# Classification of Topological Invariants Related to Corner States

Shin Hayashi<EMAIL_ADDRESS>
###### Abstract.

We discuss some bulk-surfaces gapped Hamiltonians on a lattice with corners,
and propose a periodic table for topological invariants related to corner
states aimed at studies of higher-order topological insulators. Our table is
based on four things: (1) the definition of topological invariants, (2) a
proof of their relation with corner states (3) computations of $K$-groups and
(4) a construction of explicit examples.

###### Key words and phrases:

Topologically protected corner states, higher-order topological insulators,
$K$-theory and index theory

###### 1991 Mathematics Subject Classification:

Primary 19K56; Secondary 47B35, 81V99.

AIST-TohokuU Mathematics for Advanced Materials - Open Innovation Laboratory,
National Institute of Advanced Industrial Science and Technology, 2-1-1
Katahira, Aoba, Sendai 980-8577, Japan

JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama, 332-0012, Japan

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 $KO$-Groups of $C^{*}$-algebras Associated with Half-Plane and Quarter-Plane Toeplitz Operators
  4. 4 Toeplitz Operators Associated with Subsemigroup $(\mathbb{Z}_{\geq 0})^{n}$ of $\mathbb{Z}^{n}$
  5. 5 Topological invariants and corner states in Altland–Zirnbauer classification
  6. A $\mathbb{Z}_{2}$-Spaces of Self-Adjoint/Skew-Adjoint Fredholm Operators and Boersema–Loring’s $K$-theory

##  1\. Introduction

Recent developments in condensed matter physics have greatly generalized the
bulk-boundary correspondence for topological insulators to include corner
states. Topological insulators have a gapped bulk, which incorporates some
topology that do not change unless the spectral gap of the bulk Hamiltonian
closes under deformations. Examples include the TKNN number for quantum Hall
systems [65] and the Kane-Mele $\mathbb{Z}_{2}$ index for quantum spin Hall
systems [36]. It is known that, corresponding to these bulk invariants,
gapless edge states appear, which is called the bulk-boundary correspondence
[31]. After Schnyder–Ryu–Furusaki–Ludwig’s classification of topological
insulators [60] for ten Altland–Zirnbauer classes [2], Kitaev noted the role
of $K$-theory and Bott periodicity in the classification problem and obtained
the famous periodic table [42]. Recently, some (at least bulk) gapped systems
possessing in-gap or gapless states localized around a higher codimensional
part of the boundary (corners or hinges) are studied [30, 12, 41], which are
called higher-order topological insulators (HOTIs) [59]. For example, for
second-order topological insulators, not only is the bulk gapped but also the
codimension-one boundaries (edges, surfaces), and an in-gap or a gapless state
appears around codimension-two corners or hinges. In this framework,
conventional topological insulators are regarded as first-order topological
insulators. HOTIs are now actively studied and the classification of HOTIs has
also been proposed [26, 40, 52]. Generalizing the bulk-boundary
correspondence, relations between some gapped topology and corner states are
much discussed [66, 5, 63].

Initiated by Bellissard, $K$-theory and index theory are known to provide a
powerful tool to study topological insulators. Bellissard–van Elst–Schulz-
Baldes studied quantum Hall effects by means of noncommutative geometry [10,
11], and Kellendonk–Richter–Schulz-Baldes went on to prove the bulk-boundary
correspondence by using index theory for Toeplitz operators [39]. The study of
topological insulators, especially regarding its classification and the bulk-
boundary correspondence for each of the ten Altland–Zirnbauer classes by using
$K$-theory and index theory has been much developed [39, 24, 64, 16, 29, 48,
57, 64, 17, 38, 44, 1]. In [32], three-dimensional (3-D) class A bulk periodic
systems are studied on one piece of a lattice cut by two specific hyperplanes,
which is a model for systems with corners. Based on the index theory for
quarter-plane Toeplitz operators [62, 23, 54], a topological invariant is
defined assuming the spectral gap both on the bulk Hamiltonian and two half-
space compressions of it. This gapped topological invariant is topological in
the sense that it does not change under continuous deformation of the bulk
Hamiltonians unless the spectral gap of one of the two surfaces closes. It is
proved that, corresponding to this topology gapless corner states appear. A
construction of nontrivial examples by using two first-order topological
insulators (of 2-D class A and 1-D class AIII) is also proposed. Class AIII
codimension-two systems are also studied through this method in [33] and, as
an application to HOTIs, the appearance of topological corner states in
Benalcazar–Bernevig–Hughes’ 2-D model [12] is explained based on the chiral
symmetry. The construction of examples in [33] leads to a proposal of second-
order semimetallic phase protected by the chiral symmetry [51].

The purpose of this paper is to expand the results in [32] to all
Altland–Zirnbauer classes and systems with corners of arbitrary codimension.
Since class A and class AIII systems (with codimension-two corners) were
already discussed in [32, 33] by using complex $K$-theory, we focus on the
remaining eight cases, for which we use real $K$-theory. For our expansion, a
basic scheme has already been well developed in the above previous studies,
which we mainly follow: some gapped Hamiltonian defines an element of a
$KO$-group of a real $C^{*}$-algebra, and its relation with corner states are
clarified by using index theory [39, 24, 16, 29, 44, 64, 17, 38]. Although
many techniques have already been developed in studies of topological
insulators, in our higher-codimensional cases, we still lack some basic
results at the level of $K$-theory and index theory; hence, the first half of
this paper is devoted to these $K$-theoretic preliminaries, that is, the
computation of $KO$-groups for real $C^{*}$-algebras associated with the
quarter-plane Toeplitz extension and the computation of boundary maps for the
$24$-term exact sequence of $KO$-theory associated with it, which are carried
out in Sect. 3. Since the quarter-plane Toeplitz extension [54] is a key tool
in our study of codimension-two corners, such a variant for Toeplitz operators
associated with higher-codimensional corners should be clarified, which are
carried out in Sect. 4. These variants of Toeplitz operators were discussed in
[23, 22], and the contents in Sect. 4 will be well-known to experts. Since the
author could not find an appropriate reference, especially concerning Theorem
4.1 which will play a key role in Sect. 5, the results are included for
completeness. Note that the idea there to use tensor products of the ordinary
Toeplitz extension for the study of these variants is based on the work of
Douglas–Howe [23], where these higher-codimensional generalizations are
briefly mentioned. The study of some gapped phases for systems with corners in
Altland–Zirnbauer’s classification is carried out in Sect. 5. In the framework
of the one-particle approximation, we consider $n$-D systems with a
codimension $k$ corner and take compressions of the bulk Hamiltonian onto
infinite lattices with codimension $k-1$ corners111In standard terminologies,
they will be called edges, surfaces, hinges or edge of edges depending on $n$
and $k$. In this paper, we may simply call them corners but state its
codimensions. whose intersection makes the codimension $k$ corner. We assume
that they are gapped. Note that, under this assumption, bulk, surfaces and
corners up to codimension $k-1$ which constitute the codimension $k$ corner
are also gapped. For such a system, we define two topological invariants as
elements of some $KO$-groups: one is defined for these gapped Hamiltonians
while the other one is related to in-gap or gapless codimension $k$ corner
states. We then show a relation between these two which states that
topologically protected corner states appear reflecting some gapped topology
of the system. We first study codimension-two cases (Sect. 5.1 to Sect. 5.4)
and then discuss higher-codimensional cases (Sect. 5.5). This distinction is
made because many detailed results have been obtained for codimension-two
cases by virtue of previous studies of quarter-plane Toeplitz operators [54,
35] (the shape of the corner we discuss is more general than in higher-
codimensional cases, and a relation between convex and concave corners is also
obtained in [33]). Based on these results, we propose a classification table
for topological invariants related to corner states (Table 1). Note that the
codimension-one case of Table 1 is Kitaev’s table [42] and Table 1 is also
periodic by the Bott periodicity. In order further to clarify a relation
between our invariants and corner states, in Sect. 5.6, we introduce
$\mathbb{Z}$ or $\mathbb{Z}_{2}$-valued numerical corner invariants when the
dimension of the corner is zero or one. They are defined by (roughly speaking)
counting the number of corner states. A construction of examples is discussed
in Sect. 5.7. As in [32], this is given by using pairs of Hamiltonians of two
lower-order topological insulators. In the real classes, there are $64$ pairs
of them and the results are collected in Table LABEL:producttalbe. By using
this method, we can construct nontrivial examples of each entry of Table 1,
starting from Hamiltonians of first-order topological insulators. The corner
invariant for the constructed Hamiltonian is expressed by corner (or edge)
invariants of constituent two Hamiltonians. This is given by using an exterior
product of some $KO$-groups in general, though, as in [32, 33], the formula
for numerical invariants introduced in Sect. 5.6 is also included.

Table 1. Classification of (strong) topological invariants related to corner state in Altland–Zirnbauer (AZ) classification. In this table, $n$ is the dimension of the bulk, and $k$ is the codimension of the corner. Symmetry | $n-k\mod 8$
---|---
AZ | $\Theta$ | $\Xi$ | $\Pi$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$
$\mathrm{A}$ | $0$ | $0$ | $0$ | $0$ | $\mathbb{Z}$ | $0$ | $\mathbb{Z}$ | $0$ | $\mathbb{Z}$ | $0$ | $\mathbb{Z}$
$\mathrm{AIII}$ | $0$ | $0$ | $1$ | $\mathbb{Z}$ | $0$ | $\mathbb{Z}$ | $0$ | $\mathbb{Z}$ | $0$ | $\mathbb{Z}$ | $0$
$\mathrm{AI}$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $2\mathbb{Z}$ | $0$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}$
$\mathrm{BDI}$ | $1$ | $1$ | $1$ | $\mathbb{Z}$ | $0$ | $0$ | $0$ | $2\mathbb{Z}$ | $0$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$
$\mathrm{D}$ | $0$ | $1$ | $0$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}$ | $0$ | $0$ | $0$ | $2\mathbb{Z}$ | $0$ | $\mathbb{Z}_{2}$
$\mathrm{DIII}$ | $-1$ | $1$ | $1$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}$ | $0$ | $0$ | $0$ | $2\mathbb{Z}$ | $0$
$\mathrm{AII}$ | $-1$ | $0$ | $0$ | $0$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}$ | $0$ | $0$ | $0$ | $2\mathbb{Z}$
$\mathrm{CII}$ | $-1$ | $-1$ | $1$ | $2\mathbb{Z}$ | $0$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}$ | $0$ | $0$ | $0$
$\mathrm{C}$ | $0$ | $-1$ | $0$ | $0$ | $2\mathbb{Z}$ | $0$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}$ | $0$ | $0$
$\mathrm{CI}$ | $1$ | $-1$ | $1$ | $0$ | $0$ | $2\mathbb{Z}$ | $0$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}$ | $0$

For computations of $KO$-groups and classification of such gapped systems, we
employ Boersema–Loring’s unitary picture for $KO$-theory [14] whose
definitions are collected in Sect. $2$. Basic results for some Toeplitz
operators are also included there. In Appendix A, we revisit Atiyah–Singer’s
study of spaces of skew-adjoint Fredholm operators [9] and collect necessary
results from the viewpoint of Boersema–Loring’s $K$-theory. Definitions of
some $\mathbb{Z}_{2}$-spaces, maps between them, expression of boundary maps
of $24$-term exact sequences used in this paper are collected there.

Finally, let us point out a relation with our results and the current rapidly
developing studies on HOTIs. In [26], the HOTIs are divided into two classes:
intrinsic HOTIs, which basically originate from the bulk topology protected by
a point group symmetry, and others extrinsic HOTIs. Our study will be for
extrinsic HOTIs since no point group symmetry is assumed and our
classification table (Table 1) is consistent with that of Table $1$ in [26].

##  2\. Preliminaries

In this section, we collect the necessary results and notations.

###  2.1. Boersema–Loring’s $KO$-Groups via Unitary Elements

In this subsection, we collect Boersema–Loring’s definition of $KO$-groups by
using unitaries satisfying some symmetries [14]. The basics of real
$C^{*}$-algebras and $KO$-theory can be found in [28, 61], for example.

A $C^{*,\tau}$-algebra is a pair $(\mathcal{A},\tau)$ consisting of (complex)
$C^{*}$-algebra $\mathcal{A}$ and an anti-automorphism222i.e., a complex
linear automorphism of $A$ that preserves $\ast$ and satisfies
$\tau(ab)=\tau(b)\tau(a)$. $\tau$ of $\mathcal{A}$ satisfying $\tau^{2}=1$. We
call $\tau$ the transposition and write $a^{\tau}$ for $\tau(a)$. There is a
category equivalence between the category of $C^{*,\tau}$-algebras and the
category of real $C^{*}$-algebras: for a $C^{*,\tau}$-algebra $(A,\tau)$, the
corresponding real $C^{*}$-algebra is $\mathcal{A}^{\tau}=\\{a\in\mathcal{A}\
|\ a^{\tau}=a^{*}\\}$, and its inverse is given by the complexification. A
real structure on a (complex) $C^{*}$-algebra $A$ is an antilinear
$*$-automorphism $\mathfrak{r}$ satisfying $\mathfrak{r}^{2}=1$. For a real
structure $\mathfrak{r}$, there is an associated transposition $\tau$ given by
$\tau(a)=\mathfrak{r}(a^{*})$, which gives a one-to-one correspondence between
transpositions and real structures on the $C^{*}$-algebra333Boersema–Loring
called $\tau$ the real structure in [14]. In this paper, we distinguish these
two since the antilinear structure naturally appears in our application. We
call $\tau$ the transposition following [38].. We extend the transposition
$\tau$ on $\mathcal{A}$ to the transposition (for which we simply write
$\tau$) on the matrix algebra $M_{n}(\mathcal{A})$ by
$(a_{ij})^{\tau}=(a_{ji}^{\tau})$ where $a_{ij}\in\mathcal{A}$ and $1\leq
i,j\leq n$. This induces a transposition $\tau_{\mathcal{K}}$ on
$\mathcal{K}\otimes\mathcal{A}$ where $\mathcal{K}=\mathcal{K}(\mathcal{V})$
is the $C^{*}$-algebra of compact operators on a separable complex Hilbert
space $\mathcal{V}$. Let $\sharp\otimes\tau$ be a transposition on
$M_{2}(\mathcal{A})$ defined by444 For notations of the transpositions
introduced here, we follow [14].

$\left(\begin{array}[]{cc}a_{11}&a_{12}\\\
a_{21}&a_{22}\end{array}\right)^{\sharp\otimes\tau}=\left(\begin{array}[]{cc}a_{22}^{\tau}&-a_{12}^{\tau}\\\
-a_{21}^{\tau}&a_{11}^{\tau}\end{array}\right).$

If we identify the quaternions $\mathbb{H}$ with $\mathbb{C}^{2}$ by
$x+yj\mapsto(x,y)$, the left multiplication by $j$ corresponds to
$j(x,y)=(-\bar{y},\bar{x})$. Then, we have
$\sharp\otimes\mathrm{id}=\operatorname{\mathrm{Ad}}_{j}\circ\ast$ where
$\ast$ denotes the operation of taking conjugation of matrices and the
$C^{*,\tau}$-algebra $(M_{2}(\mathbb{C}),\sharp\otimes\mathrm{id})$
corresponds to the real $C^{*}$-algebra $\mathbb{H}$ of quaternions. We extend
this transposition to $M_{2n}(\mathcal{A})$ by
$(b_{ij})^{\sharp\otimes\tau}=(b_{ji}^{\sharp\otimes\tau})$ where $1\leq
i,j\leq n$ and $b_{ij}\in M_{2}(\mathcal{A})$. On $M_{2n}(\mathcal{A})$, we
also consider a transposition $\widetilde{\sharp}\otimes\tau$ defined by

$\left(\begin{array}[]{cc}c_{11}&c_{12}\\\
c_{21}&c_{22}\end{array}\right)^{\widetilde{\sharp}\otimes\tau}=\left(\begin{array}[]{cc}c_{22}^{\tau}&-c_{12}^{\tau}\\\
-c_{21}^{\tau}&c_{11}^{\tau}\end{array}\right),$

where $c_{ij}\in M_{n}(\mathcal{A})$. For an $m\times m$ matrix $X$, we write
$X_{n}$ for the $mn\times mn$ block diagonal matrix
$\mathrm{diag}(X,\ldots,X)$. For example, we write $1_{n}$ for the $n\times n$
diagonal matrix $\mathrm{diag}(1,\ldots,1)$.

######  Definition 2.1 (Boersema–Loring [14]).

Let $(\mathcal{A},\tau)$ be a unital $C^{*,\tau}$-algebra. For
$i=-1,0,\ldots,6$, let $n_{i}$ be a positive integer, $\mathcal{R}_{i}$ be a
relation and $I^{(i)}$ be a matrix, as indicated in Table 2. Let
$U^{(i)}_{k}(\mathcal{A},\tau)$ be the set of all unitaries in $M_{n_{i}\cdot
k}(\mathcal{A})$ satisfying the relation $\mathcal{R}_{i}$. On the set
$U^{(i)}_{\infty}(\mathcal{A},\tau)=\cup_{k=1}^{\infty}U^{(i)}_{k}(\mathcal{A},\tau)$,
we consider the equivalence relation $\sim_{i}$ generated by homotopy and
stabilization given by $I^{(i)}$. We define
$KO_{i}(\mathcal{A},\tau)=U^{(i)}_{\infty}(\mathcal{A},\tau)/\sim_{i}$ which
is a group by the binary operation given by $[u]+[v]=[\mathrm{diag}(u,v)]$.

For a nonunital $C^{*,\tau}$-algebra $(A,\tau)$, the $i$-th $KO$-group
$KO_{i}(\mathcal{A},\tau)$ is defined as the kernel of $\lambda_{*}\colon
KO_{i}(\tilde{\mathcal{A}},\tau)\to KO_{i}(\mathbb{C},\mathrm{id})$, where
$\tilde{\mathcal{A}}$ is the unitization of $\mathcal{A}$ and
$\lambda\colon\tilde{\mathcal{A}}\to\mathbb{C}$ is the natural projection. In
[14], they also describe the boundary maps of the $24$-term exact sequence for
$KO$-theory associated with a short exact sequence of $C^{*,\tau}$-algebras.
In Appendix A.3, we discuss an alternative description for some of them
through exponentials.

Table 2. Boersema–Loring’s unitary picture for $KO$-theory [14] $i$ |  $KO$-group | $n_{i}$ | $\mathcal{R}_{i}$ | $I^{(i)}$
---|---|---|---|---
$-1$ | $KO_{-1}(\mathcal{A},\tau)$ | $1$ | $u^{\tau}=u$ | $1$
$0$ | $KO_{0}(\mathcal{A},\tau)$ | $2$ |  $u=u^{*}$, $u^{\tau}=u^{*}$ | $\mathrm{diag}(1,-1)$
$1$ | $KO_{1}(\mathcal{A},\tau)$ | $1$ | $u^{\tau}=u^{*}$ | $1$
$2$ | $KO_{2}(\mathcal{A},\tau)$ | $2$ |  $u=u^{*}$, $u^{\tau}=-u$ | $\left(\begin{array}[]{cc}0&i\cdot 1\\\ -i\cdot 1&0\end{array}\right)$
$3$ | $KO_{3}(\mathcal{A},\tau)$ | $2$ | $u^{\sharp\otimes\tau}=u$ | $1_{2}$
$4$ | $KO_{4}(\mathcal{A},\tau)$ | $4$ |  $u=u^{*}$, $u^{\sharp\otimes\tau}=u^{*}$ | $\mathrm{diag}(1_{2},-1_{2})$
$5$ | $KO_{5}(\mathcal{A},\tau)$ | $2$ | $u^{\sharp\otimes\tau}=u^{*}$ | $1_{2}$
$6$ | $KO_{6}(\mathcal{A},\tau)$ | $2$ |  $u=u^{*}$, $u^{\sharp\otimes\tau}=-u$ | $\left(\begin{array}[]{cc}0&i\cdot 1\\\ -i\cdot 1&0\end{array}\right)$

###  2.2. Toeplitz Operators

In this subsection, we collect the definitions and basic results for some
Toeplitz operators used in this paper [22, 54].

Let $\mathbb{T}$ be the unit circle in the complex plane $\mathbb{C}$, and let
$c$ be the complex conjugation on $\mathbb{C}$, that is, $c(z)=\bar{z}$. Let
$n$ be a positive integer. On the $n$-dimensional torus $\mathbb{T}^{n}$, we
consider an involution $\zeta$ defined as the $n$-fold product of $c$. This
induces a transposition $\tau_{\mathbb{T}}$ on $C(\mathbb{T}^{n})$ by
$(\tau_{\mathbb{T}}f)(t)=f(\zeta(t))$. Let $\mathbb{Z}_{\geq 0}$ be the set of
non-negative integers and $P_{n}$ be the orthogonal projection of
$l^{2}(\mathbb{Z}^{n})$ onto $l^{2}((\mathbb{Z}_{\geq 0})^{n})$. For a
continuous function $f\colon\mathbb{T}^{n}\to\mathbb{C}$, let $M_{f}$ be the
multiplication operator on $l^{2}(\mathbb{Z}^{n})$ generated by $f$. We
consider the operator $P_{n}M_{f}P_{n}$ on $l^{2}((\mathbb{Z}_{\geq 0})^{n})$,
which is the Toeplitz operator associated with the subsemigroup
$(\mathbb{Z}_{\geq 0})^{n}$ of $\mathbb{Z}^{n}$ of symbol $f$. We write
$\mathcal{T}^{n}$ for the $C^{*}$-subalgebra of
$\mathcal{B}(l^{2}((\mathbb{Z}_{\geq 0})^{n}))$ generated by these Toeplitz
operators. The algebra $\mathcal{T}^{1}$ is the ordinary Toeplitz algebra and
we simply write $\mathcal{T}$. Note that the algebra $\mathcal{T}^{n}$ is
isomorphic to the $n$-fold tensor product of $\mathcal{T}$. The complex
conjugation $c$ on $\mathbb{C}$ induces an antiunitary operator555An operator
$A$ on a complex Hilbert space $\mathcal{V}$ is called the antiunitary
operator if $A$ is an antilinear bijection on $\mathcal{V}$ satisfying
$\langle Av,Aw\rangle=\overline{\langle v,w\rangle}$ for any $v$ and $w$ in
$\mathcal{V}$. of order two on the Hilbert space $l^{2}(\mathbb{Z}^{n})$ by
the pointwise operation, for which we also write $c$. This induces a real
structure $\mathfrak{c}$ on $\mathcal{B}(l^{2}((\mathbb{Z}_{\geq 0})^{n}))$ by
$\mathfrak{c}(a)=\operatorname{\mathrm{Ad}}_{c}(a)=cac^{*}$. We write
$\tau_{\mathcal{T}}$ for the transposition on $\mathcal{T}^{n}$ given by its
restriction onto $\mathcal{T}^{n}$.

