††thanks: Present address: Interdisciplinary Theoretical and Mathematical
Sciences Program (iTHEMS), RIKEN, Wako 351-0198, Japan

# Atiyah-Hirzebruch spectral sequence for topological insulators and
superconductors:
$E_{2}$ pages for 1651 magnetic space groups

Ken Shiozaki  Center for Gravitational Physics and Quantum Information, Yukawa
Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
Seishiro Ono  Department of Applied Physics, University of Tokyo, Tokyo
113-8656, Japan

###### Abstract

We compute the $E_{2}$ pages of the momentum-space and real-space Atiyah-
Hirzebruch spectral sequence (AHSS) for topological crystalline insulators and
superconductors up to three spatial dimensions, considering the cell
decomposition in which if a group action fixes a cell setwise then its group
action fixes the same cell pointwise. We provide a detailed description of the
implementation for computing the $E_{2}$ pages of AHSS. Under a physically
reasonable assumption, we enumerate all possible $K$-groups that are
compatible with the $E_{2}$ pages for both momentum and real-space AHSS. As a
result, we determine the $K$-groups for approximately 59% of symmetry settings
in three spatial dimensions. All the results can be found at this http URL.

††preprint: YITP-23-17††preprint: RIKEN-iTHEMS-Report-23

###### Contents

  1. I Introduction
  2. II Symmetry and factor system
    1. II.1 MSG
    2. II.2 Factor system
    3. II.3 Symmetry action in momentum space
    4. II.4 Symmetry of gap function
    5. II.5 $SU(2)$ symmetry and class C
  3. III Preliminary for AHSS in general
    1. III.1 Symmetry in one-particle Hilbert space
      1. III.1.1 TIs
      2. III.1.2 SCs
    2. III.2 Cell decomposition
    3. III.3 Chiral symmetry and winding number
  4. IV Momentum-space AHSS
    1. IV.1 Preliminary
    2. IV.2 $E_{1}$ page
      1. IV.2.1 $\mathbb{Z}$ classification
      2. IV.2.2 $\mathbb{Z}_{2}$ classification
    3. IV.3 The first differential $d_{1}$
  5. V Real-space AHSS
    1. V.1 Preliminary
    2. V.2 $E^{1}$ page
      1. V.2.1 $\mathbb{Z}$ classification
      2. V.2.2 $\mathbb{Z}_{2}$ classification
    3. V.3 The first differential $d^{1}$
  6. VI Constraint on $K$-group by $E_{2}$-pages
    1. VI.1 Higher differential
    2. VI.2 Extension
    3. VI.3 Candidates of $K$-groups
  7. VII Other symmetry settings
  8. VIII Conclusion
  9. A An algorithm computing cell decomposition
    1. A.1 Fundamental domain
    2. A.2 Cell decomposition of the boundary of $DV(a)$
  10. B Computing irreducible characters
  11. C Computation in $\mathbb{Z}$-module
    1. C.1 ${\rm Im\,}f$
    2. C.2 ${\rm Ker\,}f$
    3. C.3 ${\rm Ker\,}g/{\rm Im\,}f$
  12. D Baer Sum

##  I Introduction

Topological crystalline insulators and superconductors are topological phases
of electronic systems protected by crystalline symmetries [1, 2, 3, 4, 5, 6,
7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. Classification of
topological crystalline phases involves listing possible higher-order
topological phases that exhibit surface, hinge, and corner states [22, 23, 24,
25, 26, 27, 28, 29, 30, 31, 32, 33]. Nowadays, it is well-known that this task
is achieved by enumerating configurations of lower-dimensional topological
phases protected by internal symmetries in real space [34, 31, 32, 28, 35].
While topological phases are generally defined in quantum many-body systems,
free fermionic systems with translation symmetry have a unique feature: their
single-particle nature allows us to use an alternative approach based on the
topology of band structures in momentum space, which is dual to the real-space
description.

A practical framework as a computational method for exhaustive classification
is provided by $K$-theory [2, 36, 37, 38, 39]. More precisely, the
classification in momentum space is described by $K$-cohomology [40, 41],
while the classification in real space is described by $K$-homology [42, 43,
44], and these $K$-groups are isomorphic to each other for topological
crystalline insulators and superconductors [45]. In the $K$-theory
classification of topological insulators and superconductors, two gapped
Hamiltonians $H_{0}$ and $H_{1}$ defined on a common set of atomic orbitals
are regarded as in the same topological phase if there is a gapped Hamiltonian
$H^{\prime}$ such that there is a path from $H_{0}\oplus H^{\prime}$ to
$H_{1}\oplus H^{\prime}$ without closing a gap. This equivalence condition is
called the stable equivalence. In the $K$-theory, the classification is given
as a $\mathbb{Z}$-module called the $K$-group. A set of pair $(H_{0},H_{1})$
represents an element of $K$-group, and we denote the equivalence class by
$[H_{0},H_{1}]$. If $[H_{0},H_{1}]\neq 0$ as an element of $K$-group, there is
no adiabatic paths between $H_{0}$ and $H_{1}$. Note that the inverse is not
true in general: Two Hamiltonians $H_{0}$ and $H_{1}$ can be stably equivalent
even if there is no adiabatic path between them.

Although the weakening of the identity condition by stable equivalence gives a
slightly coarser classification, $K$-theory has the technical advantage of
being computationally feasible. The $K$-theory is a generalized (co)homology
theory, meaning that one can apply various tools of the generalized
(co)homology theory to compute the $K$-group we are interested in. For
example, the Mayer-Vietoris sequence gives us a long exact sequence for a
decomposition of momentum/real space $X=U\cup V$, where the $K$-group over $X$
is constrained by the $K$-groups over more small spaces $U$ and $V$ [38]. A
systematic framework of this kind of bottom-up approach is the Atiyah-
Hirzebruch spectral sequence (AHSS) [46], which was introduced in [14] for the
band theory and in [28] for real-space classification. See also [35, 31, 32]
for real-space approaches. In the AHSS, $E_{1}$-page, $E_{2}$-page,
$E_{3}$-page,… are computed sequentially, and the convergent $E_{\infty}$-page
approximates the $K$-group. For three-dimensional systems, $E_{4}$-page is the
$E_{\infty}$-page. While it is currently unknown how to systematically compute
higher-order pages, such as $E_{3}$ and $E_{4}$, except for particular
symmetry classes [21], the computation of $E_{2}$-page is easier than that of
higher-order pages.

In this work, based on the physical picture and mathematical structure of the
AHSS discussed in [14] and [28], we propose an efficient and systematic
computation of $E_{2}$-pages for a certain class of decomposition of space,
and we present computed $E_{2}$-page for free fermionic insulators and
superconductors in one, two, and three dimensions. The symmetry settings we
compute are 1651 magnetic space groups (MSGs), 528 magnetic layer groups, and
393 magnetic rod groups. For superconductors, all one-dimensional
representations of pairing symmetry are considered. All the results can be
found at the following http URL.

Furthermore, we discuss a technique to find candidate $K$-groups from the
$E_{2}$-page in the momentum-space AHSS and the real-space AHSS. Although it
is generally difficult to obtain $E_{3}$\- and $E_{4}$-pages, we can tabulate
all the possible $E_{3}$\- and $E_{4}$\- pages by considering all possible
higher differentials. For each candidate of $E_{4}$-page, we can also tabulate
all the possible $K$-groups compatible with the $E_{4}$-page. Importantly, the
two sets of candidate $K$-groups are obtained from the momentum-space and
real-space AHSSs for a symmetry setting, and the true $K$-group lies in the
intersection of these two sets. These facts give us a strong constraint on
possible $K$-groups. As a result, the number of candidate $K$-groups for each
symmetry setting is limited and countable. Surprisingly, the $K$-groups are
determined for about 59% of symmetry settings we consider in three dimensions.

The organization of this paper is as follows. In Sec. II, we summarize the
crystal symmetries, factor systems for electronic fermions, pairing symmetries
in superconductors, and algebraic relations of symmetry actions targeted in
this paper. In Sec. III, we describe common preliminary matters in momentum-
space and real-space AHSS, particularly a class of cell decomposition used in
this paper and the implementation of winding numbers for each irrep. In Sec.
IV and Sec. V, we elaborate on the calculation details of the $E_{2}$ pages in
the momentum-space AHSS and real-space AHSS, respectively. In Sec. VI, we
summarize the calculation technique for imposing constraints on the possible
$K$-groups from the $E_{2}$ pages obtained by the momentum-space and real-
space AHSS. In Sec. VII, we comment on the symmetry settings other than the
MSGs calculated in this paper. We provide the conclusion in Sec. VIII. Several
computational details are summarized in Appendices of the four sections.

