | —
$\mathrm{CI}$ | $\mathrm{AI}$ | $\mathrm{C}$ | $H_{1}\otimes 1+\Pi_{1}\otimes H_{2}$ | — | $\Theta_{1}\Pi_{1}\otimes\Theta_{2}$ | —
$\mathrm{CI}$ | $\mathrm{BDI}$ | $\mathrm{CI}$ | $H_{1}\otimes\Pi_{2}+1\otimes H_{2}$ | $\Theta_{1}\otimes\Theta_{2}$ | $\Xi_{1}\otimes\Xi_{2}$ | $\Pi_{1}\otimes\Pi_{2}$
$\mathrm{CI}$ | $\mathrm{D}$ | $\mathrm{AI}$ | $H_{1}\otimes 1+\Pi_{1}\otimes H_{2}$ | $\Theta_{1}\otimes\Xi_{2}$ | — | —
$\mathrm{CI}$ | $\mathrm{DIII}$ | $\mathrm{BDI}$ | $H_{1}\otimes 1+\Pi_{1}\otimes H_{2}$ | $\Theta_{1}\otimes\Theta_{2}\Pi_{2}$ | $\Theta_{1}\Pi_{1}\otimes\Theta_{2}$ | $\Pi_{1}\otimes\Pi_{2}$
$\mathrm{CI}$ | $\mathrm{AII}$ | $\mathrm{D}$ | $H_{1}\otimes 1+\Pi_{1}\otimes H_{2}$ | — | $\Theta_{1}\Pi_{1}\otimes\Theta_{2}$ | —
$\mathrm{CI}$ | $\mathrm{CII}$ | $\mathrm{DIII}$ | $H_{1}\otimes\Pi_{2}+1\otimes H_{2}$ | $\Theta_{1}\otimes\Theta_{2}$ | $\Xi_{1}\otimes\Xi_{2}$ | $\Pi_{1}\otimes\Pi_{2}$
$\mathrm{CI}$ | $\mathrm{C}$ | $\mathrm{AII}$ | $H_{1}\otimes 1+\Pi_{1}\otimes H_{2}$ | $\Theta_{1}\otimes\Xi_{2}$ | — | —
$\mathrm{CI}$ | $\mathrm{CI}$ | $\mathrm{CII}$ | $H_{1}\otimes 1+\Pi_{1}\otimes H_{2}$ | $\Theta_{1}\otimes\Theta_{2}\Pi_{2}$ | $\Pi_{1}\Theta_{1}\otimes\Theta_{2}$ | $\Pi_{1}\otimes\Pi_{2}$
$\mathrm{A}$ | $\mathrm{A}$ | $\mathrm{AIII}$ | $H_{\bigstar}$ | — | — | $\mathrm{diag}(1,-1)$
$\mathrm{A}$ | $\mathrm{AIII}$ | $\mathrm{A}$ | $H_{1}\otimes\Pi_{2}+1\otimes H_{2}$ | — | — | —
$\mathrm{AIII}$ | $\mathrm{A}$ | $\mathrm{A}$ | $H_{1}\otimes 1+\Pi_{1}\otimes H_{2}$ | — | — | —
$\mathrm{AIII}$ | $\mathrm{AIII}$ | $\mathrm{AIII}$ | $H_{1}\otimes 1+\Pi_{1}\otimes H_{2}$ | — | — | $\Pi_{1}\otimes\Pi_{2}$

Our product formula (Theorem 5.27) and the graded ring structure of
$KO_{*}(\mathbb{C},\mathrm{id})$ (Theorem 6.9 of [7]) lead to the following
product formula for numerical corner invariants. We collect the results here
where the form of $H$ is as indicated in Table LABEL:producttalbe.

######  Corollary 5.28 (Cases of $n=k$).

The case of $k_{1}=n_{1}$ and $k_{2}=n_{2}$.

  * •

$\mathrm{BDI}\times\mathrm{BDI}\to\mathrm{BDI}$,
$\mathcal{N}^{n_{1},n_{1},\mathrm{BDI}}_{\mathrm{Gapless}}(H_{1})\cdot\mathcal{N}^{n_{2},n_{2},\mathrm{BDI}}_{\mathrm{Gapless}}(H_{2})=\mathcal{N}^{n,n,\mathrm{BDI}}_{\mathrm{Gapless}}(H)$.

  * •

$\mathrm{BDI}\times\mathrm{D}\to\mathrm{D}$,
$(\mathcal{N}^{n_{1},n_{1},\mathrm{BDI}}_{\mathrm{Gapless}}(H_{1})\bmod
2)\cdot\mathcal{N}^{n_{2},n_{2},\mathrm{D}}_{\mathrm{Gapless}}(H_{2})=\mathcal{N}^{n,n,\mathrm{D}}_{\mathrm{Gapless}}(H)$.

  * •

$\mathrm{BDI}\times\mathrm{DIII}\to\mathrm{DIII}$,
$(\mathcal{N}^{n_{1},n_{1},\mathrm{BDI}}_{\mathrm{Gapless}}(H_{1})\bmod
2)\cdot\mathcal{N}^{n_{2},n_{2},\mathrm{DIII}}_{\mathrm{Gapless}}(H_{2})=\mathcal{N}^{n,n,\mathrm{DIII}}_{\mathrm{Gapless}}(H)$.

  * •

$\mathrm{BDI}\times\mathrm{CII}\to\mathrm{CII}$,
$\mathcal{N}^{n_{1},n_{1},\mathrm{BDI}}_{\mathrm{Gapless}}(H_{1})\cdot\mathcal{N}^{n_{2},n_{2},\mathrm{CII}}_{\mathrm{Gapless}}(H_{2})=\mathcal{N}^{n,n,\mathrm{CII}}_{\mathrm{Gapless}}(H)$.

  * •

$\mathrm{D}\times\mathrm{D}\to\mathrm{DIII}$,
$\mathcal{N}^{n_{1},n_{1},\mathrm{D}}_{\mathrm{Gapless}}(H_{1})\cdot\mathcal{N}^{n_{2},n_{2},\mathrm{D}}_{\mathrm{Gapless}}(H_{2})=\mathcal{N}^{n,n,\mathrm{DIII}}_{\mathrm{Gapless}}(H)$.

  * •

$\mathrm{CII}\times\mathrm{CII}\to\mathrm{BDI}$,
$\mathcal{N}^{n_{1},n_{1},\mathrm{CII}}_{\mathrm{Gapless}}(H_{1})\cdot\mathcal{N}^{n_{2},n_{2},\mathrm{CII}}_{\mathrm{Gapless}}(H_{2})=\mathcal{N}^{n,n,\mathrm{BDI}}_{\mathrm{Gapless}}(H)$.

######  Corollary 5.29 (Cases of $n=k-1$).

The case of $k_{1}=n_{1}$ and $k_{2}=n_{2}-1$.

  * •

$\mathrm{BDI}\times\mathrm{D}\to\mathrm{D}$,
$\mathcal{N}^{n_{1},n_{1},\mathrm{BDI}}_{\mathrm{Gapless}}(H_{1})\cdot\mathcal{N}^{n_{2},n_{2}-1,\mathrm{D}}_{\mathrm{Gapless}}(H_{2})=\mathcal{N}^{n,n-1,\mathrm{D}}_{\mathrm{Gapless}}(H)$.

  * •

$\mathrm{BDI}\times\mathrm{DIII}\to\mathrm{DIII}$,
$(\mathcal{N}^{n_{1},n_{1},\mathrm{BDI}}_{\mathrm{Gapless}}(H_{1})\bmod
2)\cdot\mathcal{N}^{n_{2},n_{2}-1,\mathrm{DIII}}_{\mathrm{Gapless}}(H_{2})=\mathcal{N}^{n,n-1,\mathrm{DIII}}_{\mathrm{Gapless}}(H)$.

  * •

$\mathrm{BDI}\times\mathrm{AII}\to\mathrm{AII}$,
$(\mathcal{N}^{n_{1},n_{1},\mathrm{BDI}}_{\mathrm{Gapless}}(H_{1})\bmod
2)\cdot\mathcal{N}^{n_{2},n_{2}-1,\mathrm{AII}}_{\mathrm{Gapless}}(H_{2})=\mathcal{N}^{n,n-1,\mathrm{AII}}_{\mathrm{Gapless}}(H)$.

  * •

$\mathrm{BDI}\times\mathrm{C}\to\mathrm{C}$,
$\mathcal{N}^{n_{1},n_{1},\mathrm{BDI}}_{\mathrm{Gapless}}(H_{1})\cdot\mathcal{N}^{n_{2},n_{2}-1,\mathrm{C}}_{\mathrm{Gapless}}(H_{2})=\mathcal{N}^{n,n-1,\mathrm{C}}_{\mathrm{Gapless}}(H)$.

  * •

$\mathrm{D}\times\mathrm{D}\to\mathrm{DIII}$,
$\mathcal{N}^{n_{1},n_{1},\mathrm{D}}_{\mathrm{Gapless}}(H_{1})\cdot(\mathcal{N}^{n_{2},n_{2}-1,\mathrm{D}}_{\mathrm{Gapless}}(H_{2})\bmod
2)=\mathcal{N}^{n,n-1,\mathrm{DIII}}_{\mathrm{Gapless}}(H)$.

  * •

$\mathrm{D}\times\mathrm{DIII}\to\mathrm{AII}$,
$\mathcal{N}^{n_{1},n_{1},\mathrm{D}}_{\mathrm{Gapless}}(H_{1})\cdot\mathcal{N}^{n_{2},n_{2}-1,\mathrm{DIII}}_{\mathrm{Gapless}}(H_{2})=\mathcal{N}^{n,n-1,\mathrm{AII}}_{\mathrm{Gapless}}(H)$.