We next focus on the case of $n=2$. We consider the Hilbert space
$l^{2}(\mathbb{Z}^{2})$ and take its orthonormal basis $\\{\delta_{m,n}\ |\
(m,n)\in\mathbb{Z}^{2}\\}$, where $\delta_{m,n}$ is the characteristic
function of the point $(m,n)$ on $\mathbb{Z}^{2}$. When $f\in
C(\mathbb{T}^{2})$ is given by $f(z_{1},z_{2})=z_{1}^{m}z_{2}^{n}$, we write
$M_{m,n}$ for the multiplication operator $M_{f}$. Let $\alpha<\beta$ be real
numbers, and let $\mathcal{H}^{\alpha}$, $\mathcal{H}^{\beta}$,
$\hat{\mathcal{H}}^{\alpha,\beta}$ and $\check{\mathcal{H}}^{\alpha,\beta}$ be
closed subspaces of $l^{2}(\mathbb{Z}^{2})$ spanned by $\\{\delta_{m,n}\
|-\alpha m+n\geq 0\\}$, $\\{\delta_{m,n}\ |-\beta m+n\leq 0\\}$,
$\\{\delta_{m,n}\ |-\alpha m+n\geq 0\ \text{and}\ -\beta m+n\leq 0\\}$, and
$\\{\delta_{m,n}\ |-\alpha m+n\geq 0\ \text{or}\ -\beta m+n\leq 0\\}$,
respectively. In the following, we may take $\alpha=-\infty$ or
$\beta=\infty$, but not both. Let $P^{\alpha}$, $P^{\beta}$,
$\hat{P}^{\alpha,\beta}$ and $\check{P}^{\alpha,\beta}$ be the orthogonal
projection of $l^{2}(\mathbb{Z}^{2})$ onto $\mathcal{H}^{\alpha}$,
$\mathcal{H}^{\beta}$, $\hat{\mathcal{H}}^{\alpha,\beta}$ and
$\check{\mathcal{H}}^{\alpha,\beta}$, respectively. For $f\in
C(\mathbb{T}^{2})$, the operators $P^{\alpha}M_{f}P^{\alpha}$ on
$\mathcal{H}^{\alpha}$ and $P^{\beta}M_{f}P^{\beta}$ on $\mathcal{H}^{\beta}$
are called half-plane Toeplitz operators. The operator
$\hat{P}^{\alpha,\beta}M_{f}\hat{P}^{\alpha,\beta}$ on
$\hat{\mathcal{H}}^{\alpha,\beta}$ is called the quarter-plane Toeplitz
operator, and $\check{P}^{\alpha,\beta}M_{f}\check{P}^{\alpha,\beta}$ on
$\check{\mathcal{H}}^{\alpha,\beta}$ is its concave corner analogue. We write
$\mathcal{T}^{\alpha}$ and $\mathcal{T}^{\beta}$ for $C^{*}$-algebras
generated by these half-plane Toeplitz operators and
$\hat{\mathcal{T}}^{\alpha,\beta}$ and $\check{\mathcal{T}}^{\alpha,\beta}$
for $C^{*}$-algebras generated by the quarter-plane and concave corner
Toeplitz operators, respectively. There are $*$-homomorphisms
$\sigma^{\alpha}\colon\mathcal{T}^{\alpha}\to C(\mathbb{T}^{2})$ and
$\sigma^{\beta}\colon\mathcal{T}^{\beta}\to C(\mathbb{T}^{2})$, which map
$P^{\alpha}M_{f}P^{\alpha}$ and $P^{\beta}M_{f}P^{\beta}$ to the symbol $f$,
respectively. We define the $C^{*}$-algebra $\mathcal{S}^{\alpha,\beta}$ as a
pullback $C^{*}$-algebra of these two $*$-homomorphisms. The real structure
$c$ on $\mathcal{H}=l^{2}(\mathbb{Z}^{2})$ induces real structures
$\mathfrak{c}$ on $\mathcal{T}^{\alpha}$, $\mathcal{T}^{\beta}$,
$\hat{\mathcal{T}}^{\alpha,\beta}$, $\check{\mathcal{T}}^{\alpha,\beta}$, and
$\mathcal{S}^{\alpha,\beta}$ and thus induces transpositions $\tau_{\alpha}$,
$\tau_{\beta}$, $\hat{\tau}_{\alpha,\beta}$, $\check{\tau}_{\alpha,\beta}$ and
$\tau_{\mathcal{S}}$ on $\mathcal{T}^{\alpha}$, $\mathcal{T}^{\beta}$,
$\hat{\mathcal{T}}^{\alpha,\beta}$, $\check{\mathcal{T}}^{\alpha,\beta}$ and
$\mathcal{S}^{\alpha,\beta}$, respectively. For transpositions, we may simply
write $\tau$ when it is clear from the context. The maps $\sigma^{\alpha}$ and
$\sigma^{\beta}$ preserve the real structures and we have the following pull-
back diagram:

(2.1)
$\textstyle{(\mathcal{S}^{\alpha,\beta},\tau_{\mathcal{S}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{\beta}}$$\scriptstyle{p^{\alpha}}$$\textstyle{(\mathcal{T}^{\beta},\tau_{\beta})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma^{\beta}}$$\textstyle{(\mathcal{T}^{\alpha},\tau_{\alpha})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma^{\alpha}}$$\textstyle{(C(\mathbb{T}^{2}),\tau_{\mathbb{T}})}$

We write $\sigma$ for the composition $\sigma^{\alpha}\circ
p^{\alpha}=\sigma^{\beta}\circ p^{\beta}$. Let $\hat{\gamma}$ be a
$*$-homomorphism from $\hat{\mathcal{T}}^{\alpha,\beta}$ to
$\mathcal{S}^{\alpha,\beta}$ which maps
$\hat{P}^{\alpha,\beta}M_{f}\hat{P}^{\alpha,\beta}$ to the pair
$(P^{\alpha}M_{f}P^{\alpha},P^{\beta}M_{f}P^{\beta})$. This map $\hat{\gamma}$
preserves the real structures, and there is the following short exact sequence
of $C^{*,\tau}$-algebras (Park [54]):

(2.2)
$0\to(\mathcal{K}(\hat{\mathcal{H}}^{\alpha,\beta}),\tau_{\mathcal{K}})\to(\hat{\mathcal{T}}^{\alpha,\beta},\hat{\tau}_{\alpha,\beta})\overset{\hat{\gamma}}{\to}(\mathcal{S}^{\alpha,\beta},\tau_{\mathcal{S}})\to
0,$

where the map from
$(\mathcal{K}(\hat{\mathcal{H}}^{\alpha,\beta}),\tau_{\mathcal{K}})$ to
$(\hat{\mathcal{T}}^{\alpha,\beta},\hat{\tau}_{\alpha,\beta})$ is the
inclusion. Its concave corner analogue is studied in [33] and the following
exact sequence is obtained:

(2.3)
$0\to(\mathcal{K}(\check{\mathcal{H}}^{\alpha,\beta}),\tau_{\mathcal{K}})\to(\check{\mathcal{T}}^{\alpha,\beta},\check{\tau}_{\alpha,\beta})\overset{\check{\gamma}}{\to}(\mathcal{S}^{\alpha,\beta},\tau_{\mathcal{S}})\to
0,$

where $\check{\gamma}$ is a $*$-homomorphism mapping
$\check{P}^{\alpha,\beta}M_{f}\check{P}^{\alpha,\beta}$ to
$(P^{\alpha}M_{f}P^{\alpha},P^{\beta}M_{f}P^{\beta})$.

##  3\. $KO$-Groups of $C^{*}$-algebras Associated with Half-Plane and
Quarter-Plane Toeplitz Operators

In this section, the $KO$-theory for half-plane and quarter-plane Toeplitz
operators are discussed. In Sect. 3.1, $KO$-groups of the half-plane Toeplitz
algebra is computed. Quarter-plane Toeplitz operators are discussed in the
following sections, and the $KO$-groups of the $C^{*,\tau}$-algebra
$(\mathcal{S}^{\alpha,\beta},\tau_{\mathcal{S}})$ are computed in Sect. 3.2.
In Sect 3.3, the boundary maps of the $24$-term exact sequence for $KO$-theory
associated with the sequence (2.2) are discussed and the $KO$-groups of the
quarter-plane Toeplitz algebra
$(\hat{\mathcal{T}}^{\alpha,\beta},\hat{\tau}_{\alpha,\beta})$ are computed.

###  3.1. $KO$-Groups of $(\mathcal{T}^{\alpha},\tau_{\alpha})$

We compute the $KO$-groups of the $C^{*,\tau}$-algebra
$(\mathcal{T}^{\alpha},\tau_{\alpha})$. The discussion is divided into two
cases whether $\alpha$ is rational (or $-\infty$) or irrational.

We first consider the case when $\alpha$ is a rational number or $-\infty$.
When $\alpha\in\mathbb{Q}$, we write $\alpha=\frac{p}{q}$ where $p$ and $q$
are relatively prime integers and $q$ is positive. Let $m$ and $n$ be integers
such that $-pm+qn=1$ and let

(3.1) $\Gamma=\left(\begin{array}[]{cc}n&-m\\\ -p&q\end{array}\right)\in
SL(2,\mathbb{Z}).$

Then, the action of $\Gamma$ on $\mathbb{Z}^{2}$ induces the Hilbert space
isomorphism $\mathcal{H}^{\alpha}\cong\mathcal{H}^{0}$ and an isomorphism of
$C^{*,\tau}$-algebras
$(\mathcal{T}^{\alpha},\tau_{\alpha})\cong(\mathcal{T}^{0},\tau_{0})$. Thus,
the $C^{*,\tau}$-algebra $(\mathcal{T}^{\alpha},\tau_{\alpha})$ is isomorphic
to $(\mathcal{T},\tau_{\mathcal{T}})\otimes(C(\mathbb{T}),\tau{T})$, and its
$KO$-groups are computed as $KO_{i}(\mathcal{T}^{\alpha},\tau_{\alpha})\cong
KO_{i}(C(\mathbb{T}),\tau{T})\cong KO_{i}(\mathbb{C},\mathrm{id})\oplus
KO_{i-1}(\mathbb{C},\mathrm{id})$. For the first isomorphism, see Proposition
$1.5.1$ of [61]. Generators of the group $KO_{i}(C(\mathbb{T}),\tau{T})$ are
obtained in Example 9.2 of [14], and the unital $*$-homomorphism
$\iota\colon\mathbb{C}\to\mathcal{T}$ induces an isomorphism
$(\mathrm{id}\otimes\iota)_{*}\colon KO_{i}(C(\mathbb{T}),\tau{T})\to
KO_{i}((C(\mathbb{T}),\tau{T})\otimes(\mathcal{T},\tau_{\mathcal{T}}))\cong
KO_{i}(\mathcal{T}^{0},\tau_{0})$. Combined with them, $KO$-group
$KO_{i}(\mathcal{T}^{\alpha},\tau_{\alpha})$ and its generators are given as
follows.

  * •

$KO_{0}(\mathcal{T}^{\alpha},\tau_{\alpha})\cong\mathbb{Z}$ and its generator
is $[1_{2}]$.

  * •

$KO_{1}(\mathcal{T}^{\alpha},\tau_{\alpha})\cong\mathbb{Z}_{2}\oplus\mathbb{Z}$.
A generator of $\mathbb{Z}_{2}$ is $[-1]$ and that of $\mathbb{Z}$ is
$[P^{\alpha}M_{q,p}P^{\alpha}]$.

  * •

$KO_{2}(\mathcal{T}^{\alpha},\tau_{\alpha})\cong(\mathbb{Z}_{2})^{2}$. A
generator of one $\mathbb{Z}_{2}$ is $[-I^{(2)}]$, and that of another
$\mathbb{Z}_{2}$ is
$\left[\left(\begin{array}[]{cc}0&iP^{\alpha}M_{q,p}P^{\alpha}\\\
-iP^{\alpha}M_{-q,-p}P^{\alpha}&0\end{array}\right)\right]$.

  * •

$KO_{3}(\mathcal{T}^{\alpha},\tau_{\alpha})\cong\mathbb{Z}_{2}$ and its
generator is
$[\mathrm{diag}(P^{\alpha}M_{q,p}P^{\alpha},P^{\alpha}M_{-q,-p}P^{\alpha})]$.

  * •

$KO_{4}(\mathcal{T}^{\alpha},\tau_{\alpha})\cong\mathbb{Z}$ and its generator
is $[1_{4}]$.

  * •

$KO_{5}(\mathcal{T}^{\alpha},\tau_{\alpha})\cong\mathbb{Z}$ and its generator
is $[\mathrm{diag}(P^{\alpha}M_{q,p}P^{\alpha},P^{\alpha}M_{q,p}P^{\alpha})]$.

  * •

$KO_{6}(\mathcal{T}^{\alpha},\tau_{\alpha})=KO_{-1}(\mathcal{T}^{\alpha},\tau_{\alpha})=0.$

The case of $\alpha=-\infty$ is computed similarly, and its generators are
given by replacing $p$ and $q$ above with $-1$ and $0$, respectively.

We next consider the cases of irrational $\alpha$. In this case, complex
$K$-groups of $\mathcal{T}^{\alpha}$ are computed by Ji–Kaminker and Xia in
[34, 69].

######  Lemma 3.1.

For irrational $\alpha$ and for each $i$, we have
$KO_{i}(\mathcal{T}^{\alpha},\tau_{\alpha})\cong
KO_{i}(\mathbb{C},\mathrm{id})$, where the isomorphism is given by
$\lambda^{\alpha}_{*}$.

###### Proof.

As for complex $K$-groups, we have $K_{0}(\mathcal{T}^{\alpha})=\mathbb{Z}$
and $K_{1}(\mathcal{T}^{\alpha})=0$ by [34, 69]. We consider a split
$*$-homomorphism of $C^{*,\tau}$-algebras
$\lambda^{\alpha}\colon(\mathcal{T}^{\alpha},\tau_{\alpha})\to(\mathbb{C},\mathrm{id})$
given by the composition of
$\sigma^{\alpha}\colon(\mathcal{T}^{\alpha},\tau_{\alpha})\to(C(\mathbb{T}^{2}),\tau_{\mathbb{T}})$
and the pull-back onto a fixed point of the involution $\zeta$ on
$\mathbb{T}^{2}$. Let
$\mathcal{T}^{\alpha}_{0}=\operatorname{\mathrm{Ker}}\lambda^{\alpha}$. By the
six-term exact sequence associated with the extension
$0\to\mathcal{T}^{\alpha}_{0}\to\mathcal{T}^{\alpha}\overset{\lambda^{\alpha}}{\to}\mathbb{C}\to
0$, complex $K$-groups of $\mathcal{T}^{\alpha}_{0}$ are trivial. For a
$C^{*,\tau}$-algebra $(\mathcal{A},\tau)$, it follows from Theorem $1.12$,
Proposition $1.15$ and Theorem $1.18$ of [13] that
$KO_{*}(\mathcal{A},\tau)=0$ if and only if $K_{*}(\mathcal{A})=0$. Therefore,
$KO_{*}(\mathcal{T}^{\alpha}_{0},\tau_{\alpha})=0$. The result follows from
the $24$-term exact sequence of $KO$-theory for $C^{*,\tau}$-algebras
associated with the short exact sequence
$0\to(\mathcal{T}^{\alpha}_{0},\tau_{\alpha})\to(\mathcal{T}^{\alpha},\tau_{\alpha})\overset{\lambda^{\alpha}}{\to}(\mathbb{C},\mathrm{id})\to
0$. ∎

###  3.2. $KO$-Groups of $(\mathcal{S}^{\alpha,\beta},\tau_{\mathcal{S}})$

In this subsection, we compute the $KO$-groups of the $C^{*,\tau}$-algebra
$(\mathcal{S}^{\alpha,\beta},\tau_{\mathcal{S}})$. The basic tool is the
following Mayer–Vietoris exact sequence associated with the pull-back diagram
(2.1) (see Theorem $1.4.15$ of [61], for example):

(3.2)
$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{KO_{i+1}(C(\mathbb{T}^{2}),\tau_{\mathbb{T}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{i+1}}$$\textstyle{KO_{i}(\mathcal{S}^{\alpha,\beta},\tau_{\mathcal{S}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(p^{\alpha}_{*},p^{\beta}_{*})\hskip
25.60747pt}$$\textstyle{KO_{i}(\mathcal{T}^{\alpha},\tau_{\alpha})\oplus
KO_{i}(\mathcal{T}^{\beta},\tau_{\beta})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hskip
19.91692pt\sigma^{\beta}_{*}-\sigma^{\alpha}_{*}}$$\textstyle{KO_{i}(C(\mathbb{T}^{2}),\tau_{\mathbb{T}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{i}}$$\textstyle{KO_{i-1}(\mathcal{S}^{\alpha,\beta},\tau_{\mathcal{S}})}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

As in [54], the computation of the group
$KO_{*}(\mathcal{S}^{\alpha,\beta},\tau_{\mathcal{S}})$ is divided into three
cases corresponding to whether $\alpha$ and $\beta$ are rational (or
$\pm\infty$) or irrational. As in Sect. 3, we have a unital $*$-homomorphism
$\lambda^{\alpha}\circ
p^{\alpha}\colon(\mathcal{S}^{\alpha,\beta},\tau_{\mathcal{S}})\to(\mathbb{C},\mathrm{id})$
which splits. Correspondingly, the $KO$-group
$KO_{*}(\mathcal{S}^{\alpha,\beta},\tau_{\mathcal{S}})$ have a direct summand
corresponding to $KO_{*}(\mathbb{C},\mathrm{id})$. Noting this, these
$KO$-groups are computed by Lemma 3.1 and the sequence (3.2) when at least one
of $\alpha$ and $\beta$ is irrational. The results are collected in Tables 5
and 5. In the rest of this subsection, we focus on the cases when both
$\alpha$ and $\beta$ are rational (or $\pm\infty$).

When $\alpha,\beta\in\mathbb{Q}$, we write $\alpha=\frac{p}{q}$ and
$\beta=\frac{r}{s}$ by using mutually prime integers where $q$ and $s$ are
positive. In the following discussion, the case of $\alpha=-\infty$
corresponds to the case where $p=-1$ and $q=0$, and the case of
$\beta=+\infty$ corresponds to the case where $r=1$ and $s=0$. By using the
action of $\Gamma\in SL(2,\mathbb{Z})$ in (3.1) on $\mathbb{Z}^{2}$, there are
isomorphisms
$(\mathcal{T}^{\alpha},\tau_{\alpha})\cong(\mathcal{T}^{0},\tau_{0})$ and
$(\mathcal{T}^{\beta},\tau_{\beta})\cong(\mathcal{T}^{\gamma},\tau_{\gamma})$,
where $\gamma=\frac{t}{u}$ for $u=ns-mr$ and $t=-ps+qr$. Note that $t$ is
positive since $\alpha<\beta$. We have the following commutative diagram:

$\textstyle{KO_{i}(\mathcal{T}^{\alpha},\tau_{\alpha})\oplus
KO_{i}(\mathcal{T}^{\beta},\tau_{\beta})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hskip
28.45274pt\sigma^{\beta}_{*}-\sigma^{\alpha}_{*}}$$\scriptstyle{\cong}$$\textstyle{KO_{i}(C(\mathbb{T}^{2}),\tau_{\mathbb{T}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{KO_{i}(\mathcal{T}^{0},\tau_{0})\oplus
KO_{i}(\mathcal{T}^{\gamma},\tau_{\gamma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hskip
28.45274pt\sigma^{\gamma}_{*}-\sigma^{0}_{*}}$$\textstyle{KO_{i}(C(\mathbb{T}^{2}),\tau_{\mathbb{T}})}$

where the vertical isomorphisms are induced by the action of $\Gamma$. In the
following, we discuss the lower part of the diagram, which is enough for our
purpose since the isomorphism
$KO_{i}(\mathcal{S}^{\alpha,\beta},\tau_{\mathcal{S}})\cong
KO_{i}(\mathcal{S}^{0,\gamma},\tau_{\mathcal{S}})$ is also induced. We write
$\varphi_{i}$ for the above map $\sigma^{\gamma}_{*}-\sigma^{0}_{*}$. By the
exact sequence (3.2), we have the following short exact sequence.

(3.3) $0\to\operatorname{\mathrm{Coker}}(\varphi_{i+1})\to
KO_{i}(\mathcal{S}^{0,\gamma},\tau_{\mathcal{S}})\to\operatorname{\mathrm{Ker}}(\varphi_{i})\to
0.$

We first compute kernels and cokernels of $\varphi_{i}$. Cases for
$i=-1,0,4,6$ is easy, thus we consider the other cases.

When $i=1$, groups $KO_{1}(\mathcal{T}^{0},\tau_{0})$ and
$KO_{1}(\mathcal{T}^{\gamma},\tau_{\gamma})$ are both isomorphic to
$\mathbb{Z}_{2}\oplus\mathbb{Z}$. The $\mathbb{Z}_{2}$ direct summand is
generated by $[-1]$, and the other $\mathbb{Z}$ direct summand is generated by
$[P^{0}M_{1,0}P^{0}]$ and $[P^{\gamma}M_{u,t}P^{\gamma}]$, respectively. They
map to $[M_{1,0}]$ and $[M_{u,t}]$ in
$KO_{1}(C(\mathbb{T}^{2}),\tau_{\mathbb{T}})$ by $\sigma^{0}_{*}$ and
$\sigma^{\gamma}_{*}$, respectively. We have
$KO_{1}(C(\mathbb{T}^{2}),\tau_{\mathbb{T}})\cong\mathbb{Z}_{2}\oplus\mathbb{Z}^{2}$,
where the $\mathbb{Z}_{2}$ direct summand is generated by $[-1]$. For
$(m,n)\in\mathbb{Z}^{2}$, the element $[M_{m,n}]\in
KO_{1}(C(\mathbb{T}^{2}),\tau_{\mathbb{T}})$ corresponds to
$(0,m,n)\in\mathbb{Z}_{2}\oplus\mathbb{Z}^{2}$. Therefore,
$\operatorname{\mathrm{Ker}}(\varphi_{1})\cong\mathbb{Z}_{2}$ which is
generated by $([-1],[-1])$, and666When $\alpha=-\infty$ and
$\beta\in\mathbb{Q}$, we have
$\operatorname{\mathrm{Coker}}(\varphi_{1})\cong\mathbb{Z}_{s}$. This is the
case when $p=-1$ and $q=0$ and $t=-ps+qr=s$ in this case. A similar remark
also holds for $i=2,3,5$.
$\operatorname{\mathrm{Coker}}(\varphi_{1})\cong\mathbb{Z}_{t}$.