##  II Symmetry and factor system

In this paper, we compute AHSSs for symmetry groups that are either the MSGs
or the combination of MSGs and Particle-Hole Symmetry (PHS). In this section,
we summarize these symmetry groups and the factor system in momentum space.

We introduce an abbreviation to specify electronic insulators or
superconductors for the physical system under consideration. The electronic
insulating system is abbreviated to TI (topological insulator), and the
electric superconducting system to SC (superconductor).

###  II.1 MSG

Let ${\cal G}$ be an MSG. We denote the lattice translation group and magnetic
point group by $\Pi$ and $G={\cal G}/\Pi$, respectively. An element of ${\cal
G}$ is specified by the Seitz symbol and the homomorphism $\phi:{\cal
G}\to\mathbb{Z}_{2}=\\{\pm 1\\}$, which are introduced below. The group ${\cal
G}$ acts on the three-dimensional Euclidean space $\mathbb{E}^{3}$ as
$g(\bm{x})=p_{g}\bm{x}+\bm{t}_{g}$ for $\bm{x}\in\mathbb{E}^{3}$, where
$p_{g}\in O(3)$ is a three-dimensional rotation matrix and
$\bm{t}_{g}\in\mathbb{R}^{3}$ is a vector of (fractional) translation. We
employ the Seitz symbol $g=\\{p_{g}|\bm{t}_{g}\\}$ to specify an element
$g\in{\cal G}$. (It should be noted that $\\{p_{g}|\bm{t}_{g}\\}$ is, in fact,
a representation of the group ${\cal G}$.) They satisfy $p_{g}p_{h}=p_{gh}$
and $\bm{t}_{gh}=p_{g}\bm{t}_{h}+\bm{t}_{g}$ due to the group structure of
${\cal G}$. The homomorphism $\phi$ specifies whether $g\in{\cal G}$ acts
unitarily or antiunitarily on the one-particle Hilbert space. The symmetry
group ${\cal G}$ acts as a projective representation on the one-particle
Hilbert space: Let $u_{g}$ be representation matrices for $g\in{\cal G}$.
These matrices satisfy the group law, except for a $U(1)$ phase, as in

$\displaystyle u_{g}u_{h}^{\phi_{g}}=z_{g,h}u_{gh},\quad g,h\in{\cal G}.$ (1)

Here, we introduced a short-hand notation for matrices $A$ and the sign
$\phi_{g}$ so that $A^{\phi_{g}}=A$ for $\phi_{g}=1$ and $A^{\phi_{g}}=A^{*}$
(complex conjugate of $A$) for $\phi_{g}=-1$. The $U(1)$ phases $z_{g,h}$ are
called the factor system. We simply assume that the factor system is
independent of lattice translations. In other words,
$z_{g,h}=z_{g\tau,h\tau^{\prime}}$ holds for any lattice translations
$\tau,\tau^{\prime}\in\Pi$. (This is not the case for the so-called magnetic
translation symmetry.)

We introduce the Bravais lattice

$\displaystyle L:=\\{\tau(\bm{0})\in\mathbb{R}^{3}|\tau\in\Pi\\}$ (2)

as the orbit of the lattice translation group for a fixed center $\bm{0}$ of a
unit cell, and we specify a unit cell by $\bm{R}\in L$. To specify an MSG, it
is useful to employ a complete set of left coset representatives of $\Pi$ in
${\cal G}$, which can be written using the Seitz notation as
$\\{p_{g}|\bm{a}_{g}\\}$ for elements $g\in G$. From the group structure of
${\cal G}$, the vector

$\displaystyle p_{g}\bm{a}_{h}+\bm{a}_{g}-\bm{a}_{gh}\in L$ (3)

is in the Bravais lattice. All the data for $\phi,p_{g}$, and $\bm{a}_{g}$ of
the MSGs are available at [47], and we utilized them.

###  II.2 Factor system

Although the set of inequivalent factor systems $z_{g,h}$ is classified by the
group cohomology $H^{2}(G,U(1)_{\phi})$, where $U(1)_{\phi}$ is the group
$U(1)$ with the left $G$-action defined as $g.z=z^{\phi_{g}}$ for $z\in U(1)$,
in this paper, we restrict our scope to factor systems that are realized in
realistic electronic systems. For electrons with integer spin, $z_{g,h}=z^{\rm
sp}_{g,h}\equiv 1$. For electrons with half-integer spin, the factor system is
constructed as follows. It is enough to derive the factor system for spin 1/2.
The rotation matrix $p_{g}\in O(3)$ for $g\in\mathcal{G}$ is written as
$p_{g}=R_{\hat{n}_{g},\theta_{g}}I^{\frac{1-\det p_{g}}{2}}$, where
$R_{\hat{n}_{g},\theta_{g}}\in SO(3)$ is a rotation matrix along the
$\hat{n}_{g}$-axis by the $\theta_{g}$ angle and $I$ is the space inversion.
The space inversion trivially acts on the electrons, meaning that the space
inversion $I$ does not produce a nontrivial factor system. While the space
rotation and the time-reversal symmetry (TRS) act on the electron wave
function by elements of $Spin(3)=SU(2)$ group as

$\displaystyle
u_{g}=\begin{cases}e^{-i\theta_{g}\hat{n}_{g}\cdot\bm{\sigma}/2}&\phi_{g}=1,\\\
e^{-i\theta_{g}\hat{n}_{g}\cdot\bm{\sigma}/2}(i\sigma_{y})&\phi_{g}=-1.\\\
\end{cases}$ (4)

Here, $\bm{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$ are the Pauli matrices.
Using explicit representation matrices above, we have the factor system
$z^{\rm sp}_{g,h}$ by the defining relation $u_{g}u_{h}^{\phi_{g}}=z^{\rm
sp}_{g,h}u_{gh}$. As a result, the factor systems $z^{\rm sp}_{g,h}$ take
values in the sign $\pm 1$.

###  II.3 Symmetry action in momentum space

Let $\hat{c}_{\sigma}(\bm{R}+\delta\bm{r})$ and
$\hat{c}^{\dagger}_{\sigma}(\bm{R}+\delta\bm{r})$ be annihilation and creation
operators of a complex fermion localized at the position
$\bm{R}+\delta\bm{r}$. Here, $\bm{R}\in L$ represents the center position of
unit cells, $\delta{\bm{r}}$ is the displacement vector from the unit cell
center for the fermions, and the subscript $\sigma$ is the index for the
internal degrees of freedom like spin and orbital at
${\bm{R}}+\delta{\bm{r}}$. Let ${\cal G}$ be an MSG. An element $g\in{\cal G}$
acts on the fermions as

$\displaystyle\hat{g}\hat{c}^{\dagger}_{\sigma}(\bm{R}+\delta\bm{r})\hat{g}^{-1}$
$\displaystyle=\hat{c}^{\dagger}_{\sigma^{\prime}}(p_{g}(\bm{R}+\delta\bm{r})+\bm{t}_{g})[D_{g}]_{\sigma^{\prime}\sigma},$
(5)

Here, $\hat{g}$ is unitary when $\phi_{g}=1$, and antiunitary when
$\phi_{g}=-1$, and $D_{g}$ matrices are a set of unitary matrices satisfying

$\displaystyle D_{g}D_{h}^{\phi_{g}}=z_{g,h}D_{gh}$ (6)

with the factor system $z_{g,h}$ introduced before and does not depend on
lattice translations.