  * •

$\mathrm{DIII}\times\mathrm{D}\to\mathrm{AII}$,
$\mathcal{N}^{n_{1},n_{1},\mathrm{DIII}}_{\mathrm{Gapless}}(H_{1})\cdot(\mathcal{N}^{n_{2},n_{2}-1,\mathrm{D}}_{\mathrm{Gapless}}(H_{2})\bmod
2)=\mathcal{N}^{n,n-1,\mathrm{AII}}_{\mathrm{Gapless}}(H)$.

  * •

$\mathrm{CII}\times\mathrm{D}\to\mathrm{C}$,
$\mathcal{N}^{n_{1},n_{1},\mathrm{CII}}_{\mathrm{Gapless}}(H_{1})\cdot\mathcal{N}^{n_{2},n_{2}-1,\mathrm{D}}_{\mathrm{Gapless}}(H_{2})=\mathcal{N}^{n,n-1,\mathrm{C}}_{\mathrm{Gapless}}(H)$.

  * •

$\mathrm{CII}\times\mathrm{C}\to\mathrm{D}$,
$\mathcal{N}^{n_{1},n_{1},\mathrm{CII}}_{\mathrm{Gapless}}(H_{1})\cdot\mathcal{N}^{n_{2},n_{2}-1,\mathrm{C}}_{\mathrm{Gapless}}(H_{2})=\mathcal{N}^{n,n-1,\mathrm{D}}_{\mathrm{Gapless}}(H)$.

We also have a similar formula by exchanging $H_{1}$ and $H_{2}$ (e.g. pairs
like $\mathrm{D}\times\mathrm{BDI}\to\mathrm{D}$). Note that in the case of
$\mathrm{CII}\times\mathrm{CII}\to\mathrm{BDI}$ in Corollary 5.28, we take the
product of two even integers, which is necessarily a multiple of four. A
similar remark also holds in the case of
$\mathrm{CII}\times\mathrm{C}\to\mathrm{D}$ in Corollary 5.29.

##  Appendix A $\mathbb{Z}_{2}$-Spaces of Self-Adjoint/Skew-Adjoint Fredholm
Operators and Boersema–Loring’s $K$-theory

In this Appendix, we collect necessary results and notations used in this
paper. The results have been developed in much generality [6, 68, 9, 37, 43,
27, 15], and we contain minimal background for this paper focusing on their
relation with Boersema–Loring’s $K$-theory [14]. In Appendix A.1, we introduce
some $\mathbb{Z}_{2}$-spaces of self-adjoint and skew-adjoint Fredholm
operators following [9]. Some proofs for known results are contained simply to
fix isomorphisms used in this paper (e.g. the derivation of Table
LABEL:producttalbe). In Appendix A.2, we discuss its relation with
Boersema–Loring’s $K$-theory. In Appendix A.3, inspired by exponential maps in
[68, 9], we write the boundary maps of the $24$-term exact sequence of
$KO$-theory in Boersema–Loring’s unitary picture through exponentials. Some of
them are already expressed by exponentials in [14]; thus, we consider the
remaining cases. This form of boundary maps is useful when we discuss a
relation between our gapped invariants and gapless invariants through boundary
maps [14, 44].

###  A.1. $\mathbb{Z}_{2}$-Spaces of Self-Adjoint/Skew-Adjoint Fredholm
Operators

For non-negative integers $k$ and $l$, let $\mathit{Cl}_{k,l}$ be the Clifford
algebra that is an associative algebra with unit over $\mathbb{R}$ generated
by $k+l$ elements $e_{1},\ldots,e_{k}$ and $\epsilon_{1},\ldots,\epsilon_{l}$,
which satisfy $e_{i}^{2}=-1$ $(i=1,\ldots,k)$ and $\epsilon_{j}^{2}=1$
$(j=1,\ldots,l)$ and anticommute with each other. The following are well-known
Clifford algebra isomorphisms [47].

######  Lemma A.1.

  1. (1)

$\mathit{Cl}_{k,l+1}\cong\mathit{Cl}_{l,k+1}$.

  2. (2)

$\mathit{Cl}_{k,l}\otimes\mathit{Cl}_{1,1}\cong\mathit{Cl}_{k+1,l+1}$.

  3. (3)

$\mathit{Cl}_{k,l}\otimes\mathit{Cl}_{4,0}\cong\mathit{Cl}_{k+4,l}$ and
$\mathit{Cl}_{k,l}\otimes\mathit{Cl}_{0,4}\cong\mathit{Cl}_{k,l+4}$.

  4. (4)

$\mathit{Cl}_{k,l}\otimes\mathit{Cl}_{8,0}\cong\mathit{Cl}_{k+8,l}$ and
$\mathit{Cl}_{k,l}\otimes\mathit{Cl}_{0,8}\cong\mathit{Cl}_{k,l+8}$.

###### Proof.

  1. (1)

Let $e_{1},\ldots,e_{k}$ and $\epsilon_{1},\ldots,\epsilon_{l+1}$ be
generators of the Clifford algebra $\mathit{Cl}_{k,l+1}$. Let
$\tilde{e}_{i}=\epsilon_{i+1}\epsilon_{1}$ $(i=1,\ldots,l)$,
$\tilde{\epsilon}_{1}=\epsilon_{1}$ and
$\tilde{\epsilon}_{i}=e_{i-1}\epsilon_{1}$ $(i=2,\ldots,k+1)$. Then,
$\tilde{e}_{1},\ldots,\tilde{e}_{l}$ and
$\tilde{\epsilon}_{1},\ldots,\tilde{\epsilon}_{k+1}$ correspond to generators
of the Clifford algebra $\mathit{Cl}_{l,k+1}$.

  2. (2)

Let $e_{1},\ldots,e_{k}$ and $\epsilon_{1},\ldots,\epsilon_{l}$ be generators
of the Clifford algebra $\mathit{Cl}_{k,l}$, and let $e_{1}$ and
$\epsilon_{1}$ be those of $\mathit{Cl}_{1,1}$. We write $\omega_{1,1}$ for
$e^{\prime}_{1}\epsilon^{\prime}_{1}\in\mathit{Cl}_{1,1}$. Then,
$\tilde{e}_{i}=e_{i}\otimes\omega_{1,1}$ $(i=1,\ldots,k)$,
$\tilde{e}_{k+1}=1\otimes e^{\prime}_{1}$,
$\tilde{\epsilon}_{i}=\epsilon_{i}\otimes\omega_{1,1}$ $(i=1,\ldots,l)$ and
$\tilde{\epsilon}_{l}=1\otimes\epsilon^{\prime}_{1}$ correspond to generators
of the Clifford algebra $\mathit{Cl}_{k+1,l+1}$.

  3. (3)

We show that
$\mathit{Cl}_{k,l}\otimes\mathit{Cl}_{0,4}\cong\mathit{Cl}_{k,l+4}$; the other
is proved similarly. Let $e_{1},\ldots,e_{k}$ and
$\epsilon_{1},\ldots,\epsilon_{l}$ be generators of the Clifford algebra
$\mathit{Cl}_{k,l}$, and let $\epsilon^{\prime}_{1}$, $\epsilon^{\prime}_{2}$,
$\epsilon^{\prime}_{3}$, and $\epsilon^{\prime}_{4}$ be those of
$\mathit{Cl}_{0,4}$. We write $\omega_{0,4}$ for
$-\epsilon^{\prime}_{1}\epsilon^{\prime}_{2}\epsilon^{\prime}_{3}\epsilon^{\prime}_{4}\in\mathit{Cl}_{0,4}$.
Then, $\tilde{e}_{i}=e_{i}\otimes\omega_{0,4}$ $(i=1,\ldots,k)$,
$\tilde{\epsilon}_{i}=\epsilon_{i}\otimes\omega_{0,4}$ $(i=1,\ldots,l)$ and
$\tilde{\epsilon}_{i}=1\otimes\epsilon^{\prime}_{i}$ $(i=1,\ldots,4)$
correspond to generators of the algebra $\mathit{Cl}_{k,l+4}$.

  4. (4)

We show that
$\mathit{Cl}_{k,l}\otimes\mathit{Cl}_{0,8}\cong\mathit{Cl}_{k,l+8}$; the other
is proved similarly. Let $e_{1},\ldots,e_{k}$ and
$\epsilon_{1},\ldots,\epsilon_{l}$ be generators of $\mathit{Cl}_{k,l}$, and
let $\epsilon^{\prime}_{1},\ldots,\epsilon^{\prime}_{8}$ be those of
$\mathit{Cl}_{0,8}$. We write $\omega_{0,8}$ for
$\epsilon^{\prime}_{1}\cdots\epsilon^{\prime}_{8}\in\mathit{Cl}_{0,8}$. Then,
$\tilde{e}_{i}=e_{i}\otimes\omega_{0,8}$ $(i=1,\ldots,k)$,
$\tilde{\epsilon}_{i}=\epsilon_{i}\otimes\omega_{0,8}$ $(i=1,\ldots,l)$ and
$\tilde{\epsilon}_{i}=1\otimes\epsilon^{\prime}_{i}$ $(i=1,\ldots,8)$
correspond to generators of the algebra $\mathit{Cl}_{k,l+8}$.∎

Let $W$ be a (ungraded) complex left $\mathit{Cl}_{k,l}$-module. We say that
$W$ is a (ungraded) real (resp. quaternionic)
$\mathit{Cl}_{k,l}$-module161616Note that the “real $\mathbb{Z}_{2}$-graded
$\mathrm{Cliff}(\mathbb{R}^{k,l})$-module” introduced in [6] is the same as
the (ungraded) real $\mathit{Cl}_{l,k+1}$-module discussed in this paper. if
$W$ is equipped with an antilinear map $r\colon W\to W$ (resp. $q\colon W\to
W$), which commutes with the $\mathit{Cl}_{k,l}$-action and satisfies
$r^{2}=1$ (resp. $q^{2}=-1$). We call this $r$ (resp. $q$) the real (resp.
quaternionic) structure on the Clifford module. Since a real (resp.
quaternionic) $\mathit{Cl}_{k,l}$-module is the same thing as a module of
$\mathit{Cl}_{k,l}\otimes\mathit{Cl}_{1,1}\cong\mathit{Cl}_{k+1,l+1}$ (resp.
$\mathit{Cl}_{k,l}\otimes\mathit{Cl}_{2,0}\cong\mathit{Cl}_{l+2,k}$) over
$\mathbb{R}$, the algebra $\mathit{Cl}_{k,l}$ has one inequivalent irreducible
real or quaternionic module when $k-l\not\equiv 3\bmod 4$ and has two when
$k-l\equiv 3\bmod 4$.