We next consider the case of $i=2$. We have
$KO_{2}(C(\mathbb{T}^{2}),\tau_{\mathbb{T}})\cong\mathbb{Z}_{2}\oplus(\mathbb{Z}_{2})^{2}\oplus\mathbb{Z}$,
where the first $\mathbb{Z}_{2}$ direct summand is generated by $[-I^{(2)}]$.
For $(m,n)\in\mathbb{Z}^{2}$, the element
$\left[\left(\begin{array}[]{cc}0&iM_{m,n}\\\
-iM_{-m,-n}&0\end{array}\right)\right]$ in
$KO_{2}(C(\mathbb{T}^{2}),\tau_{\mathbb{T}})$ corresponds to $(0,m\bmod
2,n\bmod 2,0)\in\mathbb{Z}_{2}\oplus(\mathbb{Z}_{2})^{2}\oplus\mathbb{Z}$
(Example 9.2 of [14]). The groups $KO_{2}(\mathcal{T}^{0},\tau_{0})$ and
$KO_{2}(\mathcal{T}^{\gamma},\tau_{\gamma})$ and their generators are obtained
in Sect. 3 and we have

$\operatorname{\mathrm{Ker}}(\varphi_{2})\cong\left\\{\begin{aligned}
(\mathbb{Z}_{2})^{2}&\hskip 8.53581pt\text{when $t$ is even,}\\\
\mathbb{Z}_{2}\hskip 8.53581pt&\hskip 8.53581pt\text{when $t$ is
odd,}\end{aligned}\right.\ \ \
\operatorname{\mathrm{Coker}}(\varphi_{2})\cong\left\\{\begin{aligned}
\mathbb{Z}_{2}\oplus\mathbb{Z}&\hskip 8.53581pt\text{when $t$ is even,}\\\
\mathbb{Z}\hskip 9.95845pt&\hskip 8.53581pt\text{when $t$ is
odd.}\end{aligned}\right.$

When $i=3$, we have
$KO_{3}(C(\mathbb{T}^{2}),\tau_{\mathbb{T}})\cong(\mathbb{Z}_{2})^{3}$. For
$(m,n)\in\mathbb{Z}^{2}$, the element $[\mathrm{diag}(M_{m,n},M_{-m,-n})]\in
KO_{3}(C(\mathbb{T}^{2}),\tau_{\mathbb{T}})$ corresponds to $(m\bmod 2,n\bmod
2,0)\in(\mathbb{Z}_{2})^{3}$. By Sect. 3, we have

$\operatorname{\mathrm{Ker}}(\varphi_{3})\cong\left\\{\begin{aligned}
\mathbb{Z}_{2}&\hskip 8.53581pt\text{when $t$ is even,}\\\ 0\hskip
2.84526pt&\hskip 8.53581pt\text{when $t$ is odd,}\end{aligned}\right.\ \ \
\operatorname{\mathrm{Coker}}(\varphi_{3})\cong\left\\{\begin{aligned}
(\mathbb{Z}_{2})^{2}&\hskip 8.53581pt\text{when $t$ is even,}\\\
\mathbb{Z}_{2}\hskip 5.69054pt&\hskip 8.53581pt\text{when $t$ is
odd.}\end{aligned}\right.$

When $i=5$, we have
$KO_{5}(C(\mathbb{T}^{2}),\tau_{\mathbb{T}})\cong\mathbb{Z}^{2}$. For
$(m,n)\in\mathbb{Z}^{2}$, the element $[\mathrm{diag}(M_{m,n},M_{m,n})]\in
KO_{5}(C(\mathbb{T}^{2}),\tau_{\mathbb{T}})$ corresponds to
$(m,n)\in\mathbb{Z}^{2}$. By Sect. 3, we have
$\operatorname{\mathrm{Ker}}(\varphi_{5})=0$ and
$\operatorname{\mathrm{Coker}}(\varphi_{5})=\mathbb{Z}_{t}$.

Combined with the above computation and the exact sequence (3.3), $KO$-group
$KO_{i}(\mathcal{S}^{\alpha,\beta},\tau_{\mathcal{S}})$ are computed, though
some complication appears when $i=2,3$. We discuss these two cases in the
following subsections.

####  3.2.1. The Group $KO_{2}(\mathcal{S}^{\alpha,\beta},\tau_{\mathcal{S}})$

We compute the group $KO_{2}(\mathcal{S}^{0,\gamma},\tau_{\mathcal{S}})$,
which is isomorphic to
$KO_{2}(\mathcal{S}^{\alpha,\beta},\tau_{\mathcal{S}})$. The computation is
divided into two cases depending on whether $t$ is even or odd. Note that $u$
is odd when $t$ is even since $r$ and $s$ are mutually prime.

When $t$ is odd, $\operatorname{\mathrm{Ker}}(\varphi_{2})\cong\mathbb{Z}_{2}$
is generated by $([-I^{(2)}],[-I^{(2)}])$ and the sequence (3.3) splits.
Therefore,
$KO_{2}(\mathcal{S}^{\alpha,\beta},\tau_{\mathcal{S}})\cong(\mathbb{Z}_{2})^{2}$.

We next discuss the cases of even $t$. In this case, both of the kernel and
the cokernel of $\varphi_{3}$ are isomorphic to $(\mathbb{Z}_{2})^{2}$. Let
$\widetilde{KO}_{2}(\mathcal{S}^{0,\gamma},\tau_{\mathcal{S}})$ be the kernel
of the map $\lambda^{\alpha}_{*}\circ p^{\alpha}_{*}\colon
KO_{2}(\mathcal{S}^{\alpha,\beta},\tau_{\mathcal{S}})\to
KO_{2}(\mathbb{C},\mathrm{id})\cong\mathbb{Z}_{2}$ which splits. Then, the
sequence (3.3) reduces to the following extension:

(3.4)
$0\to(\mathbb{Z}_{2})^{2}\to\widetilde{KO}_{2}(\mathcal{S}^{0,\gamma},\tau_{\mathcal{S}})\to\mathbb{Z}_{2}\to
0.$

In the following, we show that this sequence (3.4) splits. We find a lift of
the generator of $\mathbb{Z}_{2}$ in
$\widetilde{KO}_{2}(\mathcal{S}^{0,\gamma},\tau_{\mathcal{S}})$ and show this
lift has order two. For $(m,n)\in\mathbb{Z}^{2}$ and $\kappa=0$ and $\gamma$,
we write $T_{m,n}^{\kappa}$ for $P^{\kappa}M_{m,n}P^{\kappa}$, and let $Q$ be
the projection $T^{\gamma}_{u,0}T^{\gamma}_{-u,0}$. Note that $1-Q$ is the
projection onto the closed subspace spanned by $\\{\delta_{m,n}\ |\
0\leq\gamma m-n<t\ \\}$. For $j=1,\ldots,t$, let $P_{j}$ be a projection in
$\mathcal{T}^{\gamma}$, defined inductively as follows:

$P_{1}=(1-Q)M_{0,-t+1}(1-Q)M_{0,t-1}(1-Q),$
$P_{j}=(1-Q)M_{0,-t+j}(1-Q)M_{0,t-j}(1-Q)-\sum^{j-1}_{k=1}P_{k}.$

Specifically, $P_{j}$ is the orthogonal projection of $\mathcal{H}^{\gamma}$
onto the closed subspace spanned by
$\\{\delta_{n,tn-j+1}\\}_{n\in\mathbb{Z}}$.

Figure 1. The case of $u=1$ and $t=4$. $1-Q$ is the projection onto the closed
subspace corresponding to lattice points in between two lines (lattice points
on the line $y=\gamma x$ are included, while that on $y=\gamma(x-1)$ are not).
$P_{j}$ is the projection onto the closed subspace spanned by
$\\{\delta_{n,4n-j+1}\ |\ n\in\mathbb{Z}\\}$. $s_{j}$ interchanges two points
in a pair up to the sign.

Note that $\sum^{t}_{j=1}P_{j}=1-Q$. For odd $j=1,3,\ldots,t-1$, let
$s_{j}=P_{j}M_{0,1}P_{j+1}-P_{j+1}M_{0,-1}P_{j}$ and
$s=\sum_{j=1,\text{odd}}^{t-1}s_{j}$. The element $s$ satisfies the relations
$\rm(\hskip 1.79993pti\hskip 1.79993pt)$ $s^{*}=-s$, $\rm(\hskip
0.80002ptii\hskip 0.80002pt)$ $s^{\tau}=-s$, $\rm(iii)$ $s^{2}=-1+Q$,
$\rm(iv)$ $Qs=sQ=0$ and $\rm(\hskip 0.59998ptv\hskip 0.59998pt)$
$sT^{\gamma}_{u,0}=T^{\gamma}_{-u,0}s=0$. Note that $\sigma^{\gamma}(s)=0$
since $\sigma^{\gamma}(1-Q)=0$. We first consider the following elements:

$a=\left(\begin{array}[]{cc}0&i\cdot 1_{\mathcal{T}^{0}}\\\ -i\cdot
1_{\mathcal{T}^{0}}&0\\\ \end{array}\right)\in M_{2}(\mathcal{T}^{0}),\ \ \
b_{\pm}=\left(\begin{array}[]{cc}\pm is&iQ\\\ -iQ&is\\\ \end{array}\right)\in
M_{2}(\mathcal{T}^{\gamma}),$

where the double-sign corresponds. Elements $a$ and $b_{\pm}$ are self-adjoint
unitaries satisfying $a^{\tau}=-a$ and $b_{\pm}^{\tau}=-b_{\pm}$, and pairs
$(a,b_{\pm})$ are elements of $M_{2}(\mathcal{S}^{0,\gamma})$; therefore, they
define the elements of $KO_{2}(\mathcal{S}^{0,\gamma},\tau_{\mathcal{S}})$.

######  Lemma 3.2.

As elements of $KO_{2}(\mathcal{S}^{0,\gamma},\tau_{\mathcal{S}})$, we have
$[(a,b_{+})]=[(a,b_{-})]=0$.

###### Proof.

We first show that $[(a,b_{+})]=0$. For $j=1,3,\ldots,t-1$, let
$r_{j}=P_{j}M_{0,1}P_{j+1}+P_{j+1}M_{0,-1}P_{j}$ and
$r=\sum_{j=1,\text{odd}}^{t-1}r_{j}$. The element $r$ satisfies $\rm(\hskip
1.79993pti\hskip 1.79993pt)$ $r^{*}=r$, $\rm(\hskip 0.80002ptii\hskip
0.80002pt)$ $r^{\tau}=r$, $\rm(iii)$ $r^{2}=1-Q$, $\rm(iv)$ $Qr=rQ=0$,
$\rm(\hskip 0.59998ptv\hskip 0.59998pt)$
$rT^{\gamma}_{u,0}=T^{\gamma}_{-u,0}r=0$ and $\rm(vi)$ $r$ anticommutes with
$s$. For $0\leq\theta\leq\frac{\pi}{2}$, let

$b_{\theta}=\left(\begin{array}[]{cc}is\cos\theta&iQ+ir\sin\theta\\\ -iQ-
ir\sin\theta&is\cos\theta\\\ \end{array}\right),\ \
d=\left(\begin{array}[]{cc}0&i\cdot 1_{\mathcal{T}^{\gamma}}\\\ -i\cdot
1_{\mathcal{T}^{\gamma}}&0\\\ \end{array}\right).$

This $b_{\theta}$ is a self-adjoint unitary satisfying
$b_{\theta}^{\tau}=-b_{\theta}$ and $b_{0}=b_{+}$. Therefore, $b_{+}$ and
$b_{\frac{\pi}{2}}$ are homotopic in
$U^{(2)}_{1}(\mathcal{T}^{\gamma},\tau_{\gamma})$. We further discuss
$b_{\frac{\pi}{2}}$. Let consider lattice points $(m,n)\in\mathbb{Z}^{2}$
satisfying $0\leq\gamma m-n<t$, as indicated in Figure 1 for the case where
$u=1$ and $t=4$. As in Figure 1, we divide these points to $\frac{t}{2}$ pairs
of lattice points: for $n\in\mathbb{Z}$ and odd $j=1,3,\ldots,t-1$, a pair
consists of $\\{(n,tn-j),(n,tn-j+1)\\}$. The action of $b_{\frac{\pi}{2}}$ is
closed on each pair of lattice points and is expressed by a $4\times 4$ matrix
(acting on $\mathbb{C}^{2}\otimes\mathbb{C}^{2}$; one $\mathbb{C}^{2}$
corresponds to a pair of lattice points, and the other $\mathbb{C}^{2}$
corresponds to the $2\times 2$ matrix we consider). Let $V$ be the following
matrix.

$V=\frac{1}{2}\left(\begin{array}[]{cccc}1&1&1&-1\\\ 1&1&-1&1\\\ 1&-1&1&1\\\
-1&1&1&1\\\ \end{array}\right).$

Then $V\in SO(4)$ and satisfies

$V\left(\begin{array}[]{cccc}0&0&0&i\\\ 0&0&i&0\\\ 0&-i&0&0\\\ -i&0&0&0\\\
\end{array}\right)V^{*}=\left(\begin{array}[]{cccc}0&0&i&0\\\ 0&0&0&i\\\
-i&0&0&0\\\ 0&-i&0&0\\\ \end{array}\right),$

where the left matrix inside the conjugation is the restriction of
$b_{\frac{\pi}{2}}$ onto the closed subspace spanned by generating functions
of these two lattice points tensor $\mathbb{C}^{2}$ and the right matrix is
that of $d$ (note that $Q=0$ on these lattice points). Let $W$ be the unitary
on $\mathcal{H}^{\gamma}\otimes\mathbb{C}^{2}$ defined by applying $V$ to
these pair of lattice points satisfying $0\leq\gamma m-n<t$ and the identity
on the lattice points satisfying $t\leq\gamma m-n$, then we have
$Wb_{\frac{\pi}{2}}W^{*}=d$. Since $SO(4)$ is path-connected, there is a path
of self-adjoint unitaries from $b_{\frac{\pi}{2}}$ to $d$ preserving the
relation of the $KO_{2}$-group. Summarizing, we have a path in
$U^{(2)}_{1}(\mathcal{T}^{\gamma},\tau_{\gamma})$ from $b_{+}$ to $d$. By its
construction, the pair of the constant path at $a\in M_{2}(\mathcal{T}^{0})$
and this path gives a path in
$U^{(2)}_{1}(\mathcal{S}^{0,\gamma},\tau_{\mathcal{S}})$ from $(a,b_{+})$ to
$(a,d)$. Therefore, we have $[(a,b_{+})]=[(a,d)]=[I^{(2)}]=0$ in
$KO_{2}(\mathcal{S}^{0,\gamma},\tau_{\mathcal{S}})$.

We next discuss the class $[(a,b_{-})]$. For $0\leq\theta\leq\frac{\pi}{2}$,
let

$b^{\prime}_{\theta}=\left(\begin{array}[]{cc}-is\cos\theta&iQ+i(1-Q)\sin\theta\\\
-iQ-i(1-Q)\sin\theta&is\cos\theta\\\ \end{array}\right).$

Then, $b^{\prime}_{\theta}$ is a self-adjoint unitary satisfying
$(b^{\prime}_{\theta})^{\tau}=-b^{\prime}_{\theta}$. We have
$b^{\prime}_{0}=b_{-}$ and $b^{\prime}_{\frac{\pi}{2}}=I^{(2)}$. Therefore,
$[(a,b_{-})]=[(a,b^{\prime}_{\frac{\pi}{2}})]=[I^{(2)}]=0$ in
$KO_{2}(\mathcal{S}^{0,\gamma},\tau_{\mathcal{S}})$. ∎

Let consider the following elements:

$v_{2}=\left(\begin{array}[]{cc}0&iT^{0}_{u,0}\\\ -iT^{0}_{-u,0}&0\\\
\end{array}\right)\in M_{2}(\mathcal{T}^{0}),\ \ \
w_{\pm}=\left(\begin{array}[]{cc}\pm is&iT^{\gamma}_{u,0}\\\
-iT^{\gamma}_{-u,0}&0\\\ \end{array}\right)\in M_{2}(\mathcal{T}^{\gamma}),$

which are self-adjoint unitaries satisfying $(v_{2})^{\tau}=-v_{2}$ and
$w_{\pm}^{\tau}=-w_{\pm}$. Since $\sigma^{0}(v_{2})=\sigma^{\gamma}(w_{\pm})$,
pairs $(v_{2},w_{\pm})$ are elements of $M_{2}(\mathcal{S}^{0,\gamma})$
satisfying $(v_{2},w_{\pm})^{\tau}=-(v_{2},w_{\pm})$ and give elements
$[(v_{2},w_{\pm})]$ of the group
$KO_{2}(\mathcal{S}^{0,\gamma},\tau_{\mathcal{S}})$.

######  Lemma 3.3.

In $KO_{2}(\mathcal{S}^{0,\gamma},\tau_{\mathcal{S}})$, we have
$[(v_{2},w_{+})]=[(v_{2},w_{-})]$.

###### Proof.

For $0\leq\theta\leq\pi$, let consider the following element in
$M_{4}(\mathcal{T}^{\gamma})$:

$R_{\theta}=\left(\begin{array}[]{cccc}is\cos\theta&iT^{\gamma}_{u,0}&-ir\sin\theta&0\\\
-iT^{\gamma}_{-u,0}&0&0&0\\\ ir\sin\theta&0&is\cos\theta&iQ\\\
0&0&-iQ&is\end{array}\right),$

Then, we have $R_{0}=w_{+}\oplus b_{+}$, $R_{\pi}=w_{-}\oplus b_{-}$ and
$R_{\theta}$ is a self-adjoint unitary satisfying
$R_{\theta}^{\tau}=-R_{\theta}$. Since
$\sigma^{\gamma}(R_{\theta})=\sigma^{0}(v_{2}\oplus a)$, the pair
$(v_{2}\oplus a,R_{\theta})$ is contained in
$U^{(2)}_{2}(\mathcal{S}^{0,\gamma},\tau_{\mathcal{S}})$ and gives a path from
$(v_{2}\oplus a,w_{+}\oplus b_{+})$ to $(v_{2}\oplus a,w_{-}\oplus b_{-})$. By
using Lemma 3.2, we obtain the following equality in
$KO_{2}(\mathcal{S}^{0,\gamma},\tau_{\mathcal{S}})$:

$[(v_{2},w_{+})]=[(v_{2}\oplus a,w_{+}\oplus b_{+})]=[(v_{2}\oplus
a,w_{-}\oplus b_{-})]=[(v_{2},w_{-})].\qed$

######  Lemma 3.4.

In $KO_{2}(\mathcal{S}^{0,\gamma},\tau_{\mathcal{S}})$, the element
$[(v_{2},w_{+})]$ has order two.

###### Proof.

For $0\leq\theta\leq\frac{\pi}{2}$, let

$A^{0}_{\theta}=\left(\begin{array}[]{cccc}0&iT^{0}_{u,0}\cos\theta&i\sin\theta&0\\\
-iT^{0}_{-u,0}\cos\theta&0&0&-i\sin\theta\\\
-i\sin\theta&0&0&iT^{0}_{u,0}\cos\theta\\\
0&i\sin\theta&-iT^{0}_{-u,0}\cos\theta&0\end{array}\right)\in
M_{4}(\mathcal{T}^{0}),$
$A^{\gamma}_{\theta}=\left(\begin{array}[]{cccc}is\cos\theta&iT^{\gamma}_{u,0}\cos\theta&i\sin\theta&0\\\
-iT^{\gamma}_{-u,0}\cos\theta&0&0&-i\sin\theta\\\
-i\sin\theta&0&-is\cos\theta&iT^{\gamma}_{u,0}\cos\theta\\\
0&i\sin\theta&-iT^{\gamma}_{-u,0}\cos\theta&0\end{array}\right)\in
M_{4}(\mathcal{T}^{\gamma}).$

Then, $A^{0}_{\theta}$ and $A^{\gamma}_{\theta}$ are self-adjoint unitaries
satisfying $(A^{0}_{\theta})^{\tau}=-A^{0}_{\theta}$ and
$(A^{\gamma}_{\theta})^{\tau}=-A^{\gamma}_{\theta}$, and their pair
$(A^{0}_{\theta},A^{\gamma}_{\theta})$ is contained in
$M_{4}(\mathcal{S}^{0,\gamma})$. Note that
$(A^{0}_{0},A^{\gamma}_{0})=(v_{2}\oplus v_{2},w_{+}\oplus w_{-})$. Therefore,
by Lemma 3.3, the following equality holds in
$KO_{2}(\mathcal{S}^{0,\gamma},\tau_{\mathcal{S}})$:

$2\cdot[(v_{2},w_{+})]=[(v_{2},w_{+})]+[(v_{2},w_{-})]=[(A^{0}_{0},A^{\gamma}_{0})]=[(A^{0}_{\frac{\pi}{2}},A^{\gamma}_{\frac{\pi}{2}})]=0.\qed$

######  Proposition 3.5.

When $\alpha$ and $\beta$ are rational numbers and $t=-ps+qr$ is even, we have
$KO_{2}(\mathcal{S}^{\alpha,\beta},\tau_{\mathcal{S}})\cong(\mathbb{Z}_{2})^{4}$.