In this paper, we introduce the fermion annihilation and creation operators in
momentum space so that they are periodic by reciprocal lattice vectors.
Namely,

$\displaystyle\hat{c}^{\dagger}_{\delta{\bm{r}},\sigma}({\bm{k}})=\sum_{{\bm{R}}\in\Pi}\hat{c}^{\dagger}_{\sigma}(\bm{R}+\delta{\bm{r}})e^{i{\bm{k}}\cdot{\bm{R}}}.$
(7)

(A reciprocal lattice vector is a vector in the dual lattice $\hat{L}:=\\{{\bf
G}\in\mathbb{R}^{3}|e^{i\bm{{\bf G}}\cdot\bm{R}}=1{\rm\ for\ all\ }\bm{R}\in
L\\}$.) This definition does not reflect the spatial position of the degrees
of freedom, which requires caution when calculating physical quantities, etc.
However, since it does not affect the classification of topological phases, we
adopt this definition in this paper. Let us introduce a permutation matrix

$\displaystyle[P_{g}]_{\delta{\bm{r}}^{\prime},\delta{\bm{r}}}=\left\\{\begin{array}[]{ll}1&\delta{\bm{r}}^{\prime}\equiv
p_{g}\delta{\bm{r}}+\bm{t}_{g}{\rm\ mod\ }L,\\\ 0&{\rm
else},\end{array}\right.$ (10)

for $g\in{\cal G}$, we have for $g\in{\cal G}$,

$\displaystyle\hat{g}\hat{c}^{\dagger}_{i}({\bm{k}})\hat{g}^{-1}=\sum_{i^{\prime}}\hat{c}^{\dagger}_{i^{\prime}}(\phi_{g}p_{g}{\bm{k}})[u_{g}({\bm{k}})]_{i^{\prime}i}$
(11)

with

$\displaystyle[u_{g}({\bm{k}})]_{i^{\prime}i}$
$\displaystyle=[P_{g}]_{\delta{\bm{r}}^{\prime}\delta{\bm{r}}}[D_{g}]_{\sigma^{\prime}\sigma}e^{-i\phi_{g}p_{g}{\bm{k}}\cdot(p_{g}\delta{\bm{r}}+\bm{t}_{g}-\delta{\bm{r}}^{\prime})},$
(12)

where the indices $\delta{\bm{r}}$ and $\sigma$ were merged as a single index
$i=(\delta{\bm{r}},\sigma)$. In particular,
$[u_{\tau}({\bm{k}})]_{i^{\prime}i}=\delta_{i^{\prime}i}e^{-i{\bm{k}}\cdot\bm{t}}$
for lattice translations $\tau=\\{1|\bm{t}\\}\in\Pi$. Note that
$u_{g}({\bm{k}})$ is periodic $u_{g}({\bm{k}}+{\bf G})=u_{g}({\bm{k}})$ for
reciprocal lattice vectors ${\bf G}$. It is straightforward to show that

$\displaystyle
u_{g}(\phi_{g}p_{g}{\bm{k}})u_{h}({\bm{k}})^{\phi_{g}}=z_{g,h}u_{gh}({\bm{k}})$
(13)

for $g,h\in{\cal G}$.

A free fermion Hamiltonian expressed in the momentum space is

$\displaystyle\hat{H}=\sum_{{\bm{k}},ij}c^{\dagger}_{i}({\bm{k}})[h({\bm{k}})]_{ij}c_{j}({\bm{k}}).$
(14)

The matrix $h({\bm{k}})$ is also called a Hamiltonian. With the definition
(7), $h({\bm{k}})$ is also periodic $h({\bm{k}}+\bm{G})=h({\bm{k}})$. For free
fermions, the MSG symmetry of the Hamiltonian $\hat{H}$ is that the matrix
$h({\bm{k}})$ satisfies

$\displaystyle
u_{g}({\bm{k}})h({\bm{k}})^{\phi_{g}}u_{g}({\bm{k}})^{-1}=h(\phi_{g}p_{g}{\bm{k}})$
(15)

for $g\in{\cal G}$. Note that the lattice translation symmetry is fulfilled as
it is written in momentum space. Only the constraint conditions coming from
the magnetic point group $G$ are meaningful.

###  II.4 Symmetry of gap function

In superconductors, the mean-field Hamiltonian in momentum space is written as

$\displaystyle\hat{H}_{\rm MF}$
$\displaystyle=\sum_{{\bm{k}},ij}c^{\dagger}_{i}({\bm{k}})[h({\bm{k}})]_{ij}c_{j}({\bm{k}})$
$\displaystyle+\frac{1}{2}\sum_{{\bm{k}},ij}(c^{\dagger}_{i}({\bm{k}})[\Delta({\bm{k}})]_{ij}c^{\dagger}_{j}(-{\bm{k}})+{\rm
h.c.}).$ (16)

The matrix $\Delta({\bm{k}})$ is the gap function and satisfies

$\displaystyle\Delta({\bm{k}})^{T}=-\Delta(-{\bm{k}})$ (17)

due to the anticommutation relation of fermion operators. The gap function
$\Delta({\bm{k}})$ is supposed to be a vector
$\Delta({\bm{k}})=\sum_{a=1}^{{\rm
dim\,}\rho}\eta_{a}\Delta_{a}({\bm{k}}),\eta_{a}\in\mathbb{C},$ of some basis
functions $\\{\Delta_{a}({\bm{k}})\\}_{a=1}^{{\rm dim\,}\rho}$ satisfying

$\displaystyle
u_{g}({\bm{k}})\Delta_{a}({\bm{k}})^{\phi_{g}}u_{g}(-{\bm{k}})^{T}$
$\displaystyle=\sum_{b=1}^{{\rm
dim\,}\rho}\Delta_{b}(\phi_{g}p_{g}{\bm{k}})[D^{\rho}_{g}]_{ba}$ (18)

for $g\in{\cal G}$. We assume $D^{\rho}_{g}$ is independent of ${\bm{k}}$ and
lattice translations, meaning that $\rho$ is a representation of the magnetic
point group $G$. From (18), the factor system for $D^{\rho}_{g}$ is
$(z_{g,h})^{2}$ and thus trivial for spinless and spinful electrons. Since
different irreps do not coexist as a solution of the gap equation in general,
$\rho$ is an irrep of the magnetic point group $G$. When $\rho$ is not the
trivial irrep of $G$, meaning that $D^{\rho}_{g}\neq 1$ for some $g\in{\cal
G}$, the gap function $\Delta({\bm{k}})$ breaks the original MSG symmetry
defined by (5). However, when $\rho$ is a one-dimensional irrep of $G$, one
can recover the MSG symmetry using $U(1)$ phase rotation $\hat{\sf
u}(e^{i\theta})c^{\dagger}_{i}\hat{\sf
u}(e^{i\theta})^{-1}=c^{\dagger}_{i}e^{i\theta}$ of complex fermions. Let us
write $D^{\rho}_{g}=\xi_{g}\in U(1)$ for a one-dimensional irrep. For each
$\xi_{g}$, we pick a sign of the square root of $\xi_{g}$ and denote it by
$\xi_{g}^{\frac{1}{2}}$. Then, the combined transformation $\hat{g}\hat{\sf
u}((\xi_{g}^{\frac{1}{2}})^{*})$ becomes symmetry of the Hamiltonian
$\hat{H}_{\rm MF}$.

The mean-field Hamiltonian $\hat{H}_{\rm MF}$ can be written as

$\displaystyle\hat{H}_{\rm MF}$
$\displaystyle=\frac{1}{2}\sum_{{\bm{k}},ij}(c^{\dagger}_{i}({\bm{k}}),c_{i}(-{\bm{k}}))$
$\displaystyle[H_{\rm BdG}({\bm{k}})]_{ij}\begin{pmatrix}c_{j}({\bm{k}})\\\
c^{\dagger}_{j}(-{\bm{k}})\\\ \end{pmatrix},$ $\displaystyle H_{\rm
BdG}({\bm{k}})=\begin{pmatrix}h({\bm{k}})&\Delta({\bm{k}})\\\
\Delta({\bm{k}})^{\dagger}&-h(-{\bm{k}})^{T}\\\ \end{pmatrix}.$ (19)

The two-component spinor
$\Psi_{i}({\bm{k}})=(c_{i}({\bm{k}}),c^{\dagger}_{i}(-{\bm{k}}))^{T}$ and the
matrix $H_{\rm BdG}({\bm{k}})$ are called the Nambu spinor and the Bogoliubov-
de Gennes (BdG) Hamiltonian, respectively. The relation (17) implies that the
BdG Hamiltonian $H_{\rm BdG}({\bm{k}})$ satisfies the following particle-hole
“symmetry”