######  Lemma A.2.

  1. (1)

Let $\Delta_{1,1}$ be a complex irreducible representation of
$\mathit{Cl}_{1,1}$. There exists a real structure $r_{1,1}$ on $\Delta_{1,1}$
that commutes with the Clifford action.

  2. (2)

Let $\Delta_{0,4}$ (resp. $\Delta_{4,0}$) be a complex irreducible
representation of $\mathit{Cl}_{0,4}$ (resp. $\mathit{Cl}_{4,0}$). There
exists a quaternionic structure $q_{0,4}$ (resp. $q_{4,0}$) on $\Delta_{0,4}$
(resp. $\Delta_{4,0}$) that commutes with the Clifford action.

  3. (3)

Let $\Delta_{0,8}$ (resp. $\Delta_{8,0}$) be a complex irreducible
representation of $\mathit{Cl}_{0,8}$ (resp. $\mathit{Cl}_{8,0}$). There
exists a real structure $r_{0,8}$ (resp. $r_{8,0}$) on $\Delta_{0,8}$ (resp.
$\Delta_{8,0}$) that commutes with the Clifford action.

For the proof of this lemma, see [25], for example. For a
$\mathbb{Z}_{2}$-space $(X,\zeta)$ with two $\mathbb{Z}_{2}$-fixed points
$x_{0},x_{1}\in X^{\zeta}$, we write $P(X;x_{0},x_{1})$ for the path space
starting from $x_{0}$ and ending at $x_{1}$, that is, the space of continuous
maps $f\colon[0,1]\to X$ satisfying $f(0)=x_{0}$ and $f(1)=x_{1}$ equipped
with the compact-open topology. On this space, we consider an involution, for
which we also write $\zeta$ by abuse of notation, defined as
$(\zeta(f))(t)=\zeta(f(t))$ for $t$ in $[0,1]$, and obtain a
$\mathbb{Z}_{2}$-space $(P(X;x_{0},x_{1}),\zeta)$. When $x_{0}=x_{1}$, we
write $\Omega_{x_{0}}X$ for $P(X;x_{0},x_{0})$, which is the based loop space
of $X$ with the base point $x_{0}$.

######  Remark A.3.

Banach $\mathbb{Z}_{2}$-spaces and its open $\mathbb{Z}_{2}$-subspaces are
$\mathbb{Z}_{2}$-absolute neighborhood retracts [3], and have the homotopy
type of $\mathbb{Z}_{2}$-CW complexes [46]. The path spaces and loop spaces we
discuss in the following also have the homotopy type of $\mathbb{Z}_{2}$-CW
complexes [67]. By the equivariant Whitehead theorem, weak
$\mathbb{Z}_{2}$-homotopy equivalences between these spaces are
$\mathbb{Z}_{2}$-homotopy equivalences [50, 4].

Let $\mathcal{V}$ be a separable infinite-dimensional complex Hilbert space.
Let $\mathcal{B}(\mathcal{V})$ be the space of bounded complex linear
operators on $\mathcal{V}$ equipped with the norm topology. Let
$GL(\mathcal{V})$, $U(\mathcal{V})$, $\mathrm{Fred}(\mathcal{V})$ and
$\mathcal{K}(\mathcal{V})$ be subspaces of $\mathcal{B}(\mathcal{V})$
consisting of invertible, unitary, Fredholm and compact operators on
$\mathcal{V}$, respectively. We assume that our Hilbert space $\mathcal{V}$
has a real structure $r$ or a quaternionic structure $q$, that is, an
antiunitary operator on $\mathcal{V}$ satisfying $r^{2}=1$ or $q^{2}=-1$,
respectively. Correspondingly, the space $\mathcal{B}(\mathcal{V})$ has an
(antilinear) involution $\mathfrak{r}=\operatorname{\mathrm{Ad}}_{r}$ or
$\mathfrak{q}=\operatorname{\mathrm{Ad}}_{q}$. These involutions induce
involutions on $GL(\mathcal{V})$, $U(\mathcal{V})$,
$\mathrm{Fred}(\mathcal{V})$ and $\mathcal{K}(\mathcal{V})$, for which we also
write $\mathfrak{r}$ or $\mathfrak{q}$. We write $a$ for $r$ or $q$ and
$\mathfrak{a}$ for $\mathfrak{r}$ or $\mathfrak{q}$. We also assume that there
is a complex linear action of the Clifford algebra $\mathit{Cl}_{k,l}$ on the
Hilbert space $\mathcal{V}$ that commutes with the real or the quaternionic
structure. For an element $v\in\mathit{Cl}_{k,l}$, we also write $v$ for its
action on $\mathcal{V}$, for simplicity. When $k-l\equiv 3\bmod 4$, we further
assume that each of the two inequivalent irreducible real or quaternionic
representations of $Cl_{k,l}$ has infinite multiplicity. In the following, we
discuss the subspaces of $\mathcal{B}(\mathcal{V})$; we may abbreviate the
Hilbert space $\mathcal{V}$ from its notation when it is clear from the
context. When the Hilbert space $\mathcal{V}$ is such a
$\mathit{Cl}_{k,l}$-module, let
$\mathcal{B}^{(\underline{k+1},l)}_{\mathrm{sk}}$ (resp.
$\mathcal{B}^{(k,\underline{l+1})}_{\mathrm{sa}}$) be the subspace of
$\mathcal{B}(\mathcal{V})$ consisting of skew-adjoint (resp. self-adjoint)
operators $A$ on $\mathcal{V}$ satisfying $e_{i}A=-Ae_{i}$ for $i=1,\ldots,k$
and $\epsilon_{j}A=-A\epsilon_{j}$ for $j=1,\ldots,l$. Let
$\mathrm{Fred}^{(\underline{k},l)}_{\mathrm{sk}}=\mathrm{Fred}\cap\mathcal{B}^{(\underline{k},l)}_{\mathrm{sk}}$
and
$\mathrm{Fred}^{(k,\underline{l})}_{\mathrm{sa}}=\mathrm{Fred}\cap\mathcal{B}^{(k,\underline{l})}_{\mathrm{sa}}$.
The involution $\mathfrak{a}$ on $\mathcal{B}(\mathcal{V})$ induces
involutions on $\mathrm{Fred}^{(\underline{k},l)}_{\mathrm{sk}}$ and
$\mathrm{Fred}^{(k,\underline{l})}_{\mathrm{sa}}$ for which we also write
$\mathfrak{a}$. Consider the space
$\mathrm{Fred}^{(\underline{k},l)}_{\mathrm{sk}}$, and let
$\Upsilon=e_{1}\cdots e_{k-1}\epsilon_{1}\cdots\epsilon_{l}$. When $k-l$ is
odd, the space $\mathrm{Fred}^{(\underline{k},l)}_{\mathrm{sk}}$ is decomposed
into three components $\mathrm{Fred}^{(\underline{k},l)}_{+}$,
$\mathrm{Fred}^{(\underline{k},l)}_{-}$ and
$\mathrm{Fred}^{(\underline{k},l)}_{*}$ corresponding to whether the following
element is essentially positive, essentially negative or neither:
$i^{-1}\Upsilon A$ when $k-l\equiv 1\bmod 4$ and $\Upsilon A$ when $k-l\equiv
3\bmod 4$ for $A\in\mathrm{Fred}^{(\underline{k},l)}_{\mathrm{sk}}$. As in
[9], each of these three components is nonempty. When $k-l\equiv 1\bmod 4$,
the involution $\mathfrak{a}$ maps $\mathrm{Fred}^{(\underline{k},l)}_{\pm}$
to $\mathrm{Fred}^{(\underline{k},l)}_{\mp}$ (double-sign corresponds), and
$\mathrm{Fred}^{(\underline{k},l)}_{*}$ is closed under the action of
$\mathfrak{a}$. When $k-l\equiv 3\bmod 4$, each of the three components is
closed under the action of $\mathfrak{a}$. The space
$\mathrm{Fred}^{(k,\underline{l})}_{\mathrm{sa}}$ is also decomposed into
three components in the same way, except that we take $e_{1}\cdots
e_{k}\epsilon_{1}\cdots\epsilon_{l-1}$ for $\Upsilon$ in this case, and we
define the space $\mathrm{Fred}^{(k,\underline{l})}_{*}$ when $k-l$ is odd.
When $k-l$ is even, we set
$\mathrm{Fred}^{(\underline{k},l)}_{*}=\mathrm{Fred}^{(\underline{k},l)}_{\mathrm{sk}}$
and
$\mathrm{Fred}^{(k,\underline{l})}_{*}=\mathrm{Fred}^{(k,\underline{l})}_{\mathrm{sa}}$.
Summarizing, we have the following $\mathbb{Z}_{2}$-spaces:

(A.1) $(\mathrm{Fred}^{(\underline{k},l)}_{*},\mathfrak{r}),\
(\mathrm{Fred}^{(k,\underline{l})}_{*},\mathfrak{r}),\
(\mathrm{Fred}^{(\underline{k},l)}_{*},\mathfrak{q}),\
(\mathrm{Fred}^{(k,\underline{l})}_{*},\mathfrak{q}).$

######  Proposition A.4.

The following $\mathbb{Z}_{2}$-homeomorphisms exist.