###### Proof.

Since $u$ is odd when $t$ is even, the pair $([v_{2}],[w_{+}])\in
KO_{2}(\mathcal{T}^{0},\tau_{0})\oplus
KO_{2}(\mathcal{T}^{\gamma},\tau_{\gamma})$ constitutes a nontrivial element
of the right $\mathbb{Z}_{2}\subset\operatorname{\mathrm{Ker}}(\varphi_{2})$
in the sequence (3.4). The element $[(v_{2},w_{+})]\in
KO_{2}(\mathcal{S}^{0,\gamma},\tau_{\mathcal{S}})$ is a lift of it. Therefore,
$[(v_{2},w_{+})]$ is nontrivial and has order two by Lemma 3.4. This element
belongs to $\widetilde{KO}_{2}(\mathcal{S}^{0,\gamma},\tau_{\mathcal{S}})$
and, by mapping $1\in\mathbb{Z}_{2}$ to $[(v_{2},w_{+})]$, we obtain a
splitting of the the sequence (3.4). Therefore,
$\widetilde{KO}_{2}(\mathcal{S}^{0,\gamma},\tau_{\mathcal{S}})\cong(\mathbb{Z}_{2})^{3}$
and the result follows. ∎

####  3.2.2. The Group $KO_{3}(\mathcal{S}^{\alpha,\beta},\tau_{\mathcal{S}})$

We next compute $KO_{3}(\mathcal{S}^{\alpha,\beta},\tau_{\mathcal{S}})$. Note
that $\operatorname{\mathrm{Ker}}(\varphi_{3})$ depends on whether $t$ is even
or odd. When $t$ is odd, $\operatorname{\mathrm{Ker}}(\varphi_{3})$ is zero
and, from the sequence (3.3), we have
$KO_{3}(\mathcal{S}^{\alpha,\beta},\tau_{\mathcal{S}})\cong\mathbb{Z}_{2}$.

We next discuss the cases of even $t$. In this case, the extension (3.3) is of
the following form:

(3.5) $0\to\mathbb{Z}_{2}\to
KO_{3}(\mathcal{S}^{0,\gamma},\tau_{\mathcal{S}})\to\mathbb{Z}_{2}\to 0.$

As in Sect. 3.2.1, we show that this sequence splits by finding a lift of the
generator of the right $\mathbb{Z}_{2}$ in
$KO_{3}(\mathcal{S}^{0,\gamma},\tau_{\mathcal{S}})$ of order two. Let consider
the following elements:

$v_{3}=\left(\begin{array}[]{cc}T^{0}_{u,0}&0\\\ 0&T^{0}_{-u,0}\\\
\end{array}\right)\in M_{2}(\mathcal{T}^{0}),\ \ \
z_{\pm}=\left(\begin{array}[]{cc}T^{\gamma}_{u,0}&\pm s\\\
0&T^{\gamma}_{-u,0}\\\ \end{array}\right)\in M_{2}(\mathcal{T}^{\gamma}),$

where the double-sign in the second equality corresponds. Pairs
$(v_{3},z_{\pm})$ are unitaries in $M_{2}(\mathcal{S}^{0,\gamma})$ satisfying
$(v_{3},z_{\pm})^{\\#\otimes\tau}=(v_{3},z_{\pm})$ and define elements
$[(v_{3},z_{\pm})]$ of the $KO$-group
$KO_{3}(\mathcal{S}^{0,\gamma},\tau_{\mathcal{S}})$.

######  Lemma 3.6.

In $KO_{3}(\mathcal{S}^{0,\gamma},\tau_{\mathcal{S}})$, we have
$[(v_{3},z_{+})]=[(v_{3},z_{-})]$.

###### Proof.

For $0\leq\theta\leq\pi$, let
$z_{\theta}=\left(\begin{array}[]{cc}T^{\gamma}_{u,0}&e^{i\theta}s\\\
0&T^{\gamma}_{-u,0}\\\ \end{array}\right)\in M_{2}(\mathcal{T}^{\gamma})$
which gives a path $\\{z_{\theta}\\}_{0\leq\theta\leq\pi}$ of unitaries
satisfying $(z_{\theta})^{\\#\otimes\tau}=z_{\theta}$. Its endpoints are
$z_{0}=z_{+}$ and $z_{\pi}=z_{-}$. The pair $(v_{3},z_{\theta})$ satisfies
$(v_{3},z_{\theta})^{\\#\otimes\tau}=(v_{3},z_{\theta})$ and gives a homotopy
between $(v_{3},z_{+})$ and $(v_{3},z_{-})$ in
$U^{(3)}_{1}(\mathcal{S}^{0,\gamma},\tau_{\mathcal{S}})$. ∎

######  Lemma 3.7.

The element $[(v_{3},z_{+})]$ in
$KO_{3}(\mathcal{S}^{0,\gamma},\tau_{\mathcal{S}})$ has order two.

###### Proof.

For $0\leq\theta\leq\frac{\pi}{2}$, let

$B^{0}_{\theta}=\left(\begin{array}[]{cccc}T^{0}_{u,0}\cos\theta&0&0&\sin\theta\\\
0&T^{0}_{-u,0}\cos\theta&\sin\theta&0\\\
0&-\sin\theta&T^{0}_{u,0}\cos\theta&0\\\
-\sin\theta&0&0&T^{0}_{-u,0}\cos\theta\end{array}\right),$
$B^{\gamma}_{\theta}=\left(\begin{array}[]{cccc}T^{\gamma}_{u,0}\cos\theta&s\cos\theta&0&\sin\theta\\\
0&T^{\gamma}_{-u,0}\cos\theta&\sin\theta&0\\\
0&-\sin\theta&T^{\gamma}_{u,0}\cos\theta&-s\cos\theta\\\
-\sin\theta&0&0&T^{\gamma}_{-u,0}\cos\theta\end{array}\right).$

For each $\theta$, matrices $B^{0}_{\theta}$ and $B^{\gamma}_{\theta}$ are
unitaries satisfying $(B^{0}_{\theta})^{\\#\otimes\tau}=B^{0}_{\theta}$ and
$(B^{\gamma}_{\theta})^{\\#\otimes\tau}=B^{\gamma}_{\theta}$. We have
$B^{\gamma}_{0}=v_{3}\oplus v_{3}$ and $B^{\gamma}_{0}=z_{+}\oplus z_{-}$.
Note that matrices $B^{0}_{\frac{\pi}{2}}$ and $B^{\gamma}_{\frac{\pi}{2}}$
are contained in $M_{4}(\mathbb{C})$, where they coincide. Since this unitary
satisfies the symmetry of the $KO_{3}$-group, this is an element of the
quaternionic unitary group $U(2,\mathbb{H})$. Since $U(2,\mathbb{H})$ is path-
connected, there is a path of unitaries in
$U_{2}^{(3)}(\mathcal{S}^{0,\gamma},\tau_{\mathcal{S}})$ connecting
$(B^{0}_{\frac{\pi}{2}},B^{\gamma}_{\frac{\pi}{2}})$ to
$(1_{\mathcal{S}})_{4}$. By using Lemma 3.6, we obtain the following equality
in $KO_{3}(\mathcal{S}^{0,\gamma},\tau_{\mathcal{S}})$:

$2\cdot[(v_{3},z_{+})]=[(v_{3}\oplus v_{3},z_{+}\oplus
z_{-})]=[(B^{0}_{\frac{\pi}{2}},B^{\gamma}_{\frac{\pi}{2}})]=[(1_{\mathcal{S}})_{4}]=0.\qed$

######  Proposition 3.8.

When $\alpha$ and $\beta$ are rational numbers and $t=-ps+qr$ is even, we have
$KO_{3}(\mathcal{S}^{\alpha,\beta},\tau_{\mathcal{S}})\cong(\mathbb{Z}_{2})^{2}$.

###### Proof.

The pair $([v_{3}],[z_{+}])\in KO_{3}(\mathcal{T}^{0},\tau_{0})\oplus
KO_{3}(\mathcal{T}^{\gamma},\tau_{\gamma})$ is contained in
$\operatorname{\mathrm{Ker}}(\varphi_{3})\cong\mathbb{Z}_{2}$ and is
nontrivial. The element $[(v_{3},z_{+})]\in
KO_{3}(\mathcal{S}^{0,\gamma},\tau_{\mathcal{S}})$ is its lift. Therefore, the
class $[(v_{3},z_{+})]$ is nontrivial and has order two by Lemma 3.7. We thus
obtain a splitting of the sequence (3.5) and the group
$KO_{3}(\mathcal{S}^{0,\gamma},\tau_{\mathcal{S}})$ is isomorphic to
$(\mathbb{Z}_{2})^{2}$. ∎

The results in this subsection are summarized in Tables 5, 5 and 5.

Table 3. $KO_{*}(\mathcal{S}^{\alpha,\beta},\tau_{\mathcal{S}})$ when both
$\alpha$ and $\beta$ are rational (or $\pm\infty$).

$i$ | $0$ | $1$ | $2$ |
---|---|---|---|---
$t=-ps+qr$ | — | even | odd | even | odd |
$KO_{i}(\mathcal{S}^{\alpha,\beta},\tau_{\mathcal{S}})$ | $\mathbb{Z}\oplus\mathbb{Z}_{t}$ | $(\mathbb{Z}_{2})^{2}\oplus\mathbb{Z}$ | $\mathbb{Z}_{2}\oplus\mathbb{Z}$ | $(\mathbb{Z}_{2})^{4}$ | $(\mathbb{Z}_{2})^{2}$ |

$3$ | $4$ | $5$ | $6$ | $7$
---|---|---|---|---
even | odd | — | — | — | —
$(\mathbb{Z}_{2})^{2}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}\oplus\mathbb{Z}_{t}$ | $\mathbb{Z}$ | $0$ | $0$

Table 4. $KO_{*}(\mathcal{S}^{\alpha,\beta},\tau_{\mathcal{S}})$ when one of
$\alpha$ and $\beta$ is rational (or $\pm\infty$) and the other is irrational.

$i$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$
---|---|---|---|---|---|---|---|---
$KO_{i}(\mathcal{S}^{\alpha,\beta},\tau_{\mathcal{S}})$ | $\mathbb{Z}^{2}$ | $(\mathbb{Z}_{2})^{2}\oplus\mathbb{Z}$ | $(\mathbb{Z}_{2})^{3}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}^{2}$ | $\mathbb{Z}$ | $0$ | $0$

Table 5. $KO_{*}(\mathcal{S}^{\alpha,\beta},\tau_{\mathcal{S}})$ when both
$\alpha$ and $\beta$ are irrational.

$i$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$
---|---|---|---|---|---|---|---|---
$KO_{i}(\mathcal{S}^{\alpha,\beta},\tau_{\mathcal{S}})$ | $\mathbb{Z}^{3}$ | $(\mathbb{Z}_{2})^{3}\oplus\mathbb{Z}$ | $(\mathbb{Z}_{2})^{4}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}^{3}$ | $\mathbb{Z}$ | $0$ | $0$

###  3.3. Boundary Maps Associated with Quarter-Plane Toeplitz Extensions and
$KO$-Groups of $(\hat{\mathcal{T}}^{\alpha,\beta},\hat{\tau}_{\alpha,\beta})$

We next consider the boundary maps777We write $\hat{\partial}_{i}$ for
boundary maps associated with (2.2) and write $\check{\partial}_{i}$ for that
with (2.3). of the $24$-term exact sequence for $KO$-theory associated with
the sequence (2.2):

(3.6) $\hat{\partial}_{i}\colon
KO_{i}(\mathcal{S}^{\alpha,\beta},\tau_{\mathcal{S}})\to
KO_{i-1}(\mathcal{K}(\hat{\mathcal{H}}^{\alpha,\beta}),\tau_{\mathcal{K}}).$

######  Proposition 3.9.

For each $i$, the boundary map $\hat{\partial}_{i}$ in (3.6) is surjective.

###### Proof.

When $i=-1,0,4,6$, the group
$KO_{i-1}(\mathcal{K}(\hat{\mathcal{H}}^{\alpha,\beta}),\tau_{\mathcal{K}})$
is trivial and the statement is obvious. We discuss the other cases. The proof
is given by constructing explicit elements of the group
$KO_{i}(\mathcal{S}^{\alpha,\beta},\tau_{\mathcal{S}})$, which maps to a
generator of the group
$KO_{i-1}(\mathcal{K}(\hat{\mathcal{H}}^{\alpha,\beta}),\tau_{\mathcal{K}})$.
As in [35], by using the action of $SL(2,\mathbb{Z})$ on $\mathbb{Z}^{2}$, we
assume $0<\alpha\leq\frac{1}{2}$ and $1\leq\beta<+\infty$ without loss of
generality. Let
$\hat{\mathcal{P}}_{m,n}=\hat{P}^{\alpha,\beta}M_{m,n}\hat{P}^{\alpha,\beta}M_{-m,-n}\hat{P}^{\alpha,\beta}$.
As in [35], we consider the following element in
$\hat{\mathcal{T}}^{\alpha,\beta}$:

(3.7)
$\hat{A}=\hat{\mathcal{P}}_{0,1}+M_{1,1}(1-\hat{\mathcal{P}}_{-1,0})+M_{1,0}(\hat{\mathcal{P}}_{-1,0}-\hat{\mathcal{P}}_{0,1}).$

The operator $\hat{A}$ is Fredholm whose kernel is trivial and has one
dimensional cokernel [35]. We also have the following.

  * •

$\hat{\gamma}(\hat{A})$ is a unitary in $\mathcal{S}^{\alpha,\beta}$.

  * •

$\hat{A}$ is a real operator, that is $\mathfrak{r}(\hat{A})=\hat{A}$, and
$\hat{A}^{\tau}=\mathfrak{r}(\hat{A}^{*})=\hat{A}^{*}$ holds.

From these preliminaries, the proof of Proposition 3.9 is parallel to the
computation in Example 9.4 of [14]. We summarize the results here.

  * •

Let $u_{1}=\hat{\gamma}(\hat{A})$. $u_{1}$ is a unitary satisfying
$u_{1}^{\tau}=u_{1}^{*}$ and gives an element $[u_{1}]\in
KO_{1}(\mathcal{S}^{\alpha,\beta},\tau_{\mathcal{S}})$.
$\hat{\partial}_{1}([u_{1}])$ is a generator of
$KO_{0}(\mathcal{K}(\hat{\mathcal{H}}^{\alpha,\beta}),\tau_{\mathcal{K}})\cong\mathbb{Z}$.

  * •

Let $u_{2}=\left(\begin{array}[]{cc}0&i\hat{\gamma}(\hat{A})\\\
-i\hat{\gamma}(\hat{A})^{*}&0\end{array}\right)$. $u_{2}$ is a self-adjoint
unitary satisfying $u_{2}^{\tau}=-u_{2}$ and gives $[u_{2}]\in
KO_{2}(\mathcal{S}^{\alpha,\beta},\tau_{\mathcal{S}})$. Its image
$\hat{\partial}_{2}[u_{2}]$ is the generator of
$KO_{1}(\mathcal{K}(\hat{\mathcal{H}}^{\alpha,\beta}),\tau_{\mathcal{K}})\cong\mathbb{Z}_{2}$.

  * •

Let $u_{3}=\mathrm{diag}(\hat{\gamma}(\hat{A}),\hat{\gamma}(\hat{A})^{*}).$
$u_{3}$ is a unitary satisfying $u_{3}^{\sharp\otimes\tau}=u_{3}$ and gives
$[u_{3}]\in KO_{3}(\mathcal{S}^{\alpha,\beta},\tau_{\mathcal{S}})$. Its image
$\hat{\partial}_{3}([u_{3}])$ is the generator of the group
$KO_{2}(\mathcal{K}(\hat{\mathcal{H}}^{\alpha,\beta}),\tau_{\mathcal{K}})\cong\mathbb{Z}_{2}$.

  * •

Let $u_{5}=\mathrm{diag}(\hat{\gamma}(\hat{A}),\hat{\gamma}(\hat{A}))$.
$u_{5}$ is a unitary satisfying $u_{5}^{\sharp\otimes\tau}=u_{5}^{*}$ and
gives $[u_{5}]\in KO_{5}(\mathcal{S}^{\alpha,\beta},\tau_{\mathcal{S}})$. Its
image $\hat{\partial}_{5}([u_{5})]$ is a generator of the group
$KO_{4}(\mathcal{K}(\hat{\mathcal{H}}^{\alpha,\beta}),\tau_{\mathcal{K}})\cong\mathbb{Z}$.
∎

######  Remark 3.10.

In the case when $\alpha$, $\beta$ are both rational (or $\pm\infty$) and
$t=-ps+qr$ is even, the group
$KO_{2}(\mathcal{S}^{0,\gamma},\tau_{\mathcal{S}})\cong(\mathbb{Z}_{2})^{4}$
is generated by $[-I^{(2)}]$, $[(v_{2},w_{+})]$, $[u_{2}]$ and
$[(w^{\prime},I_{2}^{(2)})]$, where888The matrix $Y^{(3)}_{4}$ is introduced
in Appendix A.3.

$w^{\prime}=Y^{(3)}_{4}\mathrm{diag}(1,1-2T^{0}_{0,1}T^{0}_{0,-1},2T^{0}_{0,1}T^{0}_{0,-1}-1,-1){Y^{(3)*}_{4}}.$

By the map $\sigma_{*}\colon
KO_{i}(\mathcal{S}^{\alpha,\beta},\tau_{\mathcal{S}})\to
KO_{i}(C(\mathbb{T}^{2}),\tau_{\mathbb{T}})$, components generated by
$[(v_{2},w_{+})]$ (when $i=2$) and $[(v_{3},z_{+})]$ (when $i=3$) maps
injectively.

The $KO$-groups of
$(\hat{\mathcal{T}}^{\alpha,\beta},\hat{\tau}_{\alpha,\beta})$ are computed by
the $24$-term exact sequence of $KO$-theory associated with (2.2) and
Proposition 3.9. The results are collected in Tables 8, 8 and 8.

Table 6. $KO_{*}(\hat{\mathcal{T}}^{\alpha,\beta},\hat{\tau}_{\alpha,\beta})$
when both $\alpha$, $\beta$ are rational (or $\pm\infty$).

$i$ | $0$ | $1$ | $2$ | $3$ |
---|---|---|---|---|---
$t=-ps+qr$ | — | even | odd | even | odd | even | odd |
$KO_{*}(\hat{\mathcal{T}}^{\alpha,\beta},\hat{\tau}_{\alpha,\beta})$ | $\mathbb{Z}\oplus\mathbb{Z}_{t}$ | $(\mathbb{Z}_{2})^{2}$ | $\mathbb{Z}_{2}$ | $(\mathbb{Z}_{2})^{3}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$ | $0$ |

$4$ | $5$ | $6$ | $7$
---|---|---|---
— | — | — | —
$\mathbb{Z}\oplus\mathbb{Z}_{t}$ | $0$ | $0$ | $0$

Table 7. $KO_{*}(\hat{\mathcal{T}}^{\alpha,\beta},\hat{\tau}_{\alpha,\beta})$
when one of $\alpha$ and $\beta$ is rational (or $\pm\infty$) and the other is
irrational.

$i$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$
---|---|---|---|---|---|---|---|---
$KO_{i}(\hat{\mathcal{T}}^{\alpha,\beta},\hat{\tau}_{\alpha,\beta})$ | $\mathbb{Z}^{2}$ | $(\mathbb{Z}_{2})^{2}$ | $(\mathbb{Z}_{2})^{2}$ | $0$ | $\mathbb{Z}^{2}$ | $0$ | $0$ | $0$

Table 8. $KO_{*}(\hat{\mathcal{T}}^{\alpha,\beta},\hat{\tau}_{\alpha,\beta})$
when both $\alpha$ and $\beta$ are irrational.

$i$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$
---|---|---|---|---|---|---|---|---
$KO_{i}(\hat{\mathcal{T}}^{\alpha,\beta},\hat{\tau}_{\alpha,\beta})$ | $\mathbb{Z}^{3}$ | $(\mathbb{Z}_{2})^{3}$ | $(\mathbb{Z}_{2})^{3}$ | $0$ | $\mathbb{Z}^{3}$ | $0$ | $0$ | $0$

######  Remark 3.11.

Similar results in this section also hold for convex corners. Let
$\check{A}\in\check{\mathcal{T}}^{\alpha,\beta}$ be an operator defined by
replacing $\hat{P}^{\alpha,\beta}$ in the definition of $\hat{A}$ by
$\check{P}^{\alpha,\beta}$. This $\check{A}$ is a Fredholm operator satisfying
$\check{A}^{\tau}=\check{A}^{*}$ which have one dimensional kernel and trivial
cokernel [33]. As in Proposition 3.9, by using this example, we see that the
boundary maps $\check{\partial}_{i}$ of $KO$-theory associated with the
sequence (2.3) is surjective. The $KO$-groups of
$(\check{\mathcal{T}}^{\alpha,\beta},\check{\tau}_{\alpha,\beta})$ is computed
by the $24$-term exact sequence associated with (2.3), and the results are the
same as in Tables 8, 8 and 8. Through the stabilization isomorphism, we have
two boundary maps $\hat{\partial}_{i}$ and $\check{\partial}_{i}$ from
$KO_{i}(\mathcal{S}^{\alpha,\beta},\tau_{\mathcal{S}})$ to
$KO_{i-1}(\mathbb{C},\mathrm{id})$ associated with (2.2) and (2.3). Since
$\hat{\gamma}(\hat{A})=\check{\gamma}(\check{A})$, the relation
$\hat{\partial}_{i}=-\check{\partial}_{i}$ holds, as in Corollary $1$ of [33].

##  4\. Toeplitz Operators Associated with Subsemigroup $(\mathbb{Z}_{\geq
0})^{n}$ of $\mathbb{Z}^{n}$

In this section, Toeplitz operators associated with the subsemigroup
$(\mathbb{Z}_{\geq 0})^{n}$ of $\mathbb{Z}^{n}$ for $n\geq 3$ are discussed.
They are an $n$-variable generalization of the ordinary Toeplitz and quarter-
plane Toeplitz operators and are briefly discussed in [23, 22], where a
necessary and sufficient condition for these Toeplitz operators to be Fredholm
is obtained. We revisit these operators since, in our application to condensed
matter physics, models of higher-codimensional corners are given by using
these $n$-variable generalizations. Since the Toeplitz extension (4.1) and the
quarter-plane Toeplitz extension (2.3) provide a framework for these
applications, we seek this extension for our $n$-variable cases (Theorem 4.1).
Note that we consider corners of arbitrary codimension, though of a specific
shape compared to the codimension-two case [54]. In this section, let $n$ be a
positive integer bigger than or equals to three.