$\displaystyle U_{c}H_{\rm BdG}({\bm{k}})^{*}U_{c}^{-1}=-H_{\rm
BdG}(-{\bm{k}}),$ $\displaystyle U_{c}=\begin{pmatrix}&1\\\ 1\\\
\end{pmatrix}.$ (20)

Therefore, the total symmetry group for the BdG Hamiltonian becomes ${\cal
G}\times\mathbb{Z}_{2}^{C}$ with $\mathbb{Z}_{2}^{C}$ generated by PHS.
Because $U_{C}U_{C}^{*}=1$ we call the PHS in the form (20) the class D PHS in
the Altland-Zirnbauer (AZ symmetry class [48]. The one-dimensional irrep
$\xi_{g}$ is encoded in the factor system of ${\cal
G}\times\mathbb{Z}_{2}^{C}$. On the Nambu spinor, the combined symmetry is

$\displaystyle\hat{g}\hat{\sf
u}((\xi_{g}^{\frac{1}{2}})^{*})\Psi^{\dagger}({\bm{k}})(\hat{g}\hat{\sf
u}((\xi_{g}^{\frac{1}{2}})^{*}))^{-1}$
$\displaystyle=\Psi^{\dagger}(\phi_{g}p_{g}{\bm{k}})U_{g}({\bm{k}})$ (21)

with

$\displaystyle
U_{g}({\bm{k}})=\begin{pmatrix}u_{g}({\bm{k}})(\xi_{g}^{\frac{1}{2}})^{*}\\\
&u_{g}(-{\bm{k}})^{*}\xi_{g}^{\frac{1}{2}}\\\ \end{pmatrix}$ (22)

for $g\in{\cal G}$. We find that

$\displaystyle
U_{g}(\phi_{h}p_{h}{\bm{k}})U_{h}({\bm{k}})^{\phi_{g}}=z_{g,h}z_{g,h}^{\xi}U_{gh}({\bm{k}}),$
(23) $\displaystyle
U_{c}U_{g}({\bm{k}})^{*}=U_{g}(-{\bm{k}})U_{C}^{\phi_{g}},$ (24)

with

$\displaystyle
z^{\xi}_{g,h}:=\xi_{g}^{\frac{1}{2}}(\xi_{h}^{\frac{1}{2}})^{\phi_{g}}(\xi_{gh}^{\frac{1}{2}})^{-1}\in\\{\pm
1\\}.$ (25)

In (23) we have used that $z_{g,h}$ is a sign so that $z_{g,h}^{*}=z_{g,h}$.

Alternatively, the following phase choice, which is meaningful only for BdG
Hamiltonian, is useful.

$\displaystyle U^{\prime}_{g}({\bm{k}})=\begin{pmatrix}u_{g}({\bm{k}})\\\
&u_{g}(-{\bm{k}})^{*}\xi_{g}\\\ \end{pmatrix}.$ (26)

With this,

$\displaystyle
U^{\prime}_{g}(\phi_{h}p_{h}{\bm{k}})U^{\prime}_{h}({\bm{k}})^{\phi_{g}}=z_{g,h}U^{\prime}_{gh}({\bm{k}}),$
(27) $\displaystyle
U^{\prime}_{g}(-{\bm{k}})U_{C}^{\phi_{g}}=\xi_{g}U_{C}U^{\prime}_{g}({\bm{k}})^{*}.$
(28)

###  II.5 $SU(2)$ symmetry and class C

Consider the cases in the presence of full $SU(2)$ internal symmetry of normal
state $h({\bm{k}})$ for either the spin-1/2 or a pseudo-spin-1/2 internal
degree of freedom. We denote the Pauli matrices for the (pseudo) spin-1/2
degrees of freedom by $\bm{\sigma}$. The normal part is written as
$h({\bm{k}})=\tilde{h}({\bm{k}})\otimes\sigma_{0}$. When the gap function also
preservers $SU(2)$ symmetry, the gap function is in the form

$\displaystyle\Delta({\bm{k}})=\tilde{\Delta}({\bm{k}})\otimes(i\sigma_{y}).$
(29)

The relation (17) means that

$\displaystyle\tilde{\Delta}({\bm{k}})^{T}=\tilde{\Delta}(-{\bm{k}}).$ (30)

On the basis of Nambu spinor
$\Psi^{\prime}_{i}({\bm{k}})=(c_{i}({\bm{k}}),(i\sigma_{y})c_{i}^{\dagger}({\bm{k}}))^{T}$,
the BdG hamiltonian is

$\displaystyle H_{\rm BdG}({\bm{k}})=\tilde{H}_{\rm
BdG}({\bm{k}})\otimes\sigma_{0},$ (31) $\displaystyle\tilde{H}_{\rm
BdG}({\bm{k}})=\begin{pmatrix}\tilde{h}({\bm{k}})&\tilde{\Delta}({\bm{k}})\\\
\tilde{\Delta}({\bm{k}})^{\dagger}&-\tilde{h}(-{\bm{k}})^{T}.\end{pmatrix}$
(32)

The relation (30) implies that the BdG Hamiltonian $\tilde{H}({\bm{k}})$
satisfies the following PHS

$\displaystyle\tilde{U}_{C}\tilde{H}_{\rm
BdG}({\bm{k}})^{*}\tilde{U}_{C}^{-1}=-\tilde{H}_{\rm BdG}(-{\bm{k}}),$ (33)
$\displaystyle\tilde{U}_{C}=\begin{pmatrix}&1\\\ -1\\\
\end{pmatrix},\quad\tilde{U}_{C}\tilde{U}_{C}^{*}=-1.$ (34)

This is the class C PHS [1].

If the $SU(2)$ symmetry in the electron system comes from the electron spin,
the BdG Hamiltonian $\tilde{H}_{\rm BdG}({\bm{k}})$ satisfies the factor
system for spinless electron systems. On the other hand, if the $SU(2)$
symmetry in the electron system comes from the pseudo-spin originating from an
orbital degree of freedom, the BdG Hamiltonian $H_{\rm BdG}({\bm{k}})$
satisfies the factor system for spinful electron systems. We call the former
cases the class C spinful SC and the latter class C spinless SC, respectively.

##  III Preliminary for AHSS in general

We describe common preliminaries in momentum-space and real-space AHSS,
including a class of cell decomposition used in this paper and the definition
of the symmetry-resolved winding number.

###  III.1 Symmetry in one-particle Hilbert space

Although this paper focuses only on symmetries and factor systems in electron
systems, we summarize here the more general symmetry classes [36].

Let ${\cal G}$ be a discrete group that fits into the short exact sequence

$\displaystyle\Pi\to{\cal G}\to G$ (35)

with $\Pi\cong\mathbb{Z}^{d}$ being the translational group in $d$-space
dimensions. A symmetry class is characterized by ${\cal G}$ with the quintet
$(p_{g},\bm{t}_{g},\phi_{g},c_{g},z^{\rm int}_{g,h})$ for $g,h\in{\cal G}$,
explained below. The matrix $p_{g}$ is an $O(d)$ matrix, and
$\bm{t}_{g}\in\mathbb{R}^{d}$ is a translation vector, meaning that $g\in{\cal
G}$ acts on the real space as $\bm{x}\mapsto
g(\bm{x})=p_{g}\bm{x}+\bm{t}_{g}$. We denote the one-particle Hilbert space on
a $d$-dimensional lattice by ${\cal H}$ and the symmetry action on ${\cal H}$
by $\hat{g}$. The homomorphism $\phi:{\cal G}\rightarrow\\{\pm 1\\}$ specifies
whether $\hat{g}$ is unitary or antiunitary on ${\cal H}$. We assume that
$\phi(\Pi)=\\{1\\}$, i.e., the translation group $\Pi$ is composed only of
unitary elements. The factor system $z^{\rm int}_{g,h}$ specifies how ${\cal
G}$ is represented projectively on the Hilbert space ${\cal H}$, such that
$\hat{g}\hat{h}=z^{\rm int}_{g,h}\widehat{gh}$. We assume that $z^{\rm
int}_{g,h}$ does not depend on translations in the sense that $z^{\rm
int}_{gt,gt^{\prime}}=z^{\rm int}_{g,h}$ for $\tau,\tau^{\prime}\in\Pi$,
meaning that $z$ is a two-cocycle $z^{\rm int}\in Z^{2}(G,U(1)_{\phi})$, where
$U(1)_{\phi}$ means that $g\in G$ acts on $U(1)$ as $g.z=z^{\phi_{g}}$ for
$z\in U(1)$. Let $\hat{H}$ be a Hamiltonian on ${\cal H}$. The homomorphism
$c:{\cal G}\to\\{\pm 1\\}$ specifies whether $\hat{g}$ commutes or
anticommutes with the Hamiltonian $\hat{H}$, i.e.,

$\displaystyle\hat{g}\hat{H}\hat{g}^{-1}=c_{g}\hat{H},\quad g\in{\cal G}.$
(36)

We also assume that $c(\Pi)=\\{1\\}$, i.e., lattice translations commute with
the Hamiltonian $\hat{H}$.