  1. (1)

$(\mathrm{Fred}^{(\underline{k},{l})}_{*},\mathfrak{a})\cong(\mathrm{Fred}^{(\underline{k+1},{l+1})}_{*},\mathfrak{a})$
and
$(\mathrm{Fred}^{(k,\underline{l})}_{*},\mathfrak{a})\cong(\mathrm{Fred}^{(k+1,\underline{l+1})}_{*},\mathfrak{a})$,

  2. (2)

$(\mathrm{Fred}^{(\underline{k},{l})}_{*},\mathfrak{a})\cong(\mathrm{Fred}^{(\underline{k+4},{l})}_{*},\widetilde{\mathfrak{a}})$
and
$(\mathrm{Fred}^{(k,\underline{l})}_{*},\mathfrak{a})\cong(\mathrm{Fred}^{(k+4,\underline{l})}_{*},\widetilde{\mathfrak{a}})$,

  3. (3)

$(\mathrm{Fred}^{(\underline{k},{l})}_{*},\mathfrak{a})\cong(\mathrm{Fred}^{(\underline{k},{l+4})}_{*},\widetilde{\mathfrak{a}})$
and
$(\mathrm{Fred}^{(k,\underline{l})}_{*},\mathfrak{a})\cong(\mathrm{Fred}^{(k,\underline{l+4})}_{*},\widetilde{\mathfrak{a}})$,

  4. (4)

$(\mathrm{Fred}^{(\underline{k},{l})}_{*},\mathfrak{a})\cong(\mathrm{Fred}^{(\underline{k+8},{l})}_{*},\mathfrak{a})$
and
$(\mathrm{Fred}^{(k,\underline{l})}_{*},\mathfrak{a})\cong(\mathrm{Fred}^{(k+8,\underline{l})}_{*},\mathfrak{a})$,

  5. (5)

$(\mathrm{Fred}^{(\underline{k},{l})}_{*},\mathfrak{a})\cong(\mathrm{Fred}^{(\underline{k},{l+8})}_{*},\mathfrak{a})$
and
$(\mathrm{Fred}^{(k,\underline{l})}_{*},\mathfrak{a})\cong(\mathrm{Fred}^{(k,\underline{l+8})}_{*},\mathfrak{a})$,

  6. (6)

$(\mathrm{Fred}^{(\underline{k+1},{l+1})}_{*},\mathfrak{a})\cong(\mathrm{Fred}^{(l,\underline{k+2})}_{*},\mathfrak{a})$,

where $\widetilde{\mathfrak{a}}=\mathfrak{q}$ when $\mathfrak{a}=\mathfrak{r}$
and $\widetilde{\mathfrak{a}}=\mathfrak{r}$ when $\mathfrak{a}=\mathfrak{q}$.

###### Proof.

Once the Clifford module structure on the left-hand side of these
homeomorphisms is fixed, that on the right-hand side is given following the
isomorphisms of Clifford algebras in Lemma A.1. By using Lemma A.2, the
$\mathbb{Z}_{2}$-homeomorphisms are given as follows.

  1. (1)

The map
$(\mathrm{Fred}^{(k,\underline{l})}_{*}(\mathcal{V}),\operatorname{\mathrm{Ad}}_{a})\to(\mathrm{Fred}^{(k+1,\underline{l+1})}_{*}(\mathcal{V}\otimes\Delta_{1,1}),\operatorname{\mathrm{Ad}}_{a\otimes
r_{1,1}})$ given by $A\mapsto A\otimes\omega_{1,1}$ is a
$\mathbb{Z}_{2}$-homeomorphism The other one is proved similarly.

  2. (3)

The map
$(\mathrm{Fred}^{(k,\underline{l})}_{*}(\mathcal{V}),\operatorname{\mathrm{Ad}}_{a})\to(\mathrm{Fred}^{(k,\underline{l+4})}_{*}(\mathcal{V}\otimes\Delta_{0,4}),\operatorname{\mathrm{Ad}}_{a\otimes
q_{0,4}})$ given by $A\mapsto A\otimes\omega_{0,4}$ is a
$\mathbb{Z}_{2}$-homeomorphism. The other one and (2), (4) and (5) follow in a
similar way.

  3. (6)

The map
$(\mathrm{Fred}^{(\underline{k+1},{l+1})}_{*}(\mathcal{V}),\mathfrak{a})\to(\mathrm{Fred}^{(l,\underline{k+2})}_{*}(\mathcal{V}),\mathfrak{a})$
given by $A\mapsto A\epsilon_{1}$ is a $\mathbb{Z}_{2}$-homeomorphism. ∎

######  Proposition A.5.

The following $\mathbb{Z}_{2}$-homotopy equivalences exist.

  1. (1)

$(\mathrm{Fred}^{(\underline{k+1},l)}_{*},\mathfrak{a})\simeq(\Omega_{e_{k}}\mathrm{Fred}^{(\underline{k},l)}_{*},\mathfrak{a})$,
for $k\geq 1$ and $l\geq 0$.

  2. (2)

$(\mathrm{Fred}^{(k,\underline{l+1})}_{*},\mathfrak{a})\simeq(\Omega_{\epsilon_{l}}\mathrm{Fred}^{(k,\underline{l})}_{*},\mathfrak{a})$,
for $k\geq 0$ and $l\geq 1$.

  3. (3)

$(\mathrm{Fred}^{(\underline{1},0)}_{*},\mathfrak{a})\simeq(\Omega_{1}\mathrm{Fred},\mathfrak{a})$.

Proposition A.5 is proved as in [9]. In what follows, we outline its proof
since some spaces introduced there are of our interest.

######  Proposition A.6.

The following maps are $\mathbb{Z}_{2}$-homotopy equivalences.

  1. (1)

$\alpha_{1}\colon(\mathrm{Fred}^{(\underline{k+1},l)}_{*},\mathfrak{a})\to(P(\mathrm{Fred}^{(\underline{k},l)}_{*};e_{k},-e_{k}),\mathfrak{a})$,
where
$\alpha_{1}(A)(t)=e_{k}\cos(\pi t)+A\sin(\pi t)$ for $0\leq t\leq 1$.

  2. (2)

$\alpha_{2}\colon(\mathrm{Fred}^{(k,\underline{l+1})}_{*},\mathfrak{a})\to(P(\mathrm{Fred}^{(k,\underline{l})}_{*};\epsilon_{l},-\epsilon_{l}),\mathfrak{a})$,
where
$\alpha_{2}(A)(t)=\epsilon_{l}\cos(\pi t)-A\sin(\pi t)$ for $0\leq t\leq 1$.

  3. (3)

$\alpha_{3}\colon(\mathrm{Fred}^{(\underline{1},0)}_{*},\mathfrak{a})\to(P(\mathrm{Fred};1,-1),\mathfrak{a})$,
where
$\alpha_{3}(A)(t)=\cos(\pi t)+A\sin(\pi t)$ for $0\leq t\leq 1$.

Proposition A.5 follows from Proposition A.6 since, in each case, there is a
path connecting the endpoints of each path space in the unitaries preserving
the Clifford action and the $\mathbb{Z}_{2}$-action. As in [9], the proof of
Proposition A.6 reduces to showing the $\mathbb{Z}_{2}$-homotopy equivalences
between some spaces of Fredholm operators and some spaces of unitary operators
(Proposition A.7). Let $F^{(\underline{k},l)}_{*}$ (resp.
$F^{(k,\underline{l})}_{*}$) be the subspace of
$\mathrm{Fred}^{(\underline{k},l)}_{*}$ (resp.
$\mathrm{Fred}^{(k,\underline{l})}_{*}$ ) consisting of those operators whose
essential spectra are $\\{i,-i\\}$ (resp. $\\{1,-1\\}$) and whose operator
norms are $1$. The spaces $F^{(\underline{k},l)}_{*}$ and
$F^{(k,\underline{l})}_{*}$ are closed under the action of $\mathfrak{a}$;
thus, we have $\mathbb{Z}_{2}$-spaces
$(F^{(\underline{k},l)}_{*},\mathfrak{a})$ and
$(F^{(k,\underline{l})}_{*},\mathfrak{a})$. Inclusions
$(F^{(\underline{k},l)}_{*},\mathfrak{a})\hookrightarrow(\mathrm{Fred}^{(\underline{k},l)}_{*},\mathfrak{a})$
and
$(F^{(k,\underline{l})}_{*},\mathfrak{a})\hookrightarrow(\mathrm{Fred}^{(k,\underline{l})}_{*},\mathfrak{a})$
are $\mathbb{Z}_{2}$-homotopy equivalences. Let $U_{\mathrm{cpt}}$ be the
subspace of $U(\mathcal{V})$ consisting of unitary operators of the form
$1+T$, where $T\in\mathcal{K}(\mathcal{V})$. When the Hilbert space
$\mathcal{V}$ is a $\mathit{Cl}_{k,l}$-module, let
$U^{(\underline{k},l)}_{\mathrm{cpt}}$ (resp.
$U^{(k,\underline{l})}_{\mathrm{cpt}}$) be the subspace of
$U(\mathcal{V})\cap\mathcal{B}^{(\underline{k},l)}_{\mathrm{sk}}$ (resp.
$U(\mathcal{V})\cap\mathcal{B}^{(k,\underline{l})}_{\mathrm{sa}}$) consisting
of a unitary $u$ satisfying $u^{2}=-1$ (resp. $u^{2}=1$) and $u\equiv e_{k}$
(resp. $u\equiv\epsilon_{l}$) modulo compact operators. If the Hilbert space
has a real or quaternionic structure, these spaces of unitaries are closed
under the action of $\mathfrak{a}$, and we obtain $\mathbb{Z}_{2}$-spaces.