To study such Toeplitz operators, we follow Douglas–Howe’s idea [23] to use
the tensor product of the Toeplitz extension,

(4.1)
$0\rightarrow\mathcal{K}\overset{\iota}{\rightarrow}\mathcal{T}\overset{\gamma}{\rightarrow}C(\mathbb{T})\rightarrow
0,$

where $\mathcal{K}=\mathcal{K}(l^{2}(\mathbb{Z}_{\geq 0}))$. There is a linear
splitting of the $*$-homomorphism $\gamma$ given by the compression onto
$l^{2}(\mathbb{Z}_{\geq 0})$. For a subset
$\mathcal{A}\subset\\{1,\ldots,n\\}$, let
$\mathcal{T}^{n}_{\mathcal{A}}=A_{1}\otimes\cdots\otimes A_{n}$, where $A_{i}$
is $C(\mathbb{T})$ when $i\in\mathcal{A}$ and is $\mathcal{T}$ when
$i\notin\mathcal{A}$. Note that $\mathcal{T}^{n}_{\emptyset}$ is isomorphic to
$\mathcal{T}^{n}$ introduced in Sect. 2.2. For subsets
$\mathcal{D}\subset\mathcal{R}\subset\\{1,\ldots,n\\}$, let
$\pi^{\mathcal{D}}_{\mathcal{R}}\colon\mathcal{T}^{n}_{\mathcal{D}}\to\mathcal{T}^{n}_{\mathcal{R}}$
be the $*$-homomorphism induced by $\gamma$. Specifically,
$\pi^{\mathcal{D}}_{\mathcal{R}}=a_{1}\otimes\cdots\otimes a_{n}$, where
$a_{i}$ is $\mathrm{id}_{C(\mathbb{T})}$ when $i\in\mathcal{D}$, is $\gamma$
when $i\in\mathcal{R}\setminus\mathcal{D}$ and is $\mathrm{id}_{\mathcal{T}}$
otherwise. Note that $\pi^{\mathcal{D}}_{\mathcal{R}}$ is a surjection and
$\pi^{\emptyset}_{\emptyset}=\mathrm{id}$. In the following, we use a subset
$\mathcal{A}$ of $\\{1,\ldots,n\\}$ as a label to distinguish $C^{*}$-algebras
and the morphisms between them, which we may abbreviate brackets $\\{\cdot\\}$
in our notation. For example, we write $\mathcal{T}^{n}_{1,2}$ for
$\mathcal{T}^{n}_{\\{1,2\\}}$, $\pi_{i}$ for $\pi^{\emptyset}_{\\{i\\}}$ and
$\pi^{1}_{1,2}$ for $\pi^{\\{1\\}}_{\\{1,2\\}}$. For each
$\mathcal{A}\subset\\{1,\ldots,n\\}$, the map $\pi_{\mathcal{A}}$ has a linear
splitting
$\rho_{\mathcal{A}}\colon\mathcal{T}^{n}_{\mathcal{A}}\to\mathcal{T}^{n}$
given by the compression onto $l^{2}((\mathbb{Z}_{\geq 0})^{n})$. By these
preliminaries, we consider the following $C^{*}$-subalgebra of
$\mathcal{T}^{n}_{1}\oplus\cdots\oplus\mathcal{T}^{n}_{n}$.

$\mathcal{S}^{n}=\left\\{(T_{1},\cdots,T_{n})\
\Biggl{|}\begin{array}[]{ll}\text{For}\ 1\leq i\leq n,\
T_{i}\in\mathcal{T}^{n}_{i},\\\ \text{For}\ 1\leq i<j\leq n,\
\pi^{i}_{ij}(T_{i})=\pi^{j}_{ij}(T_{j})\end{array}\right\\}.$

Let $(T_{1},\ldots,T_{n})\in\mathcal{S}^{n}$. For a nonempty subset
$\mathcal{A}\subset\\{1,\ldots,n\\}$, we take $i\in\mathcal{A}$ and consider
the element $\pi^{i}_{\mathcal{A}}(T_{i})\in\mathcal{T}^{n}_{\mathcal{A}}$.
This element does not depend on the choice of $i\in\mathcal{A}$, and we write
$T_{\mathcal{A}}=\pi^{i}_{\mathcal{A}}(T_{i})$. Let
$\rho^{\prime}\colon\mathcal{S}^{n}\to\mathcal{T}^{n}$ be a linear map defined
by

$\rho^{\prime}(T_{1},\ldots,T_{n})=\sum_{k=1}^{n}\sum_{|\mathcal{A}|=k}(-1)^{k+1}\rho_{\mathcal{A}}(T_{\mathcal{A}})\vspace{-1mm}$

for $(T_{1},\ldots,T_{n})\in\mathcal{S}^{n}$, where the second summation is
taken over all subsets $\mathcal{A}\subset\\{1,\ldots,n\\}$ consisting of $k$
elements. Let $\mathcal{K}^{n}=\mathcal{K}(l^{2}((\mathbb{Z}_{\geq 0})^{n}))$,
and let $\gamma_{n}\colon\mathcal{T}^{n}\to\mathcal{S}^{n}$ be an
$*$-homomorphism given by $\gamma_{n}(T)=(\pi_{1}(T),\ldots,\pi_{n}(T))$. Let
$\iota_{n}$ be the $n$-fold tensor product of $\iota$.

######  Theorem 4.1.

There is the following short exact sequence of $C^{*}$-algebras:

(4.2)
$0\to\mathcal{K}^{n}\overset{\iota_{n}}{\to}\mathcal{T}^{n}\overset{\gamma_{n}}{\to}\mathcal{S}^{n}\to
0,$

where the map $\gamma_{n}$ has a linear splitting given by $\rho^{\prime}$.

###### Proof.

The map $\iota_{n}$ is injective since $\iota$ is injective. We first show the
exactness at $\mathcal{T}^{n}$. Since $\gamma\circ\iota=0$, we have
$\gamma_{n}\circ\iota_{n}=0$, and thus,
$\operatorname{\mathrm{Im}}(\iota_{n})\subset\operatorname{\mathrm{Ker}}(\gamma_{n})$.
Let $T\in\operatorname{\mathrm{Ker}}(\gamma_{n})$. Since
$\pi_{1}(T)=(\gamma\otimes 1\otimes\cdots\otimes 1)(T)=0$, there exists some
$S_{1}\in\mathcal{K}\otimes\mathcal{T}\otimes\cdots\otimes\mathcal{T}$ such
that $(\iota\otimes 1\otimes\cdots\otimes 1)(S_{1})=T$. Since

$\displaystyle 0$ $\displaystyle=(1\otimes\gamma\otimes 1\otimes\cdots\otimes
1)(T)=(1\otimes\gamma\otimes 1\otimes\cdots\otimes 1)(\iota\otimes
1\otimes\cdots\otimes 1)(S_{1})$ $\displaystyle=(\iota\otimes
1\otimes\cdots\otimes 1)(1\otimes\gamma\otimes 1\otimes\cdots\otimes
1)(S_{1})$

and $\iota\otimes 1\otimes\cdots\otimes 1$ is injective,
$(1\otimes\gamma\otimes 1\otimes\cdots\otimes 1)(S_{1})=0$. Therefore, there
exists some
$S_{2}\in\mathcal{K}\otimes\mathcal{K}\otimes\mathcal{T}\otimes\cdots\otimes\mathcal{T}$
such that $S_{1}=(1\otimes\iota\otimes 1\otimes\cdots\otimes 1)(S_{2})$. By
continuing this argument, we see that there exists some
$S_{n}\in\mathcal{K}\otimes\cdots\otimes\mathcal{K}\cong\mathcal{K}^{n}$ such
that $(\iota\otimes\cdots\otimes\iota)(S_{n})=T$. Thus, we have
$\operatorname{\mathrm{Ker}}(\gamma_{n})\subset\operatorname{\mathrm{Im}}(\iota_{n})$.

For the surjectivity of $\gamma_{n}$, we see that $\rho$ is a linear splitting
of $\gamma_{n}$, that is, for $(T_{1},\ldots,T_{n})\in\mathcal{S}^{n}$ and
$1\leq i\leq n$, the relation
$\pi_{i}\circ\rho^{\prime}(T_{1},\ldots,T_{n})=T_{i}$ holds. In the following,
we show $\pi_{1}\circ\rho^{\prime}(T_{1},\ldots,T_{n})=T_{1}$ and the other
case is proved similarly. Note that

(4.3)
$\pi_{1}\circ\rho^{\prime}(T_{1},\ldots,T_{n})=\sum_{k=1}^{n}\sum_{|\mathcal{A}|=k}(-1)^{k+1}\pi_{1}\circ\rho_{\mathcal{A}}(T_{\mathcal{A}})\vspace{-1mm}$

and that $\pi_{1}\circ\rho_{1}(T_{1})=T_{1}$. Thus, it is sufficient to show
that the sum over $\mathcal{A}$ $(\neq\\{1\\})$ is zero. Note that for $2\leq
i_{1}<\cdots<i_{k-1}\leq n$, we have

$\pi_{1}\circ\rho_{i_{1},\ldots,i_{k-1}}(T_{i_{1},\ldots,i_{k-1}})=\pi_{1}\circ\rho_{1,i_{1},\ldots,i_{k-1}}(T_{1,i_{1},\ldots,i_{k-1}}).$

By using this relation, we compute the sum on the right-hand side of (4.3).
For $k=1$ and $k=2$ of the sum, we have the following:

$\displaystyle\sum_{|\mathcal{A}|=1,\
\mathcal{A}\neq\\{1\\}}\pi_{1}\circ\rho_{\mathcal{A}}(T_{\mathcal{A}})-\sum_{|\mathcal{A}|=2}\pi_{1}\circ\rho_{\mathcal{A}}(T_{\mathcal{A}})$
$\displaystyle=\sum_{i=2}^{n}\pi_{1}\circ\rho_{i}(T_{i})-\sum_{1\leq i<j\leq
n}\pi_{1}\circ\rho_{i,j}(T_{ij})=-\sum_{2\leq i<j\leq
n}\pi_{1}\circ\rho_{i,j}(T_{ij}).$

For $2<k<n$, we have

$\displaystyle(-1)^{k}\sum_{2\leq i_{1}<\cdots<i_{k-1}\leq
n}\pi_{1}\circ\rho_{i_{1},\ldots,i_{k-1}}(T_{i_{1},\ldots,i_{k-1}})$
$\displaystyle\ \ +(-1)^{k+1}\sum_{1\leq i_{1}<\cdots<i_{k}\leq
n}\pi_{1}\circ\rho_{i_{1},\ldots,i_{k}}(T_{i_{1},\ldots,i_{k}})$
$\displaystyle=(-1)^{k+1}\sum_{2\leq i_{1}<\cdots<i_{k}\leq
n}\pi_{1}\circ\rho_{i_{1},\ldots,i_{k}}(T_{i_{1},\ldots,i_{k}}),$

and since

$(-1)^{n}\pi_{1}\circ\rho_{2,\ldots,n}(T_{2,\ldots,n})+(-1)^{n+1}\pi_{1}\circ\rho_{1,\ldots,n}(T_{1,\ldots,n})=0,$

we have $\pi_{1}\circ\rho^{\prime}(T_{1},\ldots,T_{n})=T_{1}$. ∎

Theorem 4.3 leads to the necessary and sufficient condition for Toeplitz
operators associated with these codimension-$n$ corners to be Fredholm.

######  Corollary 4.2 (Theorem 18 of [22]).

Let $k$ be a positive integer. An operator $T\in M_{k}(\mathcal{T}^{n})$ is a
Fredholm operator if and only if $\gamma_{n}(T)$ is invertible in
$M_{k}(\mathcal{S}^{n})$ or, equivalently, if and only if $\pi_{i}(T)$ is
invertible for any $1\leq i\leq n$.

As in Sect. 2.2, the real structure $c$ on $l^{2}(\mathbb{Z}^{n})$ induces
real structures on $\mathcal{T}^{n}_{i}$ and $\mathcal{S}^{n}$. We write
$\tau_{\mathcal{S}}$ for the transposition on $\mathcal{S}^{n}$ associated
with this real structure. The map $\gamma_{n}$ preserves the real structure,
and we obtain the following exact sequence of $C^{*,\tau}$-algebras:

(4.4)
$0\to(\mathcal{K}^{n},\tau_{\mathcal{K}})\overset{\iota_{n}}{\to}(\mathcal{T}^{n},\tau_{\mathcal{T}})\overset{\gamma_{n}}{\to}(\mathcal{S}^{n},\tau_{\mathcal{S}})\to
0.$

We next compute the $K$-groups of the $C^{*}$-algebra $\mathcal{S}^{n}$ and
$KO$-groups of the $C^{*,\tau}$-algebra
$(\mathcal{S}^{n},\tau_{\mathcal{S}})$.

######  Proposition 4.3.

$K_{i}(\mathcal{S}^{n})\cong\mathbb{Z}$ for $i=0,1$.

###### Proof.

Note that $K_{i}(\mathcal{T}^{n})\cong K_{i}(\mathbb{C})$. The result follows
from the six-term exact sequence of $K$-theory associated with the sequence
(4.2) in Theorem 4.1. ∎

######  Proposition 4.4.

For each $i$, we have

$KO_{i}(\mathcal{S}^{n},\tau_{\mathcal{S}})\cong
KO_{i}(\mathbb{C},\mathrm{id})\oplus KO_{i-1}(\mathbb{C},\mathrm{id}).$

The results are collected in Table 9.

###### Proof.

Note that $KO_{i}(\mathcal{T}^{n},\tau_{\mathcal{T}})\cong
KO_{i}(\mathbb{C},\mathrm{id})$. The result follows from the $24$-term exact
sequence of $KO$-theory associated with the sequence (4.4). ∎

Table 9. $KO$-groups of $(\mathcal{S}^{n},\tau_{\mathcal{S}})$. $i$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$
---|---|---|---|---|---|---|---|---
$KO_{i}(\mathcal{S}^{n},\tau_{\mathcal{S}})$ | $\mathbb{Z}$ | $\mathbb{Z}\oplus\mathbb{Z}_{2}$ | $(\mathbb{Z}_{2})^{2}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}$ | $\mathbb{Z}$ | $0$ | $0$

A Fredholm Toeplitz operator associated with a codimension-$n$ corner whose
Fredholm index is one is constructed as follows.

######  Example 4.5 (A Fredholm Operator of Index One).

Let $T_{z}$ be the Toeplitz operator whose symbol
$z\colon\mathbb{T}\to\mathbb{C}$ is the inclusion. Its adjoint $T_{z}^{*}$ is
a Fredholm operator on $l^{2}(\mathbb{Z}_{\geq 0})$ of index one. Let
$p=T_{z}T_{z}^{*}$ and $q=1_{\mathcal{T}}-p$, then $p,q\in\mathcal{T}$ and are
projections onto $l^{2}(\mathbb{Z}_{\geq 1})$ and $\mathbb{C}\delta_{0}$,
respectively, where $\delta_{0}$ is the characteristic function of the point
$0\in\mathbb{Z}$. For a subset $\mathcal{A}\subset\\{1,\ldots,n\\}$, let
$P^{n}_{\mathcal{A}}=r_{1}\otimes\cdots\otimes r_{n}$, where $r_{i}$ is $p$
when $i\in\mathcal{A}$ and is $q$ otherwise. The operator
$P^{n}_{\mathcal{A}}$ is a projection which satisfies
$\sum_{\mathcal{A}}P^{n}_{\mathcal{A}}=1_{\mathcal{T}^{n}}$. Let
$\tilde{T}=T_{z}^{*}\otimes q\otimes\cdots\otimes q$ and consider the
following element in $\mathcal{T}^{n}$:

$G=\tilde{T}+\sum_{\mathcal{A}\neq\\{1\\}}P^{n}_{\mathcal{A}},$

where the sum is taken over all subsets of $\\{1,\ldots,n\\}$ except
$\\{1\\}$. Then, we can see that
$\operatorname{\mathrm{Ker}}(G)\cong\mathbb{C}$ and
$\operatorname{\mathrm{Coker}}(G)=0$, that is, $G$ is a Fredholm Toeplitz
operator associated with codimension-$n$ corners whose Fredholm index is one.

This example leads to the following result.

######  Proposition 4.6.

The boundary maps of the six-term exact sequence for $K$-theory associated
with $(\ref{multimain})$ are surjective. Moreover, the boundary maps of the
24-term exact sequence for $KO$-theory associated with $(\ref{multimainreal})$
are surjective.

###### Proof.

The result for complex $K$-theory is immediate from Example 4.5. For
$KO$-theory, since the operator $G$ in Example 4.5 satisfies $G^{\tau}=G^{*}$,
the result follows as in Proposition 3.9. ∎

Note that, we have $\gamma_{n}(G)=(\pi_{1}(G),1,\cdots,1)\in\mathcal{S}^{n}$
by using

$\pi_{1}(G)=M_{z}^{*}\otimes q\otimes\cdots\otimes
q+\sum_{\emptyset\neq\mathcal{A}\subset\\{1,\ldots,n-1\\}}1_{C(\mathbb{T})}\otimes
P^{n-1}_{\mathcal{A}},$

The element $\gamma_{n}(G)$ is a unitary and defines an element
$[\gamma_{n}(G)]$ of the group $K_{1}(\mathcal{S}^{n})$. Since the Fredholm
index of $G$ is one, this gives a generator of
$K_{1}(\mathcal{S}^{n})\cong\mathbb{Z}$. As in the proof of Proposition 3.9,
generators of the $KO$-groups $KO_{i}(\mathcal{S}^{n},\tau_{\mathcal{S}})$ are
also obtained by using $G$.

######  Remark 4.7.

Let $1\leq j\leq n$. We have the following $*$-homomorphisms:

$\mathcal{S}^{n}\longrightarrow\mathcal{T}^{n-1}\otimes
C(\mathbb{T})\overset{\gamma_{n-1}\otimes
1}{\longrightarrow}\mathcal{S}^{n-1}\otimes C(\mathbb{T}),$

where the first map maps $(T_{1},\ldots,T_{n})$ to $T_{j}$. We write
$\sigma^{n,n-1}$ for the composite of the above maps which induces the map
$\sigma^{n,n-1}_{*}\colon K_{i}(\mathcal{S}^{n})\to
K_{i}(\mathcal{S}^{n-1}\otimes C(\mathbb{T}))$. When $i=0$,
$K_{0}(\mathcal{S}^{n})\cong\mathbb{Z}$ is generated by $[1_{2}]$ and
$\sigma^{n,n-1}_{*}[1_{2}]=[1_{2}]$. When $i=1$, the map $\sigma^{n,n-1}_{*}$
is zero since, by Example 4.5, the element $[\gamma_{n}(G)]$ is a generator of
$K_{1}(\mathcal{S}^{n})\cong\mathbb{Z}$ and
$\sigma^{n,n-1}_{*}[\gamma_{n}(G)]=[1]=0$. A similar observation also holds in
real cases. The map $\sigma^{n,n-1}_{*}$ from
$KO_{i}(\mathcal{S}^{n},\tau_{\mathcal{S}})$ to
$KO_{i}(\mathcal{S}^{n-1}\otimes C(\mathbb{T}),\tau)$ maps direct summands
corresponding to $KO$-groups of a point injectively and the other components
to zero.

##  5\. Topological invariants and corner states in Altland–Zirnbauer
classification

In this section, some gapped Hamiltonians on a lattice with corners are
discussed in each of the Altland–Zirnbauer classes. Since two of them (class
$\mathrm{A}$ and $\mathrm{AIII}$) are already studied in [32, 33], we consider
the remaining cases here. The codimension of the corner will be arbitrary,
though we mainly discuss codimension-two cases, with many detailed results
being obtained by [54, 35, 33] and the results in Sect. 3. Higher-
codimensional cases are discussed in a similar way, whose results are
collected in Sect. 5.5.

###  5.1. Setup

Let $V$ be a finite rank Hermitian vector space of complex rank $N$. Let $n$
be a positive integer. Let $\Theta$ and $\Xi$ be antiunitary operators on $V$
whose squares are $+1$ or $-1$. Let $\Pi$ be a unitary operator on $V$ whose
square is one. These operators $\Theta$, $\Xi$ and $\Pi$ are naturally
extended to the operator on $l^{2}(\mathbb{Z}^{n};V)$ by the fiberwise
operation; we also denote them as $\Theta$, $\Xi$ and $\Pi$, respectively. Let
$\mathrm{Herm}(V)$ be the space of Hermitian operators on $V$. We consider a
continuous map $\mathbb{T}^{n}\to\mathrm{Herm}(V)$, $t\mapsto H(t)$, where
$t=(t_{1},t_{2},\ldots,t_{n})$ in $\mathbb{T}^{n}$. Through the Fourier
transform $L^{2}(\mathbb{T}^{n};V)\cong l^{2}(\mathbb{Z}^{n};V)$, the
multiplication operator generated by this continuous map defines a bounded
linear self-adjoint operator $H$ on the Hilbert space
$l^{2}(\mathbb{Z}^{n};V)$. We consider the lattice $\mathbb{Z}^{n}$ as a model
of the bulk and call $H$ the bulk Hamiltonian. The Hamiltonian is said to
preserve time-reversal symmetry (TRS) if it commutes with $\Theta$ (i.e.,
$\Theta H\Theta^{*}=H$), particle-hole symmetry (PHS) if it anticommutes with
$\Xi$ (i.e., $\Xi H\Xi^{*}=-H$) and chiral symmetry if it anticommutes with
$\Pi$ (i.e., $\Pi H\Pi^{*}=-H$). Furthermore, TRS or PHS is called even (resp.
odd) if $\Theta^{2}=1$ or $\Xi^{2}=1$ (resp. $\Theta^{2}=-1$ or $\Xi^{2}=-1$).
Hamiltonians may preserve both TRS and PHS. In that case, $\Theta$ and $\Xi$
are assumed to commute, and $\Pi$ is identified with $\Theta\Xi$ or
$i\Theta\Xi$ such that $\Pi^{2}=1$ is satisfied.