Now we summarize the symmetry classes discussed in this paper.

####  III.1.1 TIs

For TIs, a symmetry class is specified by an MSG ${\cal G}_{\rm MSG}$ and
whether it is spinless or spinful. The group ${\cal G}$ is an MSG ${\cal
G}_{\rm MSG}$ equipped with the data $p_{g},\bm{t}_{g},\phi_{g}$. For spinless
TIs, the factor system $z^{\rm int}_{g,h}$ is a trivial one, $z^{\rm
int}_{g,h}\equiv 1$. For spinful TIs, the factor system is that for spin-1/2
electrons, $z^{\rm int}_{g,h}=z^{\rm sp}_{g,h}\in\\{\pm 1\\}$, defined in Sec.
II.2.

####  III.1.2 SCs

For SCs, a symmetry class is specified by an MSG ${\cal G}_{\rm MSG}$, a one-
dimensional irrep $\xi$ of the magnetic point group $G_{\rm MPG}$ of ${\cal
G}_{\rm MSG}$, whether it is spinless or spinful, and the type of PHS, either
class D or class C. The total symmetry group ${\cal G}$ is the product ${\cal
G}={\cal G}_{\rm MSG}\times\mathbb{Z}_{2}^{C}$ with
$\mathbb{Z}_{2}^{C}=\\{e,C\\}$ being the group of PHS. PHS is an internal
symmetry, meaning that the PHS $C$ does not change the spatial position
$C(\bm{x})=\bm{x}$, i.e., $p_{C}={\bf 1}_{d}$ and $\bm{t}_{C}=\bm{0}$. As
discussed in Sec. II.4, PHS behaves as an antiunitary symmetry and
anticommutes with the BdG Hamiltonian, meaning that $\phi_{C}=-1$ and
$c_{C}=-1$. For generic elements $g\in{\cal G}$, $p_{g},\bm{t}_{g},\phi_{g},$
and $c_{g}$ are extended with the group structure.

For a given irrep $\xi$, let $z_{g,h}^{\xi}$ for $g,h\in{\cal G}_{\rm MSG}$ be
the factor system introduced in (25). Let $\pi_{1}:{\cal G}\to{\cal G}_{\rm
MSG}$ and $\pi_{2}:{\cal G}\to\mathbb{Z}_{2}^{C}$ be the projections onto
${\cal G}_{\rm MSG}$ and $\mathbb{Z}_{2}^{C}$, respectively. The factor system
can be summarized in the form

$\displaystyle z^{\rm int}_{g,h}=z^{\rm
sp}_{\pi_{1}(g),\pi_{1}(h)}z^{\xi}_{\pi_{1}(g),\pi_{1}(h)}z^{\rm
PHS}_{\pi_{2}(g),\pi_{2}(h)},$ $\displaystyle g,h\in{\cal G}.$ (37)

Here, $z^{\rm PHS}_{g,h}\equiv 1$ for class D, and

$\displaystyle z^{\rm PHS}_{g,h}=\left\\{\begin{array}[]{ll}-1&g=h=C,\\\
1&{\rm else},\end{array}\right.$ (40)

for class C.

###  III.2 Cell decomposition

In this section, we introduce a class of cell decompositions used for the AHSS
in this paper. Let $X$ be a space over which we want to compute the $K$-group.
The space $X$ is either the Brillouin zone (BZ) torus $T^{d}$ or the infinite
real space $\mathbb{R}^{d}$, where $d$ is the space dimension. Let $G$ be a
symmetry group acting on $X$. For $d=3$, the group $G$ is the MSG for the real
space, whereas $G$ is the magnetic point group for the momentum space. We
introduce a sequence of spaces

$\displaystyle\emptyset=X_{-1}\subset X_{0}\subset X_{1}\subset\cdots\subset
X_{d}=X$ (41)

such that each $X_{p}$ is obtained from $X_{p-1}$ by gluing $p$-cells
$D^{p}_{i}$, which are each homeomorphic to a $p$-dimensional disk $D^{p}$, to
$X_{p-1}$ along their boundary $(p-1)$-dimensional spheres $\partial
D^{p}_{i}\cong S^{p-1}$. Additionally, there is a symmetry constraint on the
$p$-cells: For each $g\in G$, $p$-cells $D^{p}_{i}$ are mapped to other
$p$-cells by $g$. In other words, for each $g\in G$, $g(D^{p}_{i})=D^{p}_{j}$
holds with some $j$. Each $X_{p}$ is called the $p$-skeleton. We refer to such
a decomposition of the space $X$ as a cell decomposition.

While the AHSS is defined for the above cell decomposition, we impose the
following additional condition on the cell decomposition in this paper:

– If $g\in G$ fixes the $p$-cell $D^{p}_{i}$ setwise, then $g$ fixes
$D^{p}_{i}$ pointwise. Namely, if $g(D^{p}_{i})=D^{p}_{i}$, then $g(x)=x$ for
all $x\in D^{p}_{i}$.

Note that even with the additional constraint, the cell decomposition is not
unique. Nevertheless, the $E_{2}$-page is known to be unique.

In addition, we assign an orientation to each $p$-cell in such a way as to
satisfy the symmetry. All orientations of 0-cells are fixed to be positive.
Orientation is not necessary for the AHSS in general, but it is required to
construct the first differential $d_{1}$, which will be developed later.

In Appendix A, we present an algorithm for computing cell decomposition such
that the condition above is satisfied and the cell of maximum dimension
($d$-cell) is a convex fundamental domain. For instance, FIGs. 1 and 2 show
fundamental domains in the momentum and real spaces for MSG
$P2_{1}1^{\prime}$.

###  III.3 Chiral symmetry and winding number

In odd spatial dimensions, one can define the winding number in the presence
of chiral symmetry. Let $G=G_{0}\coprod\gamma G_{0}$ be an internal symmetry
group composed of unitary and chiral type symmetry such that

$\displaystyle
u_{g}H({\bm{k}})u_{g}^{-1}=\left\\{\begin{array}[]{ll}H({\bm{k}})&g\in
G_{0},\\\ -H({\bm{k}})&g\in\gamma G_{0},\end{array}\right.$ (44)

where $\gamma$ is a representative element of
$\\{g\in\mathcal{G}|\phi_{g}=-c_{g}=+1\\}$. Let $\alpha$ be an irrep of
$G_{0}$. If the mapped irrep $\gamma[\alpha]$ is unitarily equivalent to
$\alpha$, one can define the winding number $W^{\alpha}_{2n-1}$ for the irrep
$\alpha$ as follows. The construction of the winding number in this section is
based on [18].