######  Proposition A.7.

The following maps are $\mathbb{Z}_{2}$-homotopy equivalences:

  1. (1)

$p_{1}\colon(F^{(\underline{k+1},l)}_{*},\mathfrak{a})\to(-U^{(\underline{k},l)}_{\mathrm{cpt}},\mathfrak{a})$,
$p_{1}(A)=e_{k}\exp(\pi Ae_{k})$, for $k\geq 1$, $l\geq 0$.

  2. (2)

$p_{2}\colon(F^{(k,\underline{l+1})}_{*},\mathfrak{a})\to(-U^{(k,\underline{l})}_{\mathrm{cpt}},\mathfrak{a})$,
$p_{2}(A)=\epsilon_{l}\exp(\pi A\epsilon_{l})$, for $k\geq 0$, $l\geq 1$.

  3. (3)

$p_{3}\colon(F^{(\underline{1},0)}_{*},\mathfrak{a})\to(-U_{\mathrm{cpt}},\mathfrak{a})$,
$p_{3}(A)=\exp(\pi A)$.

  4. (4)

$p_{4}\colon(F^{(0,\underline{1})}_{*},\mathfrak{a})\to(-U_{\mathrm{cpt}},\mathfrak{a}\circ*)$,
$p_{4}(A)=\exp(\pi iA)$.

###### Proof.

By Remark A.3, it is sufficient to show that these maps are weak
$\mathbb{Z}_{2}$-homotopy equivalences. Equivalently, to show that $p_{i}$ and
its restriction to the $\mathbb{Z}_{2}$-fixed point sets (the map
$p_{1}^{\mathbb{Z}_{2}}\colon(F^{(\underline{k+1},l)}_{*})^{\mathfrak{a}}\to(-U^{(\underline{k},l)}_{\mathrm{cpt}})^{\mathfrak{a}}$
in the case of (1)) are weak homotopy equivalences. They are proved by using
quasifibrations on some dense subspaces of contractible fibers as in [9]. ∎

######  Lemma A.8.

There is a $\mathbb{Z}_{2}$-homeomorphism
$(\mathrm{Fred},\mathfrak{a})\cong(\mathrm{Fred}_{*}^{(0,\underline{2})},\mathfrak{a})$.

###### Proof.

This is given by a $\mathbb{Z}_{2}$-map
$(\mathrm{Fred}(\mathcal{V}),\mathfrak{a})\to(\mathrm{Fred}_{*}^{(0,\underline{2})}(\mathcal{V}\oplus\mathcal{V}),\mathfrak{a}\oplus\mathfrak{a})$,
$A\mapsto\left(\begin{array}[]{cc}0&A^{*}\\\ A&0\end{array}\right)$, where the
action of $\mathit{Cl}_{0,1}$ on $\mathcal{V}\oplus\mathcal{V}$ is given by
$\epsilon_{1}=\mathrm{diag}(1,-1)$. ∎

Proposition A.4, Proposition A.5 and Lemma A.8 lead to the following.

######  Corollary A.9.

The following $\mathbb{Z}_{2}$-homotopy equivalences exist.

  1. (1)

$(\mathrm{Fred}^{(\underline{k},l)}_{*},\mathfrak{r})\simeq(\Omega^{k-l}\mathrm{Fred},\mathfrak{r})$.

  2. (2)

$(\mathrm{Fred}^{(\underline{k},l)}_{*},\mathfrak{q})\simeq(\Omega^{k-l+4}\mathrm{Fred},\mathfrak{r})$.

  3. (3)

$(\mathrm{Fred}^{(k,\underline{l})}_{*},\mathfrak{r})\simeq(\Omega^{l-k+6}\mathrm{Fred},\mathfrak{r})$.

  4. (4)

$(\mathrm{Fred}^{(k,\underline{l})}_{*},\mathfrak{q})\simeq(\Omega^{l-k+2}\mathrm{Fred},\mathfrak{r})$.

When the subscript $m$ on $\Omega^{m}$ is negative, this should be replaced by
$m+8n$ by taking a sufficiently large integer $n$ to make the subscript non-
negative.

Note that when $k$ and $l$ are relatively small, we further have the following
$\mathbb{Z}_{2}$-homeomorphisms.

######  Lemma A.10.

Multiplication by the imaginary unit $i=\sqrt{-1}$ induces the following
$\mathbb{Z}_{2}$-homeomorphisms:

  1. (1)

$(\mathrm{Fred}^{(0,\underline{2})}_{*},\operatorname{\mathrm{Ad}}_{r})\to(\mathrm{Fred}^{(\underline{1},1)}_{*},\operatorname{\mathrm{Ad}}_{\widetilde{r}})$,
where $\tilde{r}=r\epsilon_{1}$.

  2. (2)

$(\mathrm{Fred}^{(0,\underline{2})}_{*},\operatorname{\mathrm{Ad}}_{q})\to(\mathrm{Fred}^{(\underline{1},1)}_{*},\operatorname{\mathrm{Ad}}_{\widetilde{q}})$,
where $\tilde{q}=-q\epsilon_{1}$.

  3. (3)

$(\mathrm{Fred}^{(1,\underline{1})}_{*},\operatorname{\mathrm{Ad}}_{q})\to(\mathrm{Fred}^{(\underline{2},0)}_{*},\operatorname{\mathrm{Ad}}_{\widetilde{r}})$,
where $\tilde{r}=qe_{1}$.

  4. (4)

$(\mathrm{Fred}^{(1,\underline{1})}_{*},\operatorname{\mathrm{Ad}}_{r})\to(\mathrm{Fred}^{(\underline{2},0)}_{*},\operatorname{\mathrm{Ad}}_{\widetilde{q}})$,
where $\tilde{q}=-re_{1}$.

######  Remark A.11.

The $\mathbb{Z}_{2}$-spaces in Lemma A.10 appear in the study of topological
insulators. Specifically, Table 11 is obtained by taking the quantum
symmetries as real or quaternionic Clifford module structures as follows.

  * •

In class $\mathrm{BDI}$, we put $r=\Theta$ and $\epsilon_{1}=\Pi$ in (1);
then, $\tilde{r}=\Xi$.

  * •

In class $\mathrm{CII}$, we put $q=\Theta$ and $\epsilon_{1}=\Pi$ in (2);
then, $\tilde{q}=\Xi$.

  * •

In class $\mathrm{DIII}$, we put $q=\Theta$ and $e_{1}=i\Pi$ in (3); then,
$\tilde{r}=\Xi$.

  * •

In class $\mathrm{CI}$, we put $r=\Theta$ and $e_{1}=i\Pi$ in (4); then,
$\tilde{q}=\Xi$.

For $l\geq 1$, let us consider the map

(A.2)
$(\mathrm{Fred}^{(k,\underline{l+1})}_{*},\mathfrak{a})\times(\mathrm{Fred}^{(k^{\prime},\underline{l^{\prime}+1})}_{*},\mathfrak{a}^{\prime})\to(\mathrm{Fred}^{(k+k^{\prime},\underline{l+l^{\prime}})}_{*},\mathfrak{a}\otimes\mathfrak{a}^{\prime})$

defined by $(A,B)\mapsto A\otimes 1+\epsilon_{l}\otimes B$, where the Clifford
action to define $\mathrm{Fred}^{(k+k^{\prime},\underline{l+l^{\prime}})}_{*}$
is generated by $\tilde{e}_{i}=e_{i}\otimes 1$ $(i=1,\ldots,k)$,
$\tilde{e}_{k+i}=\epsilon_{l}\otimes e_{i}$ $(i=1,\ldots,k^{\prime})$,
$\tilde{\epsilon}_{i}=\epsilon_{i}\otimes 1$ $(i=1,\ldots,l-1)$ and
$\tilde{\epsilon}_{l+i-1}=\epsilon_{l}\otimes\epsilon_{i}$
$(i=1,\ldots,l^{\prime})$. This map induces the exterior product of
topological $KR$-groups as in [9].

###  A.2. Relation with Boersema–Loring’s Unitary Picture

In this subsection, we discuss a relation between these
$\mathbb{Z}_{2}$-spaces of self-adjoint/skew-adjoint Fredholm operators and
Boersema–Loring’s $K$-theory.