By taking the partial Fourier transform in the variables $t_{1}$ and $t_{2}$,
we obtain a continuous family of bounded linear self-adjoint operators
$\\{H({\bm{t}})\\}_{{\bm{t}}\in\mathbb{T}^{n-2}}$ on $\mathcal{H}\otimes V$.
By taking a compression onto $\mathcal{H}^{\alpha}\otimes V$,
$\mathcal{H}^{\beta}\otimes V$ and $\hat{\mathcal{H}}^{\alpha,\beta}\otimes
V$, we obtain a family of operators $H^{\alpha}({\bm{t}})$,
$H^{\beta}({\bm{t}})$ and $\hat{H}^{\alpha,\beta}({\bm{t}})$ parametrized by
${\bm{t}}=(t_{3},\ldots,t_{n})\in\mathbb{T}^{n-2}$. $H^{\alpha}({\bm{t}})$ and
$H^{\beta}({\bm{t}})$ are our models for two surfaces (codimension-one
boundaries), and $\hat{H}^{\alpha,\beta}({\bm{t}})$ is our model of the corner
(codimension-two corner). We assume the following spectral gap condition.

######  Assumption 5.1 (Spectral Gap Condition).

We assume that both $H^{\alpha}$ and $H^{\beta}$ are invertible.

Under this assumption, the bulk Hamiltonian $H$ is also invertible since, when
we take a basis of $V$ and identify $V$ with $\mathbb{C}^{N}$, there is a
unital $*$-homomorphism $M_{N}(\mathcal{S}^{\alpha,\beta}\otimes
C(\mathbb{T}^{n-2}))\to M_{N}(C(\mathbb{T}^{n}))$ that maps
$(H^{\alpha},H^{\beta})$ to $H$. In classes $\mathrm{AI}$ and $\mathrm{AII}$,
we further assume that the spectrum of $H$ is not contained in either
$\mathbb{R}_{>0}$ or $\mathbb{R}_{<0}$. Note that in other classes where
Hamiltonians preserve PHS or chiral symmetry, this condition follows from
Assumption 5.1. Let $h$ be the pair $(H^{\alpha},H^{\beta})$. Under Assumption
5.1, we set

(5.1) $\operatorname{\mathrm{sign}}(h)=h|h|^{-1}.$

When the bulk Hamiltonian $H$ satisfies TRS, PHS or chiral symmetry, the
operators $H^{\alpha}$, $H^{\beta}$, $\hat{H}^{\alpha,\beta}$ and
$\operatorname{\mathrm{sign}}(h)$ also satisfy the symmetry, that is, commutes
with $\Theta$ or anticommutes with $\Xi$ or $\Pi$.

###  5.2. Gapped Topological Invariants

In the following, starting from a Hamiltonian satisfying Assumption 5.1 in
each class $\mathrm{AI}$, $\mathrm{BDI}$, $\mathrm{D}$, $\mathrm{DIII}$,
$\mathrm{AII}$, $\mathrm{CII}$, $\mathrm{C}$ and $\mathrm{CI}$, we construct a
unitary and see that this unitary satisfies the relation $\mathcal{R}_{i}$ in
Table 2. By using this unitary, we define a topological invariant as an
element of some $KO$-group.

In class $\mathrm{AI}$, the Hamiltonian has even TRS. We take an orthonormal
basis of $V$ to identify $V$ with $\mathbb{C}^{N}$ and express $\Theta$ as
$\mathcal{C}=\mathrm{diag}(c,\ldots,c)$. Under our spectral gap condition, let

(5.2) $u=\left(\begin{array}[]{cc}\operatorname{\mathrm{sign}}(h)&0\\\
0&1_{N}\end{array}\right).$

This $u$ is a self-adjoint unitary satisfying
$u^{\tau}=\operatorname{\mathrm{Ad}}_{\mathcal{C}\oplus\mathcal{C}}(u^{*})=u^{*}$
by the TRS.

In class BDI, the Hamiltonian has both even TRS and even PHS. Note that the
chiral symmetry is given by $\Pi=\Theta\Xi$ and commutes with $\Theta$ and
$\Xi$. For a Hamiltonian satisfying chiral symmetry and Assumption 5.1 to
exist, the even/odd decomposition $V\cong V^{0}\oplus V^{1}$ with respect to
$\Pi$ should satisfy
$\operatorname{\mathrm{rank}}{C}V^{0}=\operatorname{\mathrm{rank}}{C}V^{1}$,
and we assume that. Then, there is an orthonormal basis of $V$ to identify $V$
with $\mathbb{C}^{N}$ such that $\Pi$ and $\Theta$ are expressed as follows:

(5.3) $\Pi=\left(\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right),\ \
\Theta=\left(\begin{array}[]{cc}\mathcal{C}&0\\\
0&\mathcal{C}\end{array}\right),$

where $\mathcal{C}=\mathrm{diag}(c,\ldots,c)$. Since the Hamiltonian $H$
anticommutes with $\Pi$, the operator $\operatorname{\mathrm{sign}}(h)$ in
(5.1) is written in the following off-diagonal form:

(5.4) $\operatorname{\mathrm{sign}}(h)=\left(\begin{array}[]{cc}0&u^{*}\\\
u&0\end{array}\right),$

where $u$ is a unitary. By the TRS, we have
$u^{\tau}=\mathcal{C}u^{*}\mathcal{C}^{*}=u^{*}$.

In class D, the Hamiltonian has even PHS. We take an orthonormal basis of $V$
to identify $V\cong\mathbb{C}^{N}$ and express $\Xi$ as
$\mathcal{C}=\mathrm{diag}(c,\ldots,c)$. Let
$u=\operatorname{\mathrm{sign}}(h)$, then we have $u^{\tau}=\Xi u\Xi^{*}=-u$
by the PHS.

In class $\mathrm{DIII}$, the Hamiltonian has both odd TRS and even PHS. Note
that the chiral symmetry is given by $\Pi=i\Theta\Xi$ and anticommutes with
$\Theta$ and $\Xi$. For such a Hamiltonian $H$ satisfying Assumption 5.1 to
exist, the complex rank of $V$ must be a multiple of $4$ since
$\mathrm{sign}(H({\bm{t}}))$, $i\Pi$, $i$ and $\Theta$ provides a
$\mathit{Cl}_{1,1}\otimes\mathit{Cl}_{2,0}\cong\mathbb{H}(2)$-module structure
on $V$. We assume $\operatorname{\mathrm{rank}}{C}V=4M$ for some positive
integer $M$.

######  Lemma 5.2.

If a Hamiltonian $H$ satisfying Assumption 5.1 exists, there is an orthonormal
basis of $V$ such that $\Pi$ and $\Theta$ are expressed as follows.

$\Pi=\left(\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right),\ \
\Theta=\left(\begin{array}[]{cc}0&\mathcal{J}\\\
\mathcal{J}&0\end{array}\right).$

We write $\mathcal{J}=\mathrm{diag}(j,\ldots,j)$, where $j$ is the
quaternionic structure on $\mathbb{H}$.

###### Proof.

By using $\Pi$, we decompose $V\cong V^{0}\oplus V^{1}$. We identify
$V^{0}\cong V^{1}\cong\mathbb{C}^{2M}\cong\mathbb{H}^{M}$, on which we
consider $\mathcal{J}=\mathrm{diag}(j,\ldots,j)$. Let
$U=\Theta\left(\begin{array}[]{cc}0&\mathcal{J}\\\
\mathcal{J}&0\end{array}\right)$. Since $U$ is a unitary and commutes with
$\Pi$, we have $U=\mathrm{diag}(u_{0},u_{1})$, where $u_{0}$ and $u_{1}$ are
unitaries on $\mathbb{C}^{2M}$. Since $\Theta^{2}=-1$, we have
$u_{1}=\operatorname{\mathrm{Ad}}_{\mathcal{J}}(u_{0}^{*})$. Let
$W=\mathrm{diag}(-u_{0}^{*},1)$, then $W\Theta
W^{*}=\left(\begin{array}[]{cc}0&\mathcal{J}\\\
\mathcal{J}&0\end{array}\right)$. ∎

We take this basis on $V$ and express $\Pi$ and $\Theta$ as above. By the
chiral symmetry, we take $u$ in (5.4). By the TRS, we have
$u^{\sharp\otimes\tau}=\mathcal{J}u^{*}\mathcal{J}^{*}=u$.

In class $\mathrm{AII}$, the Hamiltonian has odd TRS. The space $V$ has a
quaternionic structure given by $\Theta$, and the complex rank of $V$ is even,
for which we write $2M$. There is an orthonormal basis of $V$ for identifying
$V$ with $\mathbb{C}^{2M}\cong\mathbb{H}^{M}$ and expressing $\Theta$ as
$\mathcal{J}=\mathrm{diag}(j,\ldots,j)$. Let $u$ be a self-adjoint unitary in
(5.2). By the TRS, we have
$u^{\sharp\otimes\tau}=\operatorname{\mathrm{Ad}}_{\mathcal{J}\oplus\mathcal{J}}(u^{*})=u^{*}$.

In class $\mathrm{CII}$, the Hamiltonian has both odd TRS and odd PHS. The
chiral symmetry is given by $\Pi=\Theta\Xi$ and commutes with $\Theta$ and
$\Xi$. As in the class BDI case, we take an orthonormal basis of $V$ to
identify $V$ with $\mathbb{C}^{N}$ and express $\Pi$ and $\Theta$ as

(5.5) $\Pi=\left(\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right),\ \
\Theta=\left(\begin{array}[]{cc}\mathcal{J}&0\\\
0&\mathcal{J}\end{array}\right),$

where $\mathcal{J}=\mathrm{diag}(j,\ldots,j)$. By the chiral symmetry, we take
$u$ in (5.4). By the TRS, we have
$u^{\sharp\otimes\tau}=\mathcal{J}u^{*}\mathcal{J}^{*}=u^{*}$.

In class C, the Hamiltonian has odd PHS. Since $\Xi$ provides a quaternionic
structure on $V$, its complex rank is even, for which we write $2M$. We take
an orthonormal basis of $V$ to identify $V$ with
$\mathbb{C}^{2M}\cong\mathbb{H}^{M}$ and express $\Xi$ as
$\mathcal{J}=\mathrm{diag}(j,\ldots,j)$. Let
$u=\operatorname{\mathrm{sign}}(h)$, then we have
$u^{\sharp\otimes\tau}=\mathcal{J}u^{*}\mathcal{J}^{*}=-u$ by the PHS.

In class CI, the Hamiltonian has both even TRS and odd PHS. The chiral
symmetry is given by $\Pi=i\Theta\Xi$ and anticommutes with $\Theta$ and
$\Xi$. As in Lemma 5.2, we take an orthonormal basis of $V$ to express,

$\Pi=\left(\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right),\ \
\Theta=\left(\begin{array}[]{cc}0&\mathcal{C}\\\
\mathcal{C}&0\end{array}\right),$

where $\mathcal{C}=\mathrm{diag}(c,\ldots,c)$. By the chiral symmetry, we take
$u$ in (5.4). By the TRS, we have
$u^{\tau}=\mathcal{C}u^{*}\mathcal{C}^{*}=u$.

######  Definition 5.3.

For a Hamiltonian in class $\spadesuit=$ $\mathrm{AI}$, $\mathrm{BDI}$,
$\mathrm{D}$, $\mathrm{DIII}$, $\mathrm{AII}$, $\mathrm{CII}$, $\mathrm{C}$ or
$\mathrm{CI}$ satisfying Assumption 5.1, let $u$ be the unitary defined as
above. As we have seen, this unitary $u$ satisfies the relation
$\mathcal{R}_{i(\spadesuit)}$ where $i(\spadesuit)$ is as indicated in Table
10. We denote its class999We simply write $\tau$ in place of
$\tau_{\mathcal{S}}\otimes\tau{T}$. In the following, these abbreviations for
tensor products of transpositions are employed, though the meaning will be
clear from the context. $[u]$ in the $KO$-group
$KO_{i(\spadesuit)}(\mathcal{S}^{\alpha,\beta}\otimes
C(\mathbb{T}^{n-2}),\tau)$ by
$\mathcal{I}_{\mathrm{Gapped}}^{n,2,\spadesuit}(H)$.

Table 10. $i(\spadesuit)$ and $c(\spadesuit)$ for each of the Altland–Zirnbauer classes $\spadesuit$. $\spadesuit$ | $\mathrm{AI}$ | $\mathrm{BDI}$ | $\mathrm{D}$ | $\mathrm{DIII}$ | $\mathrm{AII}$ | $\mathrm{CII}$ | $\mathrm{C}$ | $\mathrm{CI}$
---|---|---|---|---|---|---|---|---
$i(\spadesuit)$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $-1$
$c(\spadesuit)$ | $1$ | $1$ | $i$ | $1$ | $1$ | $1$ | $i$ | $1$

The groups $KO_{*}(\mathcal{S}^{\alpha,\beta}\otimes
C(\mathbb{T}^{n-2}),\tau)$ are computed by results in Sect. 3.2.

######  Remark 5.4.

We expressed the symmetry operators in a specific way, though we may choose
another one. In class DIII, for example, the operator $\Theta$ can also be
expressed as $\left(\begin{array}[]{cc}0&-\mathcal{C}\\\
\mathcal{C}&0\end{array}\right)$, where
$\mathcal{C}=\mathrm{diag}(c,\ldots,c)$. Then, we obtain unitaries satisfying
$u^{\tau}=-u$, which are treated in [38].

###  5.3. Gapless Topological Invariants

We next define another topological invariant by using our model for the corner
$\hat{H}^{\alpha,\beta}$. By Assumption 5.1 and Theorem $2.6$ in [54],
$\\{\hat{H}^{\alpha,\beta}({\bm{t}})\\}_{{\bm{t}}\in\mathbb{T}^{n-2}}$ is a
continuous family of self-adjoint Fredholm operators. Corresponding to its
Altland–Zirnbauer classes, this family provides a $\mathbb{Z}_{2}$-map from
$(\mathbb{T}^{n-2},\zeta)$ to some $\mathbb{Z}_{2}$-spaces of self-adjoint or
skew-adjoint Fredholm operators introduced in Appendix A as follows.

  * •

Class AI, $\mathbb{Z}_{2}$-map
$\hat{H}^{\alpha,\beta}\colon(\mathbb{T}^{n-2},\zeta)\to(\mathrm{Fred}^{(0,\underline{1})}_{*},\mathfrak{r}_{\Theta})$.

  * •

Class BDI, let $\epsilon_{1}=\Pi$. $\mathbb{Z}_{2}$-map
$\hat{H}^{\alpha,\beta}\colon(\mathbb{T}^{n-2},\zeta)\to(\mathrm{Fred}^{(0,\underline{2})}_{*},\mathfrak{r}_{\Theta})$.

  * •

Class D, $\mathbb{Z}_{2}$-map
$i\hat{H}^{\alpha,\beta}\colon(\mathbb{T}^{n-2},\zeta)\to(\mathrm{Fred}^{(\underline{1},0)}_{*},\mathfrak{r}_{\Xi})$.

  * •

Class $\mathrm{DIII}$, let $e_{1}=i\Pi$. $\mathbb{Z}_{2}$-map
$\hat{H}^{\alpha,\beta}\colon(\mathbb{T}^{n-2},\zeta)\to(\mathrm{Fred}^{(1,\underline{1})}_{*},\mathfrak{q}_{\Theta})$.

  * •

Class $\mathrm{AII}$, $\mathbb{Z}_{2}$-map
$\hat{H}^{\alpha,\beta}\colon(\mathbb{T}^{n-2},\zeta)\to(\mathrm{Fred}^{(0,\underline{1})}_{*},\mathfrak{q}_{\Theta})$.

  * •

Class $\mathrm{CII}$, let $\epsilon_{1}=\Pi$. $\mathbb{Z}_{2}$-map
$\hat{H}^{\alpha,\beta}\colon(\mathbb{T}^{n-2},\zeta)\to(\mathrm{Fred}^{(0,\underline{2})}_{*},\mathfrak{q}_{\Theta})$.

  * •

Class C, $\mathbb{Z}_{2}$-map
$i\hat{H}^{\alpha,\beta}\colon(\mathbb{T}^{n-2},\zeta)\to(\mathrm{Fred}^{(\underline{1},0)}_{*},\mathfrak{q}_{\Xi})$.

  * •

Class CI, let $e_{1}=i\Pi$. $\mathbb{Z}_{2}$-map
$\hat{H}^{\alpha,\beta}\colon(\mathbb{T}^{n-2},\zeta)\to(\mathrm{Fred}^{(1,\underline{1})}_{*},\mathfrak{r}_{\Theta})$.

Here, we write $\mathfrak{r}_{\Theta}=\operatorname{\mathrm{Ad}}_{\Theta}$
when $\Theta^{2}=1$ and
$\mathfrak{q}_{\Theta}=\operatorname{\mathrm{Ad}}_{\Theta}$ when
$\Theta^{2}=-1$. Involutions $\mathfrak{r}_{\Xi}$ and $\mathfrak{q}_{\Xi}$ are
defined as $\operatorname{\mathrm{Ad}}_{\Xi}$ in the same way. By Corollary
A.9, the $\mathbb{Z}_{2}$-homotopy classes of $\mathbb{Z}_{2}$-maps from
$(\mathbb{T}^{n-2},\zeta)$ to the above $\mathbb{Z}_{2}$-space of self-adjoint
or skew-adjoint Fredholm operators is isomorphic to the $KO$-group
$KO_{i}(C(\mathbb{T}^{n-2}),\tau{T})$ of some degree $i$.

######  Definition 5.5.

For $\spadesuit=$ AI, BDI, D, DIII, AII, CII, C or CI, let $i(\spadesuit)$,
$c(\spadesuit)$ and $\mathrm{Fred}^{\spadesuit}$ be numbers and the
$\mathbb{Z}_{2}$-space as in Tables 10 and 11. For a Hamiltonian $H$ in class
$\spadesuit$ satisfying Assumption 5.1, we denote
$\mathcal{I}_{\mathrm{Gapless}}^{n,2,\spadesuit}(H)$ for the class
$[c(\spadesuit)\hat{H}^{\alpha,\beta}]$ in
$KO_{i(\spadesuit)-1}(C(\mathbb{T}^{n-2}),\tau{T})$. We call
$\mathcal{I}_{\mathrm{Gapless}}^{n,2,\spadesuit}(H)$ the gapless corner
invariant.

If the gapless corner invariant is nontrivial, zero is contained in the
spectrum of $\hat{H}^{\alpha,\beta}$. In Sect. 5.6, we discuss more refined
relations between the gapless corner invariant and corner states when $k=n-1$
and $n$.

Table 11. In each Altland–Zirnbauer class $\spadesuit$, gapped invariants and
gapless invariants are defined as elements of some $K$\- and $KO$-groups of
some degree, as indicated in this table. Classifying spaces for topological
$K$\- and $KR$-groups through self-adjoint or skew-adjoint Fredholm operators
and unitaries are also included. ($\mathbb{Z}_{2}$-)spaces
$\mathrm{Fred}^{\spadesuit}$ and $U^{\spadesuit}$ are introduced in Appendix
A.

AZ | Gapped | Gapless
---|---|---
$\spadesuit$ |  $K$-group |  $K$-group | $\mathrm{Fred}^{\spadesuit}$ | $U^{\spadesuit}$
$\mathrm{A}$ | $K_{0}$ | $K_{1}$ | $\mathrm{Fred}^{(0,\underline{1})}_{*}$ | $U_{\mathrm{cpt}}$
$\mathrm{AIII}$ | $K_{1}$ | $K_{0}$ | $\mathrm{Fred}$ | $U^{(0,\underline{1})}_{\mathrm{cpt}}$
$\mathrm{AI}$ | $KO_{0}$ | $KO_{-1}$ | $(\mathrm{Fred}^{(0,\underline{1})}_{*},\mathfrak{r}_{\Theta})$ | $(U_{\mathrm{cpt}},\mathfrak{r}\circ*)$
$\mathrm{BDI}$ | $KO_{1}$ | $KO_{0}$ | $(\mathrm{Fred}^{(0,\underline{2})}_{*},\mathfrak{r}_{\Theta})\ (\cong(\mathrm{Fred}^{(\underline{1},1)}_{*},\mathfrak{r}_{\Xi}))$ |  $(U^{(0,\underline{1})}_{\mathrm{cpt}},\mathfrak{r})$
$\mathrm{D}$ | $KO_{2}$ | $KO_{1}$ | $(\mathrm{Fred}^{(\underline{1},0)}_{*},\mathfrak{r}_{\Xi})$ |  $(U_{\mathrm{cpt}},\mathfrak{r})$
$\mathrm{DIII}$ | $KO_{3}$ | $KO_{2}$ | $(\mathrm{Fred}^{(1,\underline{1})}_{*},\mathfrak{q}_{\Theta})\ (\cong(\mathrm{Fred}^{(\underline{2},0)}_{*},\mathfrak{r}_{\Xi}))$ |  $(U^{(\underline{1},0)}_{\mathrm{cpt}},\mathfrak{r})$
$\mathrm{AII}$ | $KO_{4}$ | $KO_{3}$ | $(\mathrm{Fred}^{(0,\underline{1})}_{*},\mathfrak{q}_{\Theta})$ | $(U_{\mathrm{cpt}},\mathfrak{q}\circ*)$
$\mathrm{CII}$ | $KO_{5}$ | $KO_{4}$ | $(\mathrm{Fred}^{(0,\underline{2})}_{*},\mathfrak{q}_{\Theta})\ (\cong(\mathrm{Fred}^{(\underline{1},1)}_{*},\mathfrak{q}_{\Xi}))$ |  $(U^{(0,\underline{1})}_{\mathrm{cpt}},\mathfrak{q})$
$\mathrm{C}$ | $KO_{6}$ | $KO_{5}$ | $(\mathrm{Fred}^{(\underline{1},0)}_{*},\mathfrak{q}_{\Xi})$ |  $(U_{\mathrm{cpt}},\mathfrak{q})$
$\mathrm{CI}$ | $KO_{-1}$ | $KO_{6}$ | $(\mathrm{Fred}^{(1,\underline{1})}_{*},\mathfrak{r}_{\Theta})\ (\cong(\mathrm{Fred}^{(\underline{2},0)}_{*},\mathfrak{q}_{\Xi}))$ |  $(U^{(\underline{1},0)}_{\mathrm{cpt}},\mathfrak{q})$

###  5.4. Correspondence

By taking a tensor product of the extension (2.2) and
$(C(\mathbb{T}^{n-2}),\tau{T})$, we have the following short exact sequence of
$C^{*,\tau}$-algebras,

$0\to(\mathcal{K}\otimes
C(\mathbb{T}^{n-2}),\tau)\to(\hat{\mathcal{T}}^{\alpha,\beta}\otimes
C(\mathbb{T}^{n-2}),\tau)\to(\mathcal{S}^{\alpha,\beta}\otimes
C(\mathbb{T}^{n-2}),\tau)\to 0.$

Let consider the following diagram containing the boundary map of $24$-term
exact sequence for $KO$-theory associated with this sequence:

$\textstyle{KO_{i(\spadesuit)}(\mathcal{S}^{\alpha,\beta}\otimes
C(\mathbb{T}^{n-2}),\tau)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\partial}_{i(\spadesuit)}\hskip
8.53581pt}$$\scriptstyle{L}$$\textstyle{KO_{i(\spadesuit)-1}(\mathcal{K}(\hat{\mathcal{H}}^{\alpha,\beta})\otimes
C(\mathbb{T}^{n-2}),\tau)}$$\textstyle{[(\mathbb{T}^{n-2},\zeta),\mathrm{Fred}^{\spadesuit}]_{\mathbb{Z}_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{[(\mathbb{T}^{n-2},\zeta),F^{\spadesuit}]_{\mathbb{Z}_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\exp}$

where $F^{\spadesuit}$ is the $\mathbb{Z}_{2}$-subspace of
$\mathrm{Fred}^{\spadesuit}$ as in Appendix A, whose inclusion
$F^{\spadesuit}\hookrightarrow\mathrm{Fred}^{\spadesuit}$ is the
$\mathbb{Z}_{2}$-homotopy equivalence. Maps $L$ and $\exp$ are as follows.