Let $z_{g,h}\in Z^{2}(G,U(1))$ be a factor system of projective representation
appearing as $u_{g}u_{h}=z_{g,h}u_{gh}$ for $g,h\in G$. Let
$\chi^{\alpha}_{g\in G_{0}}$ be the character of the irrep $\alpha$ of
$G_{0}$. The character of the mapped irrep $\gamma[\alpha]$ is given by
$\chi^{\gamma[\alpha]}_{g\in
G_{0}}=\frac{z_{g,\gamma}}{z_{\gamma,\gamma^{-1}g\gamma}}\chi^{\alpha}_{\gamma^{-1}g\gamma}$.
If $\sum_{g\in G_{0}}(\chi^{\alpha}_{g})^{*}\chi^{\gamma[\alpha]}_{g}=1$, the
two irreps $\alpha$ and $\gamma[\alpha]$ are unitarily equivalent to each
other. If this is the case, there are exactly two irreps $\alpha+$ and
$\alpha-$ of $G_{0}\coprod\gamma G_{0}$ such that the character
$\chi^{\alpha\pm}_{g}$ of the irrep $\alpha\pm$ satisfies

$\displaystyle\chi^{\alpha\pm}_{g\in
G_{0}}=\chi^{\alpha}_{g},\quad\chi^{\alpha-}_{g\in\gamma
G_{0}}=-\chi^{\alpha+}_{g}.$ (45)

The orthogonality $\frac{1}{|G_{0}\coprod\gamma G_{0}|}\sum_{g\in
G_{0}\coprod\gamma G_{0}}(\chi^{\alpha+}_{g})^{*}\chi^{\alpha-}_{g}=0$ leads
to

$\displaystyle\frac{1}{|G_{0}|}\sum_{g\in\gamma
G_{0}}(\chi^{\alpha+}_{g})\chi^{\alpha\pm}_{g}=\pm 1.$ (46)

With these characters, we introduce the projection $P_{\alpha\pm}$ onto the
$\alpha\pm$ irrep as

$\displaystyle P_{\alpha\pm}=\frac{{\rm dim}(\alpha\pm)}{|G|}\sum_{g\in
G}(\chi^{\alpha\pm}_{g})^{*}u_{g}.$ (47)

here, ${\rm dim}(\alpha\pm)$ is the dimension of the representation
$\alpha\pm$. The chiral matrix of the irrep $\alpha$ is defined as

$\displaystyle\Gamma_{\alpha}=P_{\alpha+}-P_{\alpha-},$ (48)

and the winding number $W^{\alpha}_{2n-1}$ of the irrep $\alpha$ is given by

$\displaystyle W^{\alpha}_{2n-1}=\frac{n!}{(2\pi i)^{n}(2n)!}\int{\rm
tr\,}[(H^{-1}dH)^{2n-1}\Gamma_{\alpha}].$ (49)

For convenience, here, we have written $\alpha+$ and $\alpha-$, but there is
no way to choose one or the other as $\alpha+$ or $\alpha-$. Interchanging
$\alpha+$ and $\alpha-$ flips the sign of the chiral matrix $\Gamma_{\alpha}$
and the winding number $W^{\alpha}_{2n-1}$. Therefore, the sign of the winding
number $W^{\alpha}_{2n-1}$ depends on the choice of the sign of the chiral
matrix $\Gamma_{\alpha}$, which must be properly incorporated in the AHSS
formulated in later sections. The following expression of $W^{\alpha}_{2n-1}$
is also useful.

$\displaystyle W^{\alpha}_{2n-1}$ $\displaystyle=\frac{n!}{(2\pi
i)^{n}(2n)!}\frac{1}{|G_{0}|}\sum_{g\in\gamma G_{0}}(\chi^{\alpha+}_{g})^{*}$
$\displaystyle\times\int{\rm tr\,}[(H^{-1}dH)^{2n-1}u_{g}].$ (50)

##  IV Momentum-space AHSS

We provide a method for calculating the first differential $d_{1}$ of the
momentum-space AHSS [14]. In this section, we refer to $g,h,\dots$ as elements
of the group $G=\mathcal{G}/\Pi$.

In momentum space, we can consider the symmetry group of the Hamiltonian
$H({\bm{k}})$ as the group $G$. Let $\\{\\{p_{g}|\bm{a}_{g}\\}\\}_{g\in G}$ be
a set of left coset representatives of $\Pi$ in $\mathcal{G}$. The symmetry
constraints on the Hamiltonian $H({\bm{k}})$ are written as

$\displaystyle
u_{g}({\bm{k}})H({\bm{k}})^{\phi_{g}}u_{g}({\bm{k}})^{-1}=c_{g}H(\phi_{g}p_{g}{\bm{k}}),\quad
g\in G,$ (51)

together with the ${\bm{k}}$-dependent factor system

$\displaystyle
u_{g}(\phi_{h}p_{h}{\bm{k}})u_{h}({\bm{k}})^{\phi_{g}}=z_{g,h}({\bm{k}})u_{gh}({\bm{k}}),$
(52) $\displaystyle z_{g,h}({\bm{k}}):=z^{\rm
int}_{g,h}e^{-i\phi_{gh}p_{gh}{\bm{k}}\cdot(p_{g}\bm{a}_{h}+\bm{a}_{g}-\bm{a}_{gh})}$
(53)

for $g,h\in G$.

What we aim to compute is the twisted equivariant $K$-group
${}^{\phi}K_{G}^{(z,c)-n}(X)$ over the BZ torus $X$ by Freed and Moore [36].
Here, $n$ takes values in integers, and the $K$-group enjoys the Bott
periodicity ${}^{\phi}K_{G}^{(z,c)-(n+8)}(X)\cong{}^{\phi}K_{G}^{(z,c)-n}(X)$.
Roughly speaking, the integers $n$ have the following physical meanings:
$n=1,n=0$, and $n=-1$ correspond to gauge transformations in momentum space,
gapped Hamiltonians, and gapless Hamiltonians realized only as a boundary of
gapped Hamiltonians, respectively. In particular, the $0$th K-group
${}^{\phi}K_{G}^{(z,c)-0}(X)$ represents the classification of gapped
Hamiltonians.

Hereafter, we set the spatial dimension to three.

###  IV.1 Preliminary

For a cell decomposition $X_{0}\subset X_{1}\subset X_{2}\subset X_{3}=X$
introduced in Sec. III.2, the $E_{1}$-page $E_{1}^{p,-n}$ is defined as the
relative $K$-group

$\displaystyle E_{1}^{p,-n}:={}^{\phi}K^{(z,c)+p-n}_{G}(X_{p},X_{p-1}).$ (54)

We denote the label set of orbits of $p$-cells by $I^{p}_{\rm orb}$. Since the
$p$-skeleton is obtained by gluing $p$-cells to the $(p-1)$-skeleton
$X_{p-1}$, this is the direct sum of the $K$-groups over each orbit

$\displaystyle E_{1}^{p,-n}\cong{}^{\phi}K_{G}^{(z,c)+p-n}(\coprod_{{\sf a}\in
I^{p}_{\rm orb}}G/G_{D^{p}_{\sf a}}\times D^{p}_{\sf a},$ $\displaystyle\hskip
50.0pt\coprod_{{\sf a}\in I^{p}_{\rm orb}}G/G_{D^{p}_{\sf a}}\times\partial
D^{p}_{\sf a})$ $\displaystyle\cong\bigoplus_{{\sf a}\in I^{p}_{\rm
orb}}{}^{\phi|_{D^{p}_{\sf a}}}K^{(z|_{D^{p}_{\sf a}},c|_{D^{p}_{\sf
a}})+p-n}_{G_{D^{p}_{\sf a}}}(D^{p}_{\sf a},\partial D^{p}_{\sf a}).$ (55)

Here, $D^{p}_{\sf a}$ is a $p$-cell corresponding to a representative orbit
${\sf a}\in I^{p}_{\rm orb}$, $G_{D^{p}_{\sf a}}=\\{g\in
G|\phi_{g}p_{g}{\bm{k}}\equiv{\bm{k}}{\rm\ for\ }{\bm{k}}\in D^{p}_{\sf a}\\}$
is the little group of $G$ for the $p$-cell $D^{p}_{\sf a}$, and
$\phi|_{D^{p}_{\sf a}},c|_{D^{p}_{\sf a}},z|_{D^{p}_{\sf a}}$ are symmetry
data restricted in the $p$-cell $D^{p}_{\sf a}$. Since the little group
$G_{D^{p}_{\sf a}}$ fixes the $p$-cell $D^{p}_{\sf a}$ pointwise,
$E_{1}^{p,-n}$ is further simplified as

$\displaystyle E_{1}^{p,-n}$ $\displaystyle\cong\bigoplus_{{\sf a}\in
I^{p}_{\rm orb}}{}^{\phi|_{D^{p}_{\sf a}}}\tilde{K}^{(z|_{D^{p}_{\sf
a}},c|_{D^{p}_{\sf a}})+p-n}_{G_{D^{p}_{\sf a}}}(D^{p}_{\sf a}/\partial
D^{p}_{\sf a})$ $\displaystyle\cong\bigoplus_{{\sf a}\in I^{p}_{\rm
orb}}{}^{\phi|_{{\bm{k}}^{p}_{\sf a}}}K^{(z|_{{\bm{k}}^{p}_{\sf
a}},c|_{{\bm{k}}^{p}_{\sf a}})-n}_{G_{{\bm{k}}^{p}_{\sf
a}}}(\\{{\bm{k}}^{p}_{\sf a}\\}).$ (56)

Thus, the group $E_{1}^{p,-n}$ is the direct sum of $K$-groups over a point
${\bm{k}}^{p}_{\sf a}\in D^{p}_{\sf a}$ for each with the degree shift by
$-n$.