Let $\\{W_{i}\\}_{i\in I}$ be the set of mutually inequivalent irreducible
real (resp. quaternionic) representations of $\mathit{Cl}_{k,l}$ with
hermitian inner-products which $\\{W_{i}\\}_{i\in I}$ consists of one or two
elements corresponding to $k$ and $l$. Let $W=\oplus_{i\in I}W_{i}$ and
$\mathcal{V}=l^{2}(\mathbb{Z}_{\geq 0})\otimes W$ which has a real (resp.
quaternionic) $\mathit{Cl}_{k,l}$-module structure induced by that of
$\\{W_{i}\\}_{i\in I}$. We take a complete orthonormal basis
$\\{\delta_{j}\\}_{j\in\mathbb{Z}_{\geq 0}}$ of $l^{2}(\mathbb{Z}_{\geq 0})$
given by generating functions of each points in $\mathbb{Z}_{\geq 0}$. Let
$\mathcal{V}_{n}$ be the subspace of $\mathcal{V}$ spanned by
$\\{\delta_{j}\otimes w\ |\ 0\leq j\leq n,w\in W\\}$, which is a real (resp.
quaternionic) $\mathit{Cl}_{k,l}$-module. Let $GL_{\mathrm{cpt}}$ be the space
of invertible operators on $\mathcal{V}$ of the form $e_{k}+T$ for some
compact operator $T$. Let
$GL^{(\underline{k},l)}_{\mathrm{cpt}}=GL_{\mathrm{cpt}}\cap\mathcal{B}_{\mathrm{sk}}^{(\underline{k},l)}(\mathcal{V})$
and
$GL^{(k,\underline{l})}_{\mathrm{cpt}}=GL_{\mathrm{cpt}}\cap\mathcal{B}_{\mathrm{sa}}^{(k,\underline{l})}(\mathcal{V})$.
Let $GL^{(\underline{k},l)}_{n}$ (resp. $GL^{(k,\underline{l})}_{n}$) be the
subspace of $\mathcal{B}_{\mathrm{sk}}^{(\underline{k},l)}(\mathcal{V}_{n})$
(resp. $\mathcal{B}_{\mathrm{sa}}^{(k,\underline{l})}(\mathcal{V}_{n})$)
consisting of invertible operators, and let $U^{(\underline{k},l)}_{n}$ (resp.
$U^{(k,\underline{l})}_{n}$) be its subspace of unitaries. We have an
injection $GL^{(\underline{k},l)}_{n}\hookrightarrow
GL^{(\underline{k},l)}_{n+1}$ (resp.
$GL^{(k,\underline{l})}_{n}\hookrightarrow GL^{(k,\underline{l})}_{n+1}$)
given by mapping $A$ to $A\oplus e_{k}$ (resp. $A\oplus\epsilon_{l}$), and let
$GL^{(\underline{k},l)}_{\infty}$ (resp. $GL^{(k,\underline{l})}_{\infty}$) be
its inductive limit
$\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
<EMAIL_ADDRESS>GL^{(\underline{k},l)}_{n}$ (resp.
$\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
<EMAIL_ADDRESS>GL^{(k,\underline{l})}_{n}$). We also define $U^{(\underline{k},l)}_{\infty}$
and $U^{(k,\underline{l})}_{\infty}$ for unitaries in the same way. The space
$GL^{(\underline{k},l)}_{n}$ (resp. $GL^{(k,\underline{l})}_{n}$) is
identified with the subspace of $GL^{(\underline{k},l)}_{\mathrm{cpt}}$ (resp.
$GL^{(k,\underline{l})}_{\mathrm{cpt}}$) consisting of operators of the form
$e_{k}+T$ (resp. $\epsilon_{l}+T$), where $T\in\mathcal{B}(\mathcal{V}_{n})$,
and we have an injective $\mathbb{Z}_{2}$-map
$(GL^{(\underline{k},l)}_{\infty},\mathfrak{a})\to(GL^{(\underline{k},l)}_{\mathrm{cpt}},\mathfrak{a})$
(resp.
$(GL^{(k,\underline{l})}_{\infty},\mathfrak{a})\to(GL^{(k,\underline{l})}_{\mathrm{cpt}},\mathfrak{a})$).
As in [53], the following holds171717In [53], an upper semicontinuous function
is introduced to show that an injection $GL_{\infty}\to GL_{\mathrm{cpt}}$ is
a homotopy equivalence. In our setup, this function is
$\mathbb{Z}_{2}$-invariant, and the result follows as in [53]..

######  Proposition A.12.

The map
$(GL^{(\underline{k},l)}_{\infty},\mathfrak{a})\to(GL^{(\underline{k},l)}_{\mathrm{cpt}},\mathfrak{a})$
and the map
$(GL^{(k,\underline{l})}_{\infty},\mathfrak{a})\to(GL^{(k,\underline{l})}_{\mathrm{cpt}},\mathfrak{a})$
are $\mathbb{Z}_{2}$-homotopy equivalences.

By using a deformation of invertibles to unitaries,
$(U^{(\underline{k},l)}_{\infty},\mathfrak{a})$ and
$(U^{(k,\underline{l})}_{\infty},\mathfrak{a})$ are $\mathbb{Z}_{2}$-homotopy
equivalent to $(U^{(\underline{k},l)}_{\mathrm{cpt}},\mathfrak{a})$ and
$(U^{(k,\underline{l})}_{\mathrm{cpt}},\mathfrak{a})$, respectively. We denote
$U^{\spadesuit}_{\infty}$ for these subspaces of $U^{\spadesuit}$ as indicated
in Table 11.

These $\mathbb{Z}_{2}$-spaces of unitaries appears in Boersema–Loring’s
$KO$-theory [14]. Let $(X,\zeta)$ be a compact Hausdorff
$\mathbb{Z}_{2}$-space, and consider a $C^{*,\tau}$-algebra
$(C(X),\tau_{\zeta})$ of continuous functions on $X$, whose transposition
$\tau_{\zeta}$ is given by $f^{\tau_{\zeta}}(x)=f(\zeta(x))$. Then, the
$\mathbb{Z}_{2}$-homotopy classes
$[(X,\zeta),U^{\spadesuit}_{\infty}]_{\mathbb{Z}_{2}}$ can be identified with
the group $KO_{i(\spadesuit)-1}(C(X),\tau_{\zeta})$ where $i(\spadesuit)$ is
as indicated in Table 10. In the following, we discuss two of eight
$KO$-groups and the others are discussed in a similar way.

As for the $KO_{-1}$-group, an element of the set
$[(X,\zeta),(U_{\infty},\mathfrak{r}\circ*)]_{\mathbb{Z}_{2}}$ is represented
by a $\mathbb{Z}_{2}$-map $f\colon(X,\zeta)\to(U_{n},\mathfrak{r}\circ*)$.
This $f$ is a unitary element of $M_{n}(C(X))$ satisfying
$f(\zeta(x))=\mathfrak{r}(f(x))^{*}$ which is the same as the relation
$f^{\tau_{\zeta}}=f$ to define $KO_{-1}$-groups. Thus, the set
$[(X,\zeta),(U_{\infty},\mathfrak{r}\circ*)]_{\mathbb{Z}_{2}}$ is the same as
$KO_{-1}(C(X),\tau_{\zeta})$ by the definition of Boersema–Loring’s
$KO_{-1}$-group.

Finally, we discuss the $KO_{6}$-group. By the multiplication of $-i$, we have
a $\mathbb{Z}_{2}$-homeomorphism
$(U^{(\underline{1},0)}_{\infty},\mathfrak{q})\to(U^{(0,\underline{1})}_{\infty},-\mathfrak{q})$.
A $\mathbb{Z}_{2}$-continuous map
$f\colon(X,\zeta)\to(U^{(0,\underline{1})}_{n},-\mathfrak{q})$ is a self-
adjoint unitary in $M_{n}(C(X))$ satisfying
$f^{\sharp\otimes\tau_{\zeta}}=-f^{*}=-f$. The Clifford algebra
$\mathit{Cl}_{1,0}$ has just one irreducible quaternionic representation up to
equivalence, which is constructed as follows. On $W=\mathbb{C}^{2}$, we
consider the action $\rho$ of $\mathit{Cl}_{1,0}\otimes\mathit{Cl}_{2,0}$
defined as follows:

$\rho(1\otimes e_{1})=\left(\begin{array}[]{cc}i&0\\\ 0&i\end{array}\right),\
\ \rho(1\otimes e_{2})=\left(\begin{array}[]{cc}0&-c\\\
c&0\end{array}\right),\ \ \rho(e_{1}\otimes
1)=\left(\begin{array}[]{cc}0&-1\\\ 1&0\end{array}\right),$

where $c$ is the complex conjugation on $\mathbb{C}$. The space
$U^{(\underline{1},0)}_{\infty}$ is defined as the inductive limit of maps
$U^{(\underline{1},0)}_{n}\to U^{(\underline{1},0)}_{n+1}$, $A\mapsto A\oplus
I$ where $I=\rho(e_{1}\otimes 1)$ and the space
$U^{(0,\underline{1})}_{\infty}$ is defined as that of maps $A\mapsto A\oplus-
iI$ where $-iI=I^{(6)}$.

###  A.3. Boersema–Loring’s $K$-Theory and Exponential Maps

We describe boundary maps of the $24$-term exact sequence of $KO$-theory
(which we denote as $\partial^{\mathrm{BL}}_{i}$ in this section) in
Boersema–Loring’s unitary picture through exponential maps. The map
$\partial^{\mathrm{BL}}_{i}$ for even $i$ has already been expressed as an
exponential map in [14]; thus, we focus on $\partial^{\mathrm{BL}}_{i}$ for
odd $i$. A clue is the exponential maps given in Proposition A.7. For a short
exact sequence of $C^{*,\tau}$-algebras,

(A.3)
$0\to(\mathcal{I},\tau)\to(\mathcal{A},\tau)\overset{\varphi}{\to}(\mathcal{B},\tau)\to
0,$

and for each odd $i$, we construct a group homomorphism

(A.4) $\partial^{\exp}_{i}\colon KO_{i}(\mathcal{B},\tau)\to
KO_{i-1}(\mathcal{I},\tau)$

and show they coincides with $\partial^{\mathrm{BL}}_{i}$ up to a factor of
$-1$. Let $W_{2n}\in M_{2n}(\mathbb{C})$ and $Q_{4n}\in M_{4n}(\mathbb{C})$ be
the following matrices:

$W_{2n}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}i\cdot 1_{n}&1_{n}\\\
1_{n}&i\cdot 1_{n}\end{array}\right),\ \
Q_{4n}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}1_{2n}&-I_{n}^{(2)}\\\
I_{n}^{(2)}&1_{2n}\end{array}\right),$

and let $V_{2n}\in M_{2n}(\mathbb{R})$ and $X_{4n}\in M_{4n}(\mathbb{R})$ be
the permutation matrices satisfying181818Matrices $W_{2n}$, $Q_{4n}$, $V_{2n}$
and $X_{4n}$ are what we borrowed from Sect. $8$ of [14]. Some of the basic
formulas that they satisfy can be found there.