  * •

When $i(\spadesuit)$ is odd, for $[u]\in
KO_{i(\spadesuit)}(\mathcal{S}^{\alpha,\beta}\otimes
C(\mathbb{T}^{n-2}),\tau)$, we take a lift $a$ of $u$ and consider the matrix
$A$ as in Definition A.13, and set $L([u])=[A]$. The map $\exp$ is defined as
in Definition A.13.

  * •

When $i(\spadesuit)=0,4$, for $[u]\in
KO_{i(\spadesuit)}(\mathcal{S}^{\alpha,\beta}\otimes
C(\mathbb{T}^{n-2}),\tau)$, we take a self-adjoint lift $a$ of $u$ as in
Definition 8.3 of [14] and set $L([u])=[a]$. The map $\exp$ is defined by
$\exp([a^{\prime}])=[-\exp(\pi ia^{\prime})]$.

  * •

When $i(\spadesuit)=2,6$, for $[u]\in
KO_{i(\spadesuit)}(\mathcal{S}^{\alpha,\beta}\otimes
C(\mathbb{T}^{n-2}),\tau)$, we take a self-adjoint lift $a$ of $u$ as in
Definition 8.3 of [14] and set $L([u])=[ia]$. The map $\exp$ is defined by
$\exp([a^{\prime}])=[-\exp(\pi a^{\prime})]$.

In each case, the map $\exp$ is an isomorphism by Proposition A.7 and Sect.
A.2. For boundary maps $\hat{\partial}_{i(\spadesuit)}$, we use its
expressions through exponentials (see [14] for even $i(\spadesuit)$ and
Appendix A.3 for odd $i(\spadesuit)$) and the diagram commutes. Note that, by
Proposition 3.9, the boundary map $\hat{\partial}_{i(\spadesuit)}$ is
surjective. The following is the main result of this section.

######  Theorem 5.6.

$\hat{\partial}_{i(\spadesuit)}(\mathcal{I}_{\mathrm{Gapped}}^{n,2,\spadesuit}(H))=\mathcal{I}_{\mathrm{Gapless}}^{n,2,\spadesuit}(H)$.

###### Proof.

The operator $\hat{H}^{\alpha,\beta}$ is a self-adjoint lift of
$(H^{\alpha},H^{\beta})$ and preserves the symmetries of the class
$\spadesuit$. Therefore, we have
$L(\mathcal{I}_{\mathrm{Gapped}}^{n,2,\spadesuit}(H))=[c(\spadesuit)\hat{H}^{\alpha,\beta}]$
and the results follows from the commutativity of the above diagram. ∎

######  Remark 5.7 (Relation with bulk weak invariants).

Under Assumption 5.1, the bulk Hamiltonian $H$ is also invertible. When we
take $H$ in place of $h=(H^{\alpha},H^{\beta})$ and define the unitary
$u^{\prime}$ as in Sect. 5.2, this unitary defines an element $[u^{\prime}]$
in $KO_{i(\spadesuit)}(C(\mathbb{T}^{n}),\tau{T})$, which classifies bulk
invariants in class $\spadesuit$. A relation between our gapped invariants
$\mathcal{I}_{\mathrm{Gapped}}^{n,2,\spadesuit}(H)$ and these bulk invariants
can be discussed through the map $(\sigma\otimes 1)_{*}\colon
KO_{i(\spadesuit)}(\mathcal{S}^{\alpha,\beta}\otimes
C(\mathbb{T}^{n-2}),\tau)\to KO_{i(\spadesuit)}(C(\mathbb{T}^{n-2}),\tau{T})$,
which maps $[u]$ to $[u^{\prime}]$ and we briefly mention its consequences
here. Our gapped invariant $\mathcal{I}_{\mathrm{Gapped}}^{n,2,\spadesuit}(H)$
has no relation with bulk invariants in the sense that, under Assumption 5.1,
bulk invariants are trivial except for a component corresponding to
$KO$-groups of a point101010This component maps to zero by the boundary map
$\hat{\partial}_{i(\spadesuit)}$ and has no relation with gapless corner
invariants. and for the cases when $\alpha$, $\beta$ are both rational (or
$\pm\infty$) and $t=-ps+qr$ is even. By Remark 3.10, when $\alpha$ and $\beta$
takes these values, some bulk weak invariants can be nontrivial, though they
have no relation with $\mathcal{I}_{\mathrm{Gapless}}^{n,2,\spadesuit}(H)$,
which can be seen by comparing the above map $(\sigma\otimes 1)_{*}$ and the
boundary map $\hat{\partial}_{i(\spadesuit)}$.

######  Remark 5.8 (Convex and concave corners).

When we fix $\alpha$ and $\beta$, there exist two models of corners: convex
and concave corners ($\hat{\mathcal{H}}^{\alpha,\beta}$ and
$\check{\mathcal{H}}^{\alpha,\beta}$). We have discussed convex corners
though, as in [33], similar results also hold for concave corners by using
(2.3) in place of (2.2) in our discussion. By Remark 3.11, the gapless
invariants of these two are related by the factor $-1$.

###  5.5. Higher-Codimensional Cases

Let $n$ and $k$ be positive integers satisfying $3\leq k\leq n$. In this
subsection, we consider $n$-D system with a codimension-$k$ corner. Let
$d=n-k$. We consider a continuous map $\mathbb{T}^{n}\to\mathrm{Herm}(V)$ and
the bounded linear self-adjoint operator $H$ on $l^{2}(\mathbb{Z}^{n})$
generated by this map, which is our model of the bulk. We next introduce
models of corners of codimension $k-1$ whose intersection makes a codimension
$k$ corner. For this, we choose $d$ variables $t_{j_{1}},\ldots,t_{j_{d}}$ in
$t_{1},t_{2},\ldots,t_{n}$ and consider the partial Fourier transform in these
$d$ variables to obtain a continuous family of self-adjoint operators
$\\{H({\bm{t}})\\}_{{\bm{t}}\in\mathbb{T}^{d}}$ on $l^{2}(\mathbb{Z}^{k};V)$.
On the Hilbert space
$l^{2}(\mathbb{Z}^{k};V)\cong(l^{2}(\mathbb{Z})\otimes\cdots\otimes
l^{2}(\mathbb{Z}))\otimes V$, we consider projections $P_{k}=(P_{\geq
0}\otimes\cdots\otimes P_{\geq 0})\otimes 1_{V}$, and $P_{k,i}=(P_{\geq
0}\otimes\cdots\otimes 1\otimes\cdots\otimes P_{\geq 0})\otimes 1_{V}$ for
$1\leq i\leq k$ where inside the brackets is the tensor products of $P_{\geq
0}$ except for the $i$-th tensor product replaced by the identity. By using
these projections, we define the following operators:

$H^{c}({\bm{t}})=P_{k}H({\bm{t}})P_{k},\ \
H_{i}({\bm{t}})=P_{k,i}H({\bm{t}})P_{k,i},$

for $1\leq i\leq k$ and for ${\bm{t}}\in\mathbb{T}^{d}$. These two are our
model for a codimension $k$ corner and codimension $k-1$ corners,
respectively. When we fix a basis on $V$, we have
$(H_{1}({\bm{t}}),\ldots,H_{k}({\bm{t}}))\in M_{N}(\mathcal{S}^{k})$ by the
construction. We assume the following condition in this subsection.

######  Assumption 5.9 (Spectral Gap Condition).

We assume that our models for codimension $k-1$ corners $H_{1},\ldots,H_{k}$
are invertible.

Under this assumption, the model for the bulk, surfaces and corners of
codimension less than $k$, whose intersection makes our codimension-$k$
corner, are invertible. As in Sect. 5.1, let $h=(H_{1},\ldots,H_{k})$.

######  Definition 5.10.

For a Hamiltonian in class $\spadesuit=$ $\mathrm{AI}$, $\mathrm{BDI}$,
$\mathrm{D}$, $\mathrm{DIII}$, $\mathrm{AII}$, $\mathrm{CII}$, $\mathrm{C}$ or
$\mathrm{CI}$ satisfying Assumption 5.9, let $u$ be a unitary defined by using
this $h$ in place of that in Sect. 5.2. As in Sect. 5.2, this unitary $u$
satisfies the relation $\mathcal{R}_{i(\spadesuit)}$ where $i(\spadesuit)$ is
as indicated in Table 10. We denote its class $[u]$ in the $KO$-group
$KO_{i(\spadesuit)}(\mathcal{S}^{k}\otimes C(\mathbb{T}^{d}),\tau)$ by
$\mathcal{I}_{\mathrm{Gapped}}^{n,k,\spadesuit}(H)$.

The $KO$-groups $KO_{i}(\mathcal{S}^{k}\otimes C(\mathbb{T}^{d}),\tau)$ are
computed by using Proposition 4.4. For each ${\bm{t}}\in\mathbb{T}^{d}$, the
operator $H^{c}({\bm{t}})$ is Fredholm by Corollary 4.2.

######  Definition 5.11.

For $\spadesuit=$ AI, BDI, D, DIII, AII, CII, C or CI, let $i(\spadesuit)$,
$c(\spadesuit)$ and $\mathrm{Fred}^{\spadesuit}$ be numbers and the
$\mathbb{Z}_{2}$-space as in Tables 10 and 11. For a Hamiltonian $H$ in class
$\spadesuit$ satisfying Assumption 5.9, we denote
$\mathcal{I}_{\mathrm{Gapless}}^{n,k,\spadesuit}(H)$ for the class
$[c(\spadesuit)H^{c}]$ in $KO_{i(\spadesuit)-1}(C(\mathbb{T}^{d}),\tau{T})$.
We call $\mathcal{I}_{\mathrm{Gapless}}^{n,k,\spadesuit}(H)$ the gapless
corner invariant.

We next discuss a relation between these two topological invariants. As in
Sect. 5.4, we consider a tensor product of the extension (4.4) and
$(C(\mathbb{T}^{d}),\tau{T})$ and let $\partial_{i(\spadesuit)}\colon
KO_{i(\spadesuit)}(\mathcal{S}^{k}\otimes C(\mathbb{T}^{d}),\tau)\to
KO_{i(\spadesuit)-1}(\mathcal{K}^{k}\otimes C(\mathbb{T}^{d}),\tau)$ be the
boundary map associated with it expressed through exponentials. Since $H^{c}$
is a self-adjoint lift of $(H_{1},\ldots,H_{k})$, the following relation
holds, as in Theorem 5.6.

######  Theorem 5.12.

$\partial_{i(\spadesuit)}(\mathcal{I}_{\mathrm{Gapped}}^{n,k,\spadesuit}(H))=\mathcal{I}_{\mathrm{Gapless}}^{n,k,\spadesuit}(H).$

######  Remark 5.13.

As in Remark 5.7, under Assumption 5.9, some gapped invariants related to
corner states for corners of codimension $<k$ are also defined, though, by
Remark 4.7, they are trivial except for a component corresponding to
$KO$-groups of a point.

######  Remark 5.14.

Gapless corner invariants for each systems are elements of the group
$KO_{i}(C(\mathbb{T}^{d}),\tau{T})\cong\bigoplus_{j=0}^{d}\binom{d}{j}KO_{i-j}(\mathbb{C},\mathrm{id})$.
As in the case of (first-order) topological insulators [42], we call the
component $KO_{i-d}(\mathbb{C},\mathrm{id})$ strong and others weak.

Complex cases can also be discussed in a similar way111111In [32, 33], there
is a mistake in the computations of the group
$K_{0}(\mathcal{S}^{\alpha,\beta})$ in the case where $\alpha$ and $\beta$ are
rational numbers (there is a torsion part in general, as in
$KO_{0}(\mathcal{S}^{\alpha,\beta},\tau_{\mathcal{S}})$ computed in Sect.
$4$), which is correctly stated in [55]. The author would like to thank Guo
Chuan Thiang for pointing this mistake out.. For class A systems with a
codimension $\geq 3$ corner, under the Assumption 5.9, we define gapped and
gapless invariants as elements of $K_{0}(\mathcal{S}^{k}\otimes
C(\mathbb{T}^{d}))$ and $K_{1}(C(\mathbb{T}^{d}))$, respectively, and the
boundary map $\partial_{0}\colon K_{0}(\mathcal{S}^{k}\otimes
C(\mathbb{T}^{d}))\to K_{1}(\mathcal{K}^{k}\otimes C(\mathbb{T}^{d}))$
associated with (4.4) relates these two, which is surjective by Proposition
4.6. In class AIII systems, we use $\partial_{1}\colon
K_{1}(\mathcal{S}^{k}\otimes C(\mathbb{T}^{d}))\to
K_{0}(\mathcal{K}^{k}\otimes C(\mathbb{T}^{d}))$ instead. Gapless corner
invariants takes value in
$K_{i}(C(\mathbb{T}^{d}))\cong\bigoplus_{j=0}^{d}\binom{d}{j}K_{i-j}(\mathbb{C})$,
and we call the component $K_{i-d}(\mathbb{C})$ strong and others weak.

Strong invariants for each system are classified in Table 1.

###  5.6. Numerical Corner Invariants

Our gapless corner invariants are defined as elements of some $KO$-group. In
this subsection, we introduce $\mathbb{Z}$\- or $\mathbb{Z}_{2}$-valued
numerical corner invariants for our systems in cases where $k=n$ and $k=n-1$
to make the relation between our gapped invariants and corner states more
explicit. From Table 11, we discuss Hamiltonians in classes $\mathrm{BDI}$,
$\mathrm{D}$, $\mathrm{DIII}$, and $\mathrm{CII}$ when $k=n$ and $\mathrm{D}$,
$\mathrm{DIII}$, $\mathrm{AII}$ and $\mathrm{CII}$ when $k=n-1$ satisfying our
spectral gap condition.

####  5.6.1. Case of $k=n$

In this case, our model of the corner $H^{c}$ is a self-adjoint Fredholm
operator which has some symmetry corresponding to its Altland–Zirnbauer
class121212In what follows, we also write $H^{c}$ for $\hat{H}^{\alpha,\beta}$
in $k=2$ case.. An appropriate definition of numerical topological invariants
is introduced in [9] and we put them in our framework.

In class BDI, the operator $H^{c}$ is an element of the fixed point set
$(\mathrm{Fred}^{(0,\underline{2})}_{*})^{\mathfrak{r}_{\Theta}}$ of the
involution $\mathfrak{r}_{\Theta}$, where the Clifford action of
$\mathit{Cl}_{0,1}$ on the Hilbert space is given by $\epsilon_{1}=\Pi$ (see
also Lemma A.10 and Remark A.11). We express $\Pi$ and $\Theta$ as in (5.3)
and express $H^{c}$ as follows.

(5.6) $H^{c}=\left(\begin{array}[]{cc}0&(h^{c})^{*}\\\
h^{c}&0\end{array}\right).$

The operator $h^{c}$ is a Fredholm operator that commutes with $\mathcal{C}$
and thus is a real Fredholm operator. Its Fredholm index is

$\operatorname{\mathrm{ind}}(h^{c})=\operatorname{\mathrm{rank}}{C}\operatorname{\mathrm{Ker}}(h^{c})-\operatorname{\mathrm{rank}}{C}\operatorname{\mathrm{Coker}}(h^{c})=\mathrm{Tr}(\Pi|_{\operatorname{\mathrm{Ker}}(H^{c})}),$

where the right-hand side is the trace of $\Pi$ restricted to
$\operatorname{\mathrm{Ker}}(H^{c})$. The Fredholm index induces an
isomorphism
$\operatorname{\mathrm{ind}}^{\mathrm{BDI}}\colon[(\mathrm{pt},\mathrm{id}),(\mathrm{Fred}^{(0,\underline{2})}_{*},\mathfrak{r}_{\Theta})]_{\mathbb{Z}_{2}}\to\mathbb{Z}$.

In class D, $iH^{c}$ commutes with the real structure $\Xi$ and is a real
skew-adjoint Fredholm operator. Its mod $2$ index [9] is

$\operatorname{\mathrm{ind}}_{1}(iH^{c})=\operatorname{\mathrm{rank}}{C}\operatorname{\mathrm{Ker}}(H^{c})\mod
2,$

which induces the isomorphism
$\operatorname{\mathrm{ind}}^{\mathrm{D}}\colon[(\mathrm{pt},\mathrm{id}),(\mathrm{Fred}^{(\underline{1},0)}_{*},\mathfrak{r}_{\Xi})]_{\mathbb{Z}_{2}}\to\mathbb{Z}_{2}.$

In class $\mathrm{DIII}$, $H^{c}$ is an element of
$(\mathrm{Fred}^{(1,\underline{1})}_{*})^{\mathfrak{q}_{\Theta}}$, where the
action of $\mathit{Cl}_{1,0}$ is given by $e_{1}=i\Pi$. The operator $iH^{c}$
and $e_{1}$ commute with the real structure $\Xi$; thus, $iH^{c}$ is a real
skew-adjoint Fredholm operator that anticommutes with $e_{1}$. Its mod $2$
index [9] is

$\operatorname{\mathrm{ind}}_{2}(iH^{c})=\frac{1}{2}\operatorname{\mathrm{rank}}{C}\operatorname{\mathrm{Ker}}(H^{c})\mod
2,$

which induces the isomorphism
$\operatorname{\mathrm{ind}}^{\mathrm{DIII}}\colon[(\mathrm{pt},\mathrm{id}),(\mathrm{Fred}^{(1,\underline{1})}_{*},\mathfrak{q}_{\Theta})]_{\mathbb{Z}_{2}}\to\mathbb{Z}_{2}.$

In class $\mathrm{CII}$, the operator $H^{c}$ is an element of
$(\mathrm{Fred}^{(0,\underline{2})}_{*})^{\mathfrak{q}_{\Theta}}$, where the
Clifford action of $\mathit{Cl}_{0,1}$ is given by $\epsilon_{1}=\Pi$. We
express $\Theta$ and $\Pi$ as in (5.5) and express $H^{c}$ as in (5.6). The
operator $h^{c}$ commutes with $\mathcal{J}$ and is a quaternionic Fredholm
operator. Its Fredholm index $\operatorname{\mathrm{ind}}(h^{c})$ is an even
integer that induces an isomorphism
$\operatorname{\mathrm{ind}}^{\mathrm{CII}}\colon[(\mathrm{pt},\mathrm{id}),(\mathrm{Fred}^{(0,\underline{2})}_{*},\mathfrak{q}_{\Theta})]_{\mathbb{Z}_{2}}\to
2\mathbb{Z}$.

######  Definition 5.15.

For $n$-D systems with codimension-$n$ corners in classes $\mathrm{BDI}$,
$\mathrm{D}$, $\mathrm{DIII}$ and $\mathrm{CII}$, we define the numerical
corner invariant as follows.

  * •

In class $\mathrm{BDI}$, let
$\mathcal{N}^{n,n,\mathrm{BDI}}_{\mathrm{Gapless}}(H)=\operatorname{\mathrm{ind}}(h^{c})\in\mathbb{Z}$.

  * •

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

  * •

In class $\mathrm{DIII}$, let
$\mathcal{N}^{n,n,\mathrm{DIII}}_{\mathrm{Gapless}}(H)=\operatorname{\mathrm{ind}}_{2}(iH^{c})\in\mathbb{Z}_{2}$.

  * •

In class $\mathrm{CII}$, let
$\mathcal{N}^{n,n,\mathrm{CII}}_{\mathrm{Gapless}}(H)=\operatorname{\mathrm{ind}}(h^{c})\in
2\mathbb{Z}$.

Note that by these definitions, they are images of gapless corner invariants
$\mathcal{I}^{n,n,\spadesuit}_{\mathrm{Gapless}}(H)$ for each class
$\spadesuit=\mathrm{BDI}$, $\mathrm{D}$, $\mathrm{DIII}$ and $\mathrm{CII}$
through the isomorphism $\operatorname{\mathrm{ind}}^{\spadesuit}$. In each
case, the numerical corner invariant is computed through
$\operatorname{\mathrm{Ker}}(H^{c})$ and is related to the number of corner
states.

####  5.6.2. Case of $k=n-1$

In this case, $\\{H^{c}({\bm{t}})\\}_{{\bm{t}}\in\mathbb{T}}$ is a continuous
family of self-adjoint Fredholm operators preserving some symmetry. The
numerical corner invariants are given by using ($\mathbb{Z}$-valued) spectral
flow [8] and its $\mathbb{Z}_{2}$-valued variants [19, 18]. We first review
$\mathbb{Z}$\- and $\mathbb{Z}_{2}$-valued spectral flow.

Spectral flow is, roughly speaking, the net number of crossing points of
eigenvalues of a continuous family of self-adjoint Fredholm operators with
zero [8]. The following definition of spectral flow is due to Phillips [56].