There are several ways to represent the $K$-group ${}^{\phi_{{\bm{k}}^{p}_{\sf
a}}}K^{(z|_{{\bm{k}}^{p}_{\sf a}},c|_{{\bm{k}}^{p}_{\sf
a}})-n}_{G_{{\bm{k}}^{p}_{\sf a}}}(\\{{\bm{k}}^{p}_{\sf a}\\})$. One is adding
$n$ chiral symmetries to the symmetry group $G_{{\bm{k}}^{p}_{\sf a}}$ for the
gapped Hamiltonians $H({\bm{k}}^{p}_{\sf a})$ over the point
${\bm{k}}^{p}_{\sf a}$ [38]. In the following, instead of adding $n$ chiral
symmetries, we use suspension isomorphism

$\displaystyle{}^{\phi|_{{\bm{k}}^{p}_{\sf a}}}K^{(z|_{{\bm{k}}^{p}_{\sf
a}},c|_{{\bm{k}}^{p}_{\sf a}})-n}_{G_{{\bm{k}}^{p}_{\sf
a}}}(\\{{\bm{k}}^{p}_{\sf a}\\})$
$\displaystyle\cong{}^{\phi|_{{\bm{k}}^{p}_{\sf
a}}}\tilde{K}^{(z|_{{\bm{k}}^{p}_{\sf a}},c|_{{\bm{k}}^{p}_{\sf
a}})}_{G_{{\bm{k}}^{p}_{\sf a}}}(\\{{\bm{k}}^{p}_{\sf
a}\\}\times\tilde{S}^{n})$ (57)

to represent the $K$-group by gapped Hamiltonians on the $n$-dimensional
sphere $\tilde{S}^{n}$ without changing the symmetry group
$G_{{\bm{k}}^{p}_{\sf a}}$, where $G_{{\bm{k}}^{p}_{\sf a}}$ trivially acts on
the sphere $\tilde{S}^{n}$.

Now, we develop the detail of the calculation. First, we note that for the
calculation of the first differential $d_{1}$, it is sufficient to consider
only the $p$-cells that intersect with the closure of the fundamental domain
of the BZ. The first differential $d_{1}^{p,-n}$ is defined as the composition
of homomorphisms

$\displaystyle
d_{1}^{p,-n}:{}^{\phi}K_{G}^{(z,c)+p-n}(X_{p},X_{p-1})\to{}^{\phi}K_{G}^{(z,c)+p-n}(X_{p})$
$\displaystyle\to{}^{\phi}K_{G}^{(z,c)+p-n+1}(X_{p+1},X_{p})=E_{1}^{p+1,-n}.$
(58)

The first line is induced homomorphism of the inclusion
$X_{p}\to(X_{p},X_{p-1})$, and the second line is the connected homomorphism.
The relation

$\displaystyle d_{1}^{p+1,-n}\circ d_{1}^{p,-n}=0$ (59)

holds. The $E_{2}$-page is defined as

$\displaystyle E_{2}^{p,-n}:={\rm Ker\,}d_{1}^{p,-n}/{\rm
Im\,}d_{1}^{p-1,-n}.$ (60)

(For $p=0$, $E_{2}^{0.-n}={\rm Ker\,}d_{1}^{0,-n}$.) The physical meaning of
$d_{1}^{p,-n}$ is that the gap closes inside $p$-cells and creates gapless
points in adjacent $(p+1)$-cells. Therefore, for a $(p+1)$-cell, only adjacent
$p$-cells contribute to $d_{1}^{p,-n}$. Starting from the 3-cell, the
fundamental domain, only the 2-cells on the boundary of the 3-cell contribute
to $d_{1}^{2,-n}$, only the 1-cells on the boundary of these 2-cells
contribute to $d_{1}^{1,-n}$, and finally, only the $0$-cells on the boundary
of these 1-cells contribute to $d_{1}^{0,-n}$. This means that one can compute
$d_{1}^{p,-n}$ with the $p$-cells adjacent to the fundamental domain.

Figure 1:  A cell decomposition of BZ for MSG $P2_{1}1^{\prime}$. The figure
shows the fundamental domain (3-cell). The 0-, 1-, and 2-cells are shown with
black, red, and blue Arabic numerals, respectively. The arrows of the 1- and
2-cells show orientations, which respect symmetry. The momenta of 0-cells are
$(0,0,-\pi)$, $(0,\pi,-\pi)$, $(0,0,\pi)$, $(0,\pi,\pi)$, $(\pi,\pi,-\pi)$,
$(\pi,\pi,\pi)$, $(\pi,0,-\pi)$, $(\pi,0,\pi)$, $(0,\pi,0)$, $(0,0,0)$,
$(\pi,\pi,0)$, $(\pi,0,0)$ in order.

Based on the above consideration, we introduce the integer lattices
$E_{0}^{p}$ for $p=0,1,2,3$ as follows. Let $D^{3}$ be the fundamental domain.
We define the set of relevant $p$-cells by

$\displaystyle{\cal
C}_{p}=\\{D^{p}_{i}|D^{p}_{i}\cap\overline{D^{3}}\neq\emptyset\\},\quad
p=0,1,2,$ (61) $\displaystyle{\cal C}_{3}=\\{D^{3}\\}.$ (62)

Here, $\overline{D^{p}_{i}}$ is the closure of the open $p$-cell $D^{p}_{i}$.
Note that ${\cal C}_{p}$ includes equivalent $p$-cells as independent ones.
(For example, see Fig. 1 for a choice of fundamental domain for the MSG
$P2_{1}1^{\prime}$ together with the boundary 0-, 1-, and 2-cells.) Also, we
regard cells that are related to other cells by reciprocal lattice vectors as
different cells. For each $p$-cell, we pick a representative point
${\bm{k}}^{p}_{i}\in D^{p}_{i}$ for each $p$-cell. For ${\bm{k}}\in X$,
introduce the left coset decomposition

$\displaystyle G_{{\bm{k}}}=G_{{\bm{k}}}^{0}\coprod{\sf
t}G_{\bm{k}}^{0}\coprod{\sf c}G_{\bm{k}}^{0}\coprod\gamma G_{\bm{k}}^{0},$
(63)

where $G_{\bm{k}}^{0}={\rm Ker\,}\phi\cap{\rm Ker\,}c\cap G_{\bm{k}}$, and

$\displaystyle{\sf t}\in\\{g\in G_{\bm{k}}|-\phi_{g}=c_{g}=1\\},$ (64)
$\displaystyle{\sf c}\in\\{g\in G_{\bm{k}}|-\phi_{g}=-c_{g}=1\\},$ (65)
$\displaystyle\gamma\in\\{g\in G_{\bm{k}}|\phi_{g}=-c_{g}=1\\},$ (66)

are representatives. In general, ${\sf t},{\sf c}$, and $\gamma$ depend on the
${\bm{k}}\in X$, but ${\bm{k}}$ is omitted unless misunderstandings can arise.
At ${\bm{k}}$, $\\{\alpha_{j}({\bm{k}})\\}_{j=1}^{N_{\bm{k}}}$ denotes the set
of irreps of $G_{\bm{k}}^{0}$ with the factor system $z_{g,h}({\bm{k}})$. (See
Appendix B for a derivation of irreducible characters.) The lattice
$E_{0}^{p}$ is defined as the $\mathbb{Z}$-module generated by irreps

$\displaystyle E_{0}^{p}:=\Braket{\bigcup_{D^{p}_{i}\in{\cal
C}_{p}}\bigcup_{r=1}^{N_{{\bm{k}}^{p}_{i}}}\alpha_{r}({\bm{k}}^{p}_{i})},$
(67)

the $\mathbb{Z}$-module generated by the set of irreps
$\bigcup_{D^{p}_{i}\in{\cal
C}_{p}}\bigcup_{r=1}^{N_{{\bm{k}}^{p}_{i}}}\alpha_{r}({\bm{k}}^{p}_{i})$.