$\displaystyle
V_{2n}\mathrm{diag}(x_{1},\ldots,x_{2n})V_{2n}^{*}=\mathrm{diag}(x_{1},x_{n+1},x_{2},x_{n+2},\ldots,x_{n},x_{2n}),$
$\displaystyle X_{4n}\mathrm{diag}(x_{1},\ldots,x_{4n})X_{4n}^{*}\hskip
170.71652pt$
$\displaystyle=\mathrm{diag}(x_{1},x_{2},x_{2n+1},x_{2n+2},x_{3},x_{4},x_{2n+3},x_{2n+4},\ldots,x_{4n}).$

As in [14], let $Y_{2n}^{(-1)}=V_{2n}W_{2n}$, $Y_{2n}^{(1)}=V_{2n}$,
$Y_{4n}^{(3)}=V_{4n}Q_{4n}W_{4n}$ and $Y_{4n}^{(5)}=X_{4n}$.

######  Definition A.13.

Suppose we have a short exact sequence of $C^{*,\tau}$-algebras as in (A.3).
We assume $\mathcal{I}=\operatorname{\mathrm{Ker}}(\varphi)$ and identify the
unit in $\widetilde{\mathcal{I}}$ with that of $\widetilde{\mathcal{A}}$. For
$i\in\\{-1,1,3,5\\}$, suppose $[u]\in KO_{i}(\mathcal{B},\tau)$, where $u\in
M_{n_{i}\cdot n}(\widetilde{\mathcal{B}})$ is a unitary satisfying the
relation $\mathcal{R}_{i}$ and $\lambda(u)=I_{n}^{(i)}$, for which $n_{i}$,
$\mathcal{R}_{i}$ and $I^{(i)}$ are as in Table 2. Let $a$ in $M_{n_{i}\cdot
n}(\widetilde{\mathcal{A}})$ be a lift of $u$ satisfying the relation
$\mathcal{R}_{i}$ and $\|a\|\leq 1$. Then, define

$\partial^{\exp}_{i}([u])=\left[-Y_{2n_{i}\cdot
n}^{(i)}\bigl{(}\epsilon_{1}\exp(\pi A\epsilon_{1})\bigl{)}Y_{2n_{i}\cdot
n}^{(i)*}\right]\in KO_{i}(\mathcal{I},\tau),$

where $A=\left(\begin{array}[]{cc}0&a^{*}\\\ a&0\end{array}\right)$ and
$\epsilon_{1}=\mathrm{diag}(1_{n_{i}\cdot n},-1_{n_{i}\cdot n})$.

######  Lemma A.14.

$\partial^{\exp}_{i}$ for odd $i$ are well-defined group homomorphisms.

###### Proof.

We need to show that (a) the unitaries constructed all satisfy the correct
relation, (b) the choice of lift is not important, (c) some lift is always
available, (d) homotopy is respected, (e) compatible with respect to the
stabilization by $I^{(i)}$ and (f) the addition is respected. (c), (b) and (d)
are proved in the same way as in Lemma $8.2$ of [14] and we discuss the other
parts. For convenience, let $C(a)=-\epsilon_{1}\exp(\pi A\epsilon_{1})$ and
$C^{\prime}(a)=Y_{2n_{i}\cdot n}^{(i)}C(a)Y_{2n_{i}\cdot n}^{(i)*}$.

  1. (1)

We first consider the case of $i=1$. Let $u\in M_{n}(\widetilde{\mathcal{B}})$
be a unitary satisfying $u^{\tau}=u^{*}$ and $\lambda_{n}(u)=I_{n}^{(1)}$. We
take a lift $a\in M_{n}(\widetilde{\mathcal{A}})$ of $u$ such that $\|a\|\leq
1$ and $a^{\tau}=a^{*}$. Since
$\varphi(C^{\prime}(a))=V_{2n}\epsilon_{1}V_{2n}^{*}=I^{(0)}_{n}$, we have
$C^{\prime}(a)\in M_{n}(\widetilde{I})$ and
$\lambda(C^{\prime}(a))=I^{(0)}_{n}$. Since $A^{\tau}=A$, we have,

$C(a)^{\tau}=-\exp(\pi
A\epsilon_{1})^{\tau}\epsilon_{1}^{\tau}=-\epsilon_{1}^{2}\exp(\pi\epsilon_{1}^{\tau}A^{\tau})\epsilon_{1}^{*}=-\epsilon_{1}\exp(\pi\epsilon_{1}^{2}A\epsilon_{1}^{*})=C(a).$

Since $Y_{2n}^{(1)}=V_{2n}$ is the orthogonal matrix,
$(Y_{2n}^{(1)})^{\tau}=Y_{2n}^{(1)*}$, and thus,
$C^{\prime}(a)^{\tau}=C^{\prime}(a)$ holds. When $u=1$, we can take $a=1$ and
$C^{\prime}(1)=I^{(1)}$ in this case. Combined with this, the proof is
completed once we have checked that $\partial^{\exp}_{1}$ preserves the
addition. Let $u\in M_{m}(\widetilde{\mathcal{B}})$ and $v\in
M_{n}(\widetilde{\mathcal{B}})$. We take their lift $a$ and $b$ such that
$a^{\tau}=a^{*}$ and $b^{\tau}=b^{*}$. Then, we have
$C^{\prime}(\mathrm{diag}(a,b))=\mathrm{diag}(C^{\prime}(a),C^{\prime}(b))$
since

$C\left(\left(\begin{array}[]{cc}a&0\\\
0&b\end{array}\right)\right)=-\left(\begin{array}[]{cc}1_{m+n}&0\\\
0&-1_{m+n}\end{array}\right)\exp\Biggl{(}\pi\left(\begin{array}[]{cc}0&\mathrm{diag}(-a^{*},-b^{*})\\\
\mathrm{diag}(a,b)&0\end{array}\right)\Biggl{)},$
$\displaystyle\operatorname{\mathrm{Ad}}_{V_{2m+2n}}\exp\Biggl{(}\pi\left(\begin{array}[]{cc}0&\mathrm{diag}(-a^{*},-b^{*})\\\
\mathrm{diag}(a,b)&0\end{array}\right)\Biggl{)}$
$\displaystyle=\exp\left(\pi\cdot\mathrm{diag}\left(\operatorname{\mathrm{Ad}}_{V_{2m}}\left(\begin{array}[]{cc}0&-a^{*}\\\
a&0\\\
\end{array}\right),\operatorname{\mathrm{Ad}}_{V_{2n}}\left(\begin{array}[]{cc}0&-b^{*}\\\
b&0\\\ \end{array}\right)\right)\right),$

which shows that
$\partial^{\exp}_{1}([u]+[v])=\partial^{\exp}_{1}([u])+\partial^{\exp}_{1}([v])$.

  2. (2)

We next consider the case of $i=-1$. Let $u\in M_{n}(\widetilde{\mathcal{B}})$
be a unitary satisfying $u^{\tau}=u$ and $\lambda_{n}(u)=I_{n}^{(-1)}$. We
take a lift $a\in M_{n}(\widetilde{\mathcal{A}})$ of $u$ such that $\|a\|\leq
1$ and $a^{\tau}=a$. Since $\varphi(C^{\prime}(a))=I^{(6)}_{n}$, we have
$C^{\prime}(a)\in M_{n}(\widetilde{\mathcal{I}})$ and
$\lambda(C^{\prime}(a))=I^{(6)}_{n}$. Since
$A^{\widetilde{\sharp}\otimes\tau}=-A$, we have

$\displaystyle C(a)^{\widetilde{\sharp}\otimes\tau}$ $\displaystyle=-\exp(\pi
A\epsilon_{1})^{\widetilde{\sharp}\otimes\tau}\epsilon_{1}^{\widetilde{\sharp}\otimes\tau}=\exp(\pi\epsilon_{1}^{\widetilde{\sharp}\otimes\tau}A^{\widetilde{\sharp}\otimes\tau})\epsilon_{1}$
$\displaystyle=\exp(\pi\epsilon_{1}A)\epsilon_{1}=\epsilon_{1}\exp(\pi
A\epsilon_{1})=-C(a).$

Since
$(V_{2n}xV_{2n}^{*})^{\sharp\otimes\tau}=V_{2n}x^{\widetilde{\sharp}\otimes\tau}V_{2n}^{*}$
and $W_{2n}^{\widetilde{\sharp}\otimes\tau}=-W_{2n}^{*}$, we have
$C^{\prime}(a)^{\sharp\otimes\tau}=-C^{\prime}(a)$. For $u=1$, we take $a=1$
and $C^{\prime}(1)=I^{(6)}$ holds. Therefore, as in (1), all we have to show
is the additivity of $\partial_{-1}^{\exp}$. Let $a\in
M_{m}(\widetilde{\mathcal{A}})$ and $b\in M_{n}(\widetilde{\mathcal{A}})$ be
lifts of the unitaries $u$ and $v$. Then,
$C^{\prime}(\mathrm{diag}(a,b))=\mathrm{diag}(C^{\prime}(a),C^{\prime}(b))$
follows from