######  Definition 5.16 (Spectral flow).

Let $A\colon[-1,1]\rightarrow\mathrm{Fred}^{(0,\underline{1})}_{*}$ be a
continuous map. We choose a partition $-1=s_{0}<s_{1}<\cdots<s_{n}=1$ and
positive numbers $c_{1},c_{2},\ldots,c_{n}$ so that for each $i=1,2,\ldots,n$,
the function $t\mapsto\chi_{[-c_{i},c_{i}]}(A_{s})$ is continuous and finite
rank on $[s_{i-1},s_{i}]$, where $\chi_{[a,b]}$ is the characteristic function
of $[a,b]$. We define the spectral flow of $A$ as follows.

$\operatorname{\mathrm{sf}}(A)=\sum^{n}_{i=1}(\operatorname{\mathrm{rank}}{C}(\chi_{[0,c_{i}]}(A_{s_{i}}))-\operatorname{\mathrm{rank}}{C}(\chi_{[0,c_{i}]}(A_{s_{i-1}}))\in\mathbb{Z}.$

Spectral flow is independent of the choice made and depends only on the
homotopy class of the path $A$ leaving the endpoints fixed. Thus the spectral
flow induces a map
$\operatorname{\mathrm{sf}}\colon[\mathbb{T},\mathrm{Fred}^{(0,\underline{1})}_{*}]\to\mathbb{Z}$
which is a group isomorphism.

We next discuss $\mathbb{Z}_{2}$-valued spectral flow. Let $\zeta_{0}$ be an
involution on the interval $[-1,1]$ given by $\zeta_{0}(s)=-s$. Let $A$ be a
$\mathbb{Z}_{2}$-map from $([-1,1],\zeta_{0})$ to
$(\mathrm{Fred}^{(0,\underline{1})}_{*},\mathfrak{q})$. Then, the spectrum
$\mathrm{sp}(A_{s})$ of $A_{s}$ is symmetric with respect to $\zeta_{0}$, and
roughly speaking, $\mathbb{Z}_{2}$-valued spectral flow counts the mod $2$ of
the net number of pairs of crossing points of $\mathrm{sp}(A_{s})$ with zero.
$\mathbb{Z}_{2}$-valued spectral flow is studied in [19, 18, 21, 15] and we
give one definition following [56, 19].

######  Definition 5.17 ($\mathbb{Z}_{2}$-Valued Spectral Flow).

Let
$A\colon([-1,1],\zeta_{0})\rightarrow(\mathrm{Fred}^{(0,\underline{1})}_{*},\mathfrak{q})$
be a $\mathbb{Z}_{2}$-map. We choose a partition
$0=s_{0}<s_{1}<\cdots<s_{n}=1$ of $[0,1]$ and positive numbers
$c_{1},c_{2},\ldots,c_{n}$ so that for each $i=1,2,\ldots,n$, the map
$t\mapsto\chi_{[-c_{i},c_{i}]}(A_{s})$ is continuous and finite rank on
$[s_{i-1},s_{i}]$. We define the $\mathbb{Z}_{2}$-valued spectral flow
$\operatorname{\mathrm{sf_{2}}}(A)$ of $A$ as follows.

$\operatorname{\mathrm{sf_{2}}}(A)=\sum^{n}_{i=1}(\operatorname{\mathrm{rank}}{C}(\chi_{[0,c_{k}]}(A_{s_{i}}))+\operatorname{\mathrm{rank}}{C}(\chi_{[0,c_{i}]}(A_{s_{i-1}})))\hskip
5.69054pt\bmod{2}\ \in\mathbb{Z}_{2}.$

$\mathbb{Z}_{2}$-valued spectral flow is independent of the choice made and
depends only on the $\mathbb{Z}_{2}$-homotopy class of the
$\mathbb{Z}_{2}$-map $A$ leaving the endpoints fixed or leaving these points
in the $\mathbb{Z}_{2}$-fixed point set
$(\mathrm{Fred}^{(0,\widehat{1})}_{*})^{\mathfrak{q}}$. Thus
$\mathbb{Z}_{2}$-valued spectral flow induces a group homomorphism
$\operatorname{\mathrm{sf_{2}}}\colon[(\mathbb{T},\zeta_{0}),(\mathrm{Fred}^{(0,\underline{1})}_{*},\mathfrak{q})]_{\mathbb{Z}_{2}}\to\mathbb{Z}_{2}$.
By Appendix A, the $\mathbb{Z}_{2}$-homotopy classes
$[(\mathbb{T},\zeta_{0}),(\mathrm{Fred}^{(0,\underline{1})}_{*},\mathfrak{q})]_{\mathbb{Z}_{2}}$
is isomorphic to $KO_{3}(C(\mathbb{T}),\tau{T})\cong\mathbb{Z}_{2}$.

######  Example 5.18.

On $\mathbb{C}^{2}$, let consider a family of self-adjoint operators given by
$B_{s}=\mathrm{diag}(s,-s)$ for $s\in[-1,1]$, and an antiunitary $j$ given by
$j(x,y)=(-\bar{y},\bar{x})$. Then, we have a $\mathbb{Z}_{2}$-map
$B\colon([-1,1],\zeta_{0})\to(M_{2}(\mathbb{C}),\operatorname{\mathrm{Ad}}_{j})$
whose $\mathbb{Z}_{2}$-valued spectral flow
$\operatorname{\mathrm{sf_{2}}}(B)$ is one. We extend this finite-dimensional
example to an infinite-dimensional one to give an example of a family
parametrized by the circle of nontrivial $\mathbb{Z}_{2}$-valued spectral
flow. Let $\mathcal{V}$ be a separable infinite-dimensional complex Hilbert
space equipped with a quaternionic structure $q$. On
$\mathcal{V}^{\prime}=\mathbb{C}^{2}\oplus\mathcal{V}\oplus\mathcal{V}$, we
consider a quaternionic structure $q^{\prime}=j\oplus q\oplus q$ and a family
self-adjoint Fredholm operators given by
$C_{s}=\mathrm{diag}(B_{s},1_{\mathcal{V}},-1_{\mathcal{V}})$. Let
$U^{(0,\underline{1})}_{*}(\mathcal{V}^{\prime})$ the space of unitaries on
$\mathcal{V}^{\prime}$ whose spectrum is $\\{\pm 1\\}$ equipped with the norm
topology. Then, its endpoints $C_{\pm 1}$ are contained in
$U^{(0,\underline{1})}_{*}(\mathcal{V}^{\prime})$. Through an identification
$(\mathcal{V}\oplus\mathcal{V},q\oplus
q)\cong(\mathcal{V}^{\prime},q^{\prime})$, the operator
$\mathrm{diag}(1_{\mathcal{V}},-1_{\mathcal{V}})$ gives an element $v_{0}\in
U^{(0,\underline{1})}_{*}(\mathcal{V}^{\prime})$ which satisfies
$\operatorname{\mathrm{Ad}}_{q}^{\prime}(v_{0})=v_{0}$. The space
$U^{(0,\underline{1})}_{*}(\mathcal{V}^{\prime})$ is homeomorphic to the
homogeneous space
$U(\mathcal{V}^{\prime})/(U(\mathbb{C}\oplus\mathcal{V})\times
U(\mathbb{C}\oplus\mathcal{V}))$, which is contractible by Kuiper’s theorem
[45]. Thus, there is a path $l\colon[0,1]\to
U^{(0,\underline{1})}_{*}(\mathcal{V}^{\prime})$ whose endpoints are
$l(0)=v_{0}$ and $l(1)=C_{1}$. We extend $l$ to a $\mathbb{Z}_{2}$-map
$l^{\prime}\colon([-1,1],\zeta_{0})\to(U^{(0,\underline{1})}_{*}(\mathcal{V}),\operatorname{\mathrm{Ad}}_{q^{\prime}})$
by $l^{\prime}(s)=\operatorname{\mathrm{Ad}}_{q^{\prime}}(l(-s))$ for
$s\in[-1,0]$. Since $l^{\prime}(\pm 1)=C_{\pm 1}$, we connect the endpoints of
$C$ and $l^{\prime}$ to construct a $\mathbb{Z}_{2}$-map
$C^{\prime}\colon(\mathbb{T},\zeta)\to(\mathrm{Fred}^{(0,\widehat{1})}_{*},\mathfrak{q}^{\prime})$,
where $\mathfrak{q}^{\prime}=\operatorname{\mathrm{Ad}}_{q^{\prime}}$. Then,
$\operatorname{\mathrm{sf_{2}}}(C^{\prime})=\operatorname{\mathrm{sf_{2}}}(B)=1$.

In class $\mathrm{D}$, we have a $\mathbb{Z}_{2}$-map
$iH^{c}\colon(\mathbb{T},\zeta)\to(\mathrm{Fred}^{(\underline{1},0)}_{*},\mathfrak{r}_{\Xi})$.
The $\mathbb{Z}_{2}$-homotopy classes
$[(\mathbb{T},\zeta),(\mathrm{Fred}^{(\underline{1},0)}_{*},\mathfrak{r}_{\Xi})]_{\mathbb{Z}_{2}}$
is isomorphic to
$KO_{1}(C(\mathbb{T}),\tau{T})\cong\mathbb{Z}_{2}\oplus\mathbb{Z}.$ By
forgetting the $\mathbb{Z}_{2}$-actions and multiplying by $-i$, there is a
map from
$[(\mathbb{T},\zeta),(\mathrm{Fred}^{(\underline{1},0)}_{*},\mathfrak{r}_{\Xi})]_{\mathbb{Z}_{2}}$
to $[\mathbb{T},\mathrm{Fred}^{(0,\underline{1})}_{*}]$. Combined with this
map and the spectral flow
$\operatorname{\mathrm{sf}}\colon[\mathbb{T},\mathrm{Fred}^{(0,\underline{1})}_{*}]\to\mathbb{Z}$,
we obtain a homomorphism

$\operatorname{\mathrm{sf}}^{\mathrm{D}}\colon[(\mathbb{T},\zeta),(\mathrm{Fred}^{(\underline{1},0)}_{*},\mathfrak{r}_{\Xi})]_{\mathbb{Z}_{2}}\to\mathbb{Z},\
\ [A]\mapsto\operatorname{\mathrm{sf}}(-iA).$

######  Example 5.19.

Let
$B^{\prime}\colon([-1,1],\zeta_{0})\to(\mathbb{C},\operatorname{\mathrm{Ad}}_{c})$
be a $\mathbb{Z}_{2}$-map defined by $B^{\prime}_{s}=is$. Then,
$\operatorname{\mathrm{sf}}^{\mathrm{D}}(B^{\prime})$ is defined and
$\operatorname{\mathrm{sf}}^{\mathrm{D}}(B^{\prime})=1$.

In class $\mathrm{DIII}$,
$H^{c}\colon(\mathbb{T},\zeta)\to(\mathrm{Fred}^{(1,\underline{1})}_{*},\mathfrak{q}_{\Theta})$
is a $\mathbb{Z}_{2}$-map, where the action of the Clifford algebra on the
right-hand side is given by $e_{1}=i\Pi$. The $\mathbb{Z}_{2}$-homotopy
classes
$[(\mathbb{T},\zeta),(\mathrm{Fred}^{(1,\underline{1})}_{*},\mathfrak{q}_{\Theta})]_{\mathbb{Z}_{2}}$
is isomorphic to
$KO_{2}(C(\mathbb{T}),\tau{T})\cong\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$. Since
$(\mathrm{Fred}^{(1,\underline{1})}_{*},\mathfrak{q}_{\Theta})$ is a
$\mathbb{Z}_{2}$-subspace of
$(\mathrm{Fred}^{(0,\underline{1})}_{*},\mathfrak{q}_{\Theta})$, the inclusion
induces a map from
$[(\mathbb{T},\zeta),(\mathrm{Fred}^{(1,\underline{1})}_{*},\mathfrak{q}_{\Theta})]_{\mathbb{Z}_{2}}$
to
$[(\mathbb{T},\zeta),(\mathrm{Fred}^{(0,\underline{1})}_{*},\mathfrak{q}_{\Theta})]_{\mathbb{Z}_{2}}$.
Combined with the $\mathbb{Z}_{2}$-valued spectral flow, we obtain the
following map:

$\operatorname{\mathrm{sf}}^{\mathrm{DIII}}\colon[(\mathbb{T},\zeta),(\mathrm{Fred}^{(1,\underline{1})}_{*},\mathfrak{q}_{\Theta})]_{\mathbb{Z}_{2}}\to\mathbb{Z}_{2},\
\ [A]\mapsto\operatorname{\mathrm{sf_{2}}}(A).$

For $b=1$ or $-1$, let $i_{b}$ be the inclusion
$\\{b\\}\hookrightarrow\mathbb{T}$, and let $w_{b}$ be the composite of the
following maps:

$[(\mathbb{T},\zeta),(\mathrm{Fred}^{(1,\underline{1})}_{*},\mathfrak{q}_{\Theta})]_{\mathbb{Z}_{2}}\overset{i_{b}^{*}}{\longrightarrow}[(\\{\pm
1\\},\mathrm{id}),(\mathrm{Fred}^{(1,\underline{1})}_{*},\mathfrak{q}_{\Theta})]_{\mathbb{Z}_{2}}\overset{\operatorname{\mathrm{ind}}_{2}}{\longrightarrow}\mathbb{Z}_{2}.$

######  Example 5.20.

Let $j$, $\mathcal{V}$, $\mathcal{V}^{\prime}$, $q$, $q^{\prime}$, $B_{s}$ and
$C_{s}$ be as in Example 5.18. Let $e_{1}=\left(\begin{array}[]{cc}0&i\\\
i&0\end{array}\right)$ and
$e^{\prime}_{1}=\left(\begin{array}[]{cc}0&-1_{\mathcal{V}}\\\
1_{\mathcal{V}}&0\end{array}\right)$ which gives a $\mathit{Cl}_{1,0}$-module
structure on $\mathbb{C}^{2}$ and $\mathcal{V}\oplus\mathcal{V}$,
respectively. Then, $C_{s}=\mathrm{diag}(B_{s},1,-1)$ gives a
$\mathbb{Z}_{2}$-map from $([-1,1],\zeta_{0})$ to
$(\mathrm{Fred}^{(1,\underline{1})}_{*},\mathfrak{q}^{\prime})$. The operator
$C_{1}$ is contained in the space of self-adjoint unitaries on
$\mathcal{V}^{\prime}$ that anticommutes with $e_{1}\oplus e_{1}^{\prime}$. As
in [9], this space of unitaries is contractible by Kuiper’s theorem. We embed
$[-1,1]$ into $\mathbb{T}$ by $s\mapsto\exp(\frac{\pi is}{2})$ and, as in
Example 5.18, extend $C$ onto $\mathbb{T}$ through this contractible space of
unitaries to obtain a $\mathbb{Z}_{2}$-map
$D\colon(\mathbb{T},\zeta)\to(\mathrm{Fred}^{(1,\underline{1})}_{*},\mathfrak{q}^{\prime})$.
For this example, we have
$\operatorname{\mathrm{sf}}^{\mathrm{DIII}}(D)=\operatorname{\mathrm{sf}}^{\mathrm{DIII}}(B)=1$,
$w_{+}(D)=1$ and $w_{-}(D)=0$. If we take $D^{\prime}$ as
$D^{\prime}_{s}=D_{-s}$, then $D^{\prime}$ is also such a $\mathbb{Z}_{2}$-map
and its invariants are
$\operatorname{\mathrm{sf}}^{\mathrm{DIII}}(D^{\prime})=1$,
$w_{+}(D^{\prime})=0$ and $w_{-}(D^{\prime})=1$.

In class $\mathrm{AII}$,
$H^{c}\colon(\mathbb{T},\zeta)\to(\mathrm{Fred}^{(0,\underline{1})}_{*},\mathfrak{q}_{\Theta})$
is a $\mathbb{Z}_{2}$-map and its $\mathbb{Z}_{2}$-valued spectral flow is
defined. We denote $\operatorname{\mathrm{sf}}^{\mathrm{AII}}$ for
$\operatorname{\mathrm{sf}}_{2}$.

In class $\mathrm{C}$, we have a $\mathbb{Z}_{2}$-map
$iH^{c}\colon(\mathbb{T},\zeta)\to(\mathrm{Fred}^{(\underline{1},0)}_{*},\mathfrak{q}_{\Xi})$.
Note that the $\mathbb{Z}_{2}$-homotopy classes
$[(\mathbb{T},\zeta),(\mathrm{Fred}^{(\underline{1},0)}_{*},\mathfrak{q}_{\Xi})]_{\mathbb{Z}_{2}}$
is isomorphic to $KO_{5}(C(\mathbb{T}),\tau_{\zeta})\cong\mathbb{Z}$. By
forgetting the $\mathbb{Z}_{2}$-actions and multiplying by $-i$, there is a
map from
$[(\mathbb{T},\zeta),(\mathrm{Fred}^{(\underline{1},0)}_{*},\mathfrak{q}_{\Xi})]_{\mathbb{Z}_{2}}$
to $[\mathbb{T},\mathrm{Fred}^{(0,\underline{1})}_{*}]$. Combined with the
spectral flow, we obtain a homomorphism

$\operatorname{\mathrm{sf}}^{\mathrm{C}}\colon[(\mathbb{T},\zeta),(\mathrm{Fred}^{(\underline{1},0)}_{*},\mathfrak{q}_{\Xi})]_{\mathbb{Z}_{2}}\to
2\mathbb{Z},\ \ [A]\mapsto\operatorname{\mathrm{sf}}(-iA).$

Note that image the image of $\operatorname{\mathrm{sf}}^{\mathrm{C}}$ are
even integers since each eigenspace corresponding to the crossing points of
the spectrum of $-iA_{t}$ with zero has a quaternionic vector space structure
given by $\Xi$.

######  Example 5.21.

For $s\in[-1,1]$, let $B^{\prime\prime}_{s}=\mathrm{diag}(is,is)$, and let $j$
be the quaternionic structure in Example 5.18. Then,
$B^{\prime\prime}\colon([-1,1],\zeta)\to(M_{2}(\mathbb{C}),\operatorname{\mathrm{Ad}}_{j})$
is a $\mathbb{Z}_{2}$-map, and we have
$\operatorname{\mathrm{sf}}^{\mathrm{C}}(B^{\prime\prime})=\operatorname{\mathrm{sf}}(-iB^{\prime\prime})=2$.

######  Lemma 5.22.

  1. (1)

$\operatorname{\mathrm{sf}}^{\mathrm{AII}}\colon[(\mathbb{T},\zeta),(\mathrm{Fred}^{(0,\underline{1})}_{*},\mathfrak{q})]_{\mathbb{Z}_{2}}\to\mathbb{Z}_{2}$
is an isomorphism.

  2. (2)

$\operatorname{\mathrm{sf}}^{\mathrm{D}}\colon[(\mathbb{T},\zeta),(\mathrm{Fred}^{(\underline{1},0)}_{*},\mathfrak{r})]_{\mathbb{Z}_{2}}\to\mathbb{Z}$
is surjective.

  3. (3)

$\operatorname{\mathrm{sf}}^{\mathrm{DIII}},\ w_{+},\
w_{-}\colon[(\mathbb{T},\zeta),(\mathrm{Fred}^{(1,\underline{1})}_{*},\mathfrak{q})]_{\mathbb{Z}_{2}}\to\mathbb{Z}_{2}$
are surjective

  4. (4)

$\operatorname{\mathrm{sf}}^{\mathrm{C}}\colon[(\mathbb{T},\zeta),(\mathrm{Fred}^{(\underline{1},0)}_{*},\mathfrak{q})]_{\mathbb{Z}_{2}}\to
2\mathbb{Z}$ is an isomorphism.

###### Proof.

It is sufficient to find examples of $\mathbb{Z}_{2}$-maps which maps to
generators of $\mathbb{Z}$, $\mathbb{Z}_{2}$ and $2\mathbb{Z}$. Therefore, (1)
and (3) follows from Example 5.18 and Example 5.20. For (2) and (4), we can
construct such examples from Example 5.19 and Example 5.21, as in Example
5.18. ∎

In class DIII cases, we have three surjections
$\operatorname{\mathrm{sf}}^{\mathrm{DIII}}$, $w_{+}$ and $w_{-}$ from
$\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$ to $\mathbb{Z}_{2}$. There is the
following relation between them.

######  Lemma 5.23.

$\operatorname{\mathrm{sf}}^{\mathrm{DIII}}=w_{+}+w_{-}$.

###### Proof.

Let $D$ and $D^{\prime}$ be $\mathbb{Z}_{2}$-maps in Example 5.20. Let
$D^{\prime\prime}=D\oplus D^{\prime}$, then we have
$\operatorname{\mathrm{sf}}^{\mathrm{DIII}}(D^{\prime\prime})=0$,
$w_{+}(D^{\prime\prime})=1$ and $w_{-}(D^{\prime\prime})=1$. Invariants
$\operatorname{\mathrm{sf}}^{\mathrm{DIII}}$, $w_{-}$ and $w_{+}$ for $D$,
$D^{\prime}$ and $D^{\prime\prime}$ tell that non-trivial three elements in
the group
$[(\mathbb{T},\zeta),(\mathrm{Fred}^{(1,\underline{1})}_{*},\mathfrak{q})]_{\mathbb{Z}_{2}}$
consists of classes of $D$, $D^{\prime}$ and $D^{\prime\prime}$. Therefore, we
computed three maps $\operatorname{\mathrm{sf}}^{\mathrm{DIII}}$, $w_{-}$ and
$w_{+}$, from which the result follows. ∎

######  Remark 5.24.

For our class $\mathrm{DIII}$ systems, $\mathbb{Z}_{2}$-valued spectral flow
counts the strong invariant. This corresponds to one direct summand of
$\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$, while the other corresponds to a weak
invariant. When $w_{+}\neq w_{-}$, the strong invariant is nonzero. When
$w_{+}=w_{-}=1$, the strong invariant is zero and the weak invariant is
nonzero. When $w_{+}=w_{-}=0$, both of them are zero.

######  Definition 5.25.

For $n$-D systems with codimension $n-1$ corners in classes $\mathrm{D}$,
$\mathrm{DIII}$, $\mathrm{AII}$ and $\mathrm{C}$, we define the numerical
corner invariant as follows.

  * •

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

  * •

In class $\mathrm{DIII}$, let