$\displaystyle
V_{2m+2n}W_{2m+2n}\left(\begin{array}[]{cc}0&\mathrm{diag}(-a^{*},-b^{*})\\\
\mathrm{diag}(a,b)&0\end{array}\right)W_{2m+2n}^{*}V_{2m+2n}^{*}$
$\displaystyle=\mathrm{diag}\left(V_{2m}W_{2m}\left(\begin{array}[]{cc}0&-a^{*}\\\
a&0\\\
\end{array}\right)W_{2m}^{*}V_{2m}^{*},V_{2n}W_{2n}\left(\begin{array}[]{cc}0&-b^{*}\\\
b&0\\\ \end{array}\right)W_{2n}^{*}V_{2n}^{*}\right).$

  3. (3)

Let us consider the case of $i=5$. Let $u\in M_{2n}(\widetilde{\mathcal{B}})$
be a unitary satisfying $u^{\sharp\otimes\tau}=u^{*}$ and
$\lambda_{2n}(u)=I_{n}^{(5)}=1_{2n}$. We take a lift $a\in
M_{2n}(\widetilde{\mathcal{A}})$ of $u$ such that $\|a\|\leq 1$ and
$a^{\sharp\otimes\tau}=a^{*}$. Since
$(X_{4n}xX_{4n})^{\sharp\otimes\tau}=X_{4n}x^{\sharp\otimes\tau}X_{4n}$,
$A^{\sharp\otimes\tau}=A$ and $\epsilon_{1}^{\sharp\otimes\tau}=\epsilon_{1}$,
the relation

$C^{\prime}(a)^{\sharp\otimes\tau}=-X_{4n}\exp(\pi\epsilon_{1}A^{\sharp\otimes\tau})\epsilon_{1}X_{4n}^{*}=-X_{4n}\exp(\pi\epsilon_{1}A)\epsilon_{1}X_{4n}^{*}=C^{\prime}(a)$

holds. We have $C^{\prime}(1_{2})=I^{(4)}$, and for the additivity of
$\partial^{\exp}_{5}$, note that

$\displaystyle
X_{4m+4n}\left(\begin{array}[]{cc}0&\mathrm{diag}(-a^{*},-b^{*})\\\
\mathrm{diag}(a,b)&0\end{array}\right)X_{4m+4n}^{*}$
$\displaystyle=\mathrm{diag}\left(X_{4m}\left(\begin{array}[]{cc}0&-a^{*}\\\
a&0\\\ \end{array}\right)X_{4m}^{*},X_{4n}\left(\begin{array}[]{cc}0&-b^{*}\\\
b&0\\\ \end{array}\right)X_{4n}^{*}\right).$

  4. (4)

Consider the case of $i=3$. Let $u\in M_{2n}(\widetilde{\mathcal{B}})$ be a
unitary satisfying $u^{\sharp\otimes\tau}=u$ and
$\lambda_{2n}(u)=I_{n}^{(3)}=1_{2n}$. We take a lift $a\in
M_{2n}(\widetilde{\mathcal{A}})$ of $u$ such that $\|a\|\leq 1$ and
$a^{\sharp\otimes\tau}=a$. Since
$A^{\widetilde{\sharp}\otimes\sharp\otimes\tau}=-A$ and
$\epsilon_{1}^{\widetilde{\sharp}\otimes\sharp\otimes\tau}=-\epsilon_{1}$, the
relation $C(a)^{\widetilde{\sharp}\otimes\sharp\otimes\tau}=-C(a)$ holds.
Since
$(Q_{4n}xQ_{4n}^{*})^{\tau}=Q_{4n}x^{\widetilde{\sharp}\otimes\sharp\otimes\tau}Q_{4n}^{*}$
and $W_{4n}^{\widetilde{\sharp}\otimes\sharp}=-W_{4n}^{*}$, we have
$C^{\prime}(a)^{\tau}=-C^{\prime}(a)$. For the remaining part, we note that
$C^{\prime}(1_{2})=I^{(2)}_{2}$ and

$\displaystyle
Y_{4m+4n}^{(3)}\left(\begin{array}[]{cc}0&\mathrm{diag}(-a^{*},-b^{*})\\\
\mathrm{diag}(a,b)&0\end{array}\right)Y_{4m+4n}^{(3)*}$
$\displaystyle=\mathrm{diag}\left(Y_{4m}^{(3)}\left(\begin{array}[]{cc}0&-a^{*}\\\
a&0\\\
\end{array}\right)Y_{4m}^{(3)*},Y_{4n}^{(3)}\left(\begin{array}[]{cc}0&-b^{*}\\\
b&0\\\ \end{array}\right)Y_{4n}^{(3)*}\right).\qed$

######  Lemma A.15.

Each $\partial_{i}^{\exp}$ is natural with respect to the morphisms of short
exact sequences of $C^{*}$-algebras. That is, suppose we have the following
commutative diagram of exact lows:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathcal{I}_{1},\tau)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{(\mathcal{A}_{1},\tau)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{1}}$$\scriptstyle{\alpha}$$\textstyle{(\mathcal{B}_{1},\tau)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathcal{I}_{2},\tau)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathcal{A}_{2},\tau)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{2}}$$\textstyle{(\mathcal{B}_{2},\tau)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

Then, we have
$\iota_{*}\circ\partial_{i}^{\exp}=\partial_{i}^{\exp}\circ\beta_{*}$.

###### Proof.

As in Lemma $8.5$ of [14], this lemma is proved by following the definition of
each map. We assume $\operatorname{\mathrm{Ker}}(\varphi_{j})=\mathcal{I}_{j}$
$(j=1,2)$ for simplicity. Let $[u_{1}]\in KO_{i}^{u}(\mathcal{B}_{1},\tau)$ be
an element represented by a unitary $u_{1}\in M_{n_{i}\cdot
n}(\widetilde{\mathcal{B}}_{1})$ satisfying the symmetry relation
$\mathcal{R}_{i}$. Let $a_{1}\in M_{n_{i}\cdot n}(\widetilde{\mathcal{A}})$ be
a lift of $u_{1}$ such that $||a_{1}||\leq 1$, and satisfy the relation
$\mathcal{R}_{i}$. Then, $a_{2}=\alpha(a_{1})$ is a lift of $u_{2}$ satisfying
the symmetry, and thus, $\partial_{i}^{\exp}([u_{2}])=[C^{\prime}(a_{2})]$
holds. Since
$\alpha(C^{\prime}(a_{1}))=C^{\prime}(\alpha(a_{1}))=C^{\prime}(\alpha_{2})$,
we have

$\iota_{*}\circ\partial_{i}^{\exp}([u_{1}])=\iota_{*}[C^{\prime}(a_{1})]=[\alpha(C^{\prime}(a_{1})]\\\
=\partial_{i}^{\exp}([u_{2}])=\partial_{i}^{\exp}\circ\beta_{*}[u_{1}].\qed$

######  Proposition A.16.

$\partial_{i}^{\mathrm{BL}}=-\partial_{i}^{\exp}$ for odd $i$.

As in the proof of Theorem $8.9$ of [14], we can reduce the proof to the
complex case, and Proposition A.16 follows from the lemma below. In the
complex case, the boundary map $\partial_{1}^{\exp}\colon
K_{1}(\mathcal{B})\to K_{0}(\mathcal{I})$ is defined by forgetting the real
structure in the case of $i=1$ of Definition A.13.

######  Lemma A.17.

The boundary maps $\partial_{1}^{\mathrm{BL}}$ and $\partial_{1}^{\exp}$ from
$K_{1}(\mathcal{B})$ to $K_{0}(\mathcal{I})$ satisfy the relation
$\partial_{1}^{\mathrm{BL}}=-\partial_{1}^{\exp}$.

###### Proof.

Suppose that $[u]\in K_{1}(\mathcal{B})$ where $u\in
M_{n}(\tilde{\mathcal{B}})$ and $\lambda(u)=1_{n}$. We take a lift $a$ of $u$
in $M_{n}(\tilde{A})$ satisfying $||a||\leq 1$. Consider the partial isometry
$v=\left(\begin{array}[]{cc}a&0\\\ \sqrt{1-a^{*}a}&0\end{array}\right)$ and
let $V=\left(\begin{array}[]{cc}0&v^{*}\\\ v&0\end{array}\right)$.
$\partial_{1}^{\exp}([u])$ is computed as

$\displaystyle\partial_{1}^{\exp}([u])$
$\displaystyle=\left[-Y_{4n}^{(1)}\bigl{(}\epsilon_{1}\exp(\pi
V\epsilon_{1})\bigl{)}Y_{4n}^{(1)*}\right]$
$\displaystyle=[Y_{4n}^{(1)}(1-2vv^{*})Y_{4n}^{(1)^{*}}]-[Y_{4n}^{(1)}(1-2v^{*}v)Y_{4n}^{(1)^{*}}].$

As in [58], $\partial_{1}^{\mathrm{BL}}([u])$ is also expressed by using $v$,
which is $-\partial_{1}^{\exp}([u])$. ∎

### Acknowledgments

The author thanks Takeshi Nakanishi, Ryo Okugawa, Guo Chuan Thiang and
Yukinori Yoshimura for the many discussions. This work was supported by JSPS
KAKENHI (Grant Nos. JP17H06461, JP19K14545) and JST PRESTO (Grant No.
JPMJPR19L7).

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