Introduce the integer $n_{i,i^{\prime}}\in\\{\pm 1,0\\}$ such that
$n_{i,i^{\prime}}=1(-1)$ if $D^{p}_{i^{\prime}}\in\partial D^{p+1}_{i}$ and
the orientation of $D^{p}_{i^{\prime}}$ agrees (disagrees) with $\partial
D^{p+1}_{i}$, and $n_{i,i^{\prime}}=0$ if $D^{p}_{i^{\prime}}\notin\partial
D^{p+1}_{i}$. We denote by $\chi^{\alpha_{r}({\bm{k}}^{p}_{i})}_{g\in
G_{{\bm{k}}^{p}_{i}}}$ the irreducible character of the irrep
$\alpha_{r}({\bm{k}}^{p}_{i})$. We define the homomorphism
$\delta_{p}:E_{0}^{p}\to E_{0}^{p+1}$ as how irreps at $p$-cells decompose
into irreps at $(p+1)$-cells. Explicitly,
$[\delta_{p}]_{ir,i^{\prime}r^{\prime}}=0$ when
$D^{p}_{i^{\prime}}\notin\partial D^{p+1}_{i}$, and

$\displaystyle[\delta_{p}]_{ir,i^{\prime}r^{\prime}}$
$\displaystyle:=n_{i,i^{\prime}}\times\frac{1}{|G^{0}_{{\bm{k}}^{p+1}_{i}}|}\sum_{g\in
G^{0}_{{\bm{k}}^{p+1}_{i}}}(\chi^{\alpha_{r}({\bm{k}}^{p+1}_{i})}_{g})^{*}$
$\displaystyle\times\chi^{\alpha_{r^{\prime}}({\bm{k}}^{p}_{i^{\prime}})}_{g}e^{-i\phi_{g}p_{g}({\bm{k}}^{p+1}_{i}-{\bm{k}}^{p}_{i^{\prime}})\cdot\bm{a}_{g}}$
(68)

when $D^{p}_{i^{\prime}}\in\partial D^{p+1}_{i}$. The last factor in (IV.1) is
needed to match the factor systems between ${\bm{k}}^{p}_{i^{\prime}}$ and
${\bm{k}}^{p+1}_{i}$.

$\begin{array}[]{c|ccc|ccccccccccc}&W^{\alpha}_{T}&W^{\alpha}_{C}&W^{\alpha}_{\Gamma}&n=0&n=1&n=2&n=3&n=4&n=5&n=6&n=7\\\
\hline\cr{\rm
A}&0&0&0&(\mathbb{Z},1)&0&(\mathbb{Z},2)&0&(\mathbb{Z},4)&0&(\mathbb{Z},8)&0\\\
{\rm
AIII}&0&0&1&0&(\mathbb{Z},2)&0&(\mathbb{Z},4)&0&(\mathbb{Z},8)&0&(\mathbb{Z},16)\\\
\hline\cr{\rm
AI}&1&0&0&(\mathbb{Z},1)&(\mathbb{Z}_{2},2)&(\mathbb{Z}_{2},4)&0&(\mathbb{Z},8)&0&0&0\\\
{\rm
BDI}&1&1&1&(\mathbb{Z}_{2},2)&(\mathbb{Z}_{2},4)&0&(\mathbb{Z},8)&0&0&0&(\mathbb{Z},16)\\\
{\rm
D}&0&1&0&(\mathbb{Z}_{2},2)&0&(\mathbb{Z},4)&0&0&0&(\mathbb{Z},8)&(\mathbb{Z}_{2},16)\\\
{\rm
DIII}&-1&1&1&0&(\mathbb{Z},4)&0&0&0&(\mathbb{Z},8)&(\mathbb{Z}_{2},16)&(\mathbb{Z}_{2},32)\\\
{\rm
AII}&-1&0&0&(\mathbb{Z},2)&0&0&0&(\mathbb{Z},4)&(\mathbb{Z}_{2},8)&(\mathbb{Z}_{2},16)&0\\\
{\rm
CII}&-1&-1&1&0&0&0&(\mathbb{Z},4)&(\mathbb{Z}_{2},8)&(\mathbb{Z}_{2},16)&0&(\mathbb{Z},32)\\\
{\rm
C}&0&-1&0&0&0&(\mathbb{Z},2)&(\mathbb{Z}_{2},4)&(\mathbb{Z}_{2},8)&0&(\mathbb{Z},16)&0\\\
{\rm
CI}&1&-1&1&0&(\mathbb{Z},2)&(\mathbb{Z}_{2},4)&(\mathbb{Z}_{2},8)&0&(\mathbb{Z},16)&0&0\\\
\hline\cr{\rm mapped\
rep.}&&&&c_{h}&s_{h}\phi_{h}&\phi_{h}c_{h}&s_{h}&c_{h}&s_{h}\phi_{h}&\phi_{h}c_{h}&s_{h}\\\
\end{array}$ Table 1: Wignar criteria, AZ classes, and classification of
gapped Hamiltonian over the momentum-space $n$-sphere. In the table,
$0,\mathbb{Z}$ and $\mathbb{Z}_{2}$ indicate the classification, and when the
classification is nontrivial, the matrix size of the generator Dirac
Hamiltonian is shown together. The last row shows the sign change of the
$\mathbb{Z}$ topological invariant under the action of the group element $h$.

For each irrep $\alpha({\bm{k}})$ at ${\bm{k}}$, we identify the AZ class by
the Wigner criteria [14]

$\displaystyle
W^{\alpha({\bm{k}})}_{T}:=\frac{1}{|G^{0}_{\bm{k}}|}\sum_{g\in{\sf
t}G_{\bm{k}}^{0}}z_{g,g}({\bm{k}})\chi^{\alpha({\bm{k}})}_{g^{2}}\in\\{\pm
1,0\\},$ (69) $\displaystyle
W^{\alpha({\bm{k}})}_{C}:=\frac{1}{|G^{0}_{\bm{k}}|}\sum_{g\in{\sf
c}G_{\bm{k}}^{0}}z_{g,g}({\bm{k}})\chi^{\alpha({\bm{k}})}_{g^{2}}\in\\{\pm
1,0\\},$ (70) $\displaystyle
W^{\alpha({\bm{k}})}_{\Gamma}:=\frac{1}{|G^{0}_{\bm{k}}|}\sum_{g\in
G_{\bm{k}}^{0}}(\chi^{\alpha({\bm{k}})}_{g})^{*}\frac{z_{g,\gamma}({\bm{k}})}{z_{\gamma,\gamma^{-1}g\gamma}({\bm{k}})}\chi^{\alpha({\bm{k}})}_{\gamma^{-1}g\gamma}$
$\displaystyle\hskip 50.0pt\in\\{1,0\\}.$ (71)

We extend the notation as follows: $W^{\alpha({\bm{k}})}_{T}=0$ also indicates
the case where ${\sf t}\in G_{\bm{k}}$ does not exists. The same notation for
$W^{\alpha({\bm{k}})}_{C}$ and $W^{\alpha({\bm{k}})}_{\Gamma}$ is used. For
each
$(W^{\alpha({\bm{k}})}_{T},W^{\alpha({\bm{k}})}_{C},W^{\alpha({\bm{k}})}_{\Gamma})$,
the AZ class and the corresponding classification are listed in TABLE 1.

For each ${\bm{k}}$, if $W^{\alpha({\bm{k}})}_{\Gamma}=1$, we pick an irrep
$\alpha({\bm{k}})+$ of $G^{0}_{\bm{k}}\coprod\gamma G^{0}_{\bm{k}}$ so that
$\chi^{\alpha({\bm{k}})+}_{g\in G^{0}_{\bm{k}}}=\chi^{\alpha({\bm{k}})}_{g}$.
We introduce another homomorphism $\delta^{\Gamma}_{p}:E_{0}^{p}\to
E_{0}^{p+1}$ as

$\displaystyle[\delta^{\Gamma}_{p}]_{ir,i^{\prime}r^{\prime}}$
$\displaystyle:=n_{i,i^{\prime}}\times\frac{1}{|G^{0}_{{\bm{k}}^{p+1}_{i}}|}\sum_{g\in\gamma
G^{0}_{{\bm{k}}^{p+1}_{i}}}(\chi^{\alpha_{r}({\bm{k}}^{p+1}_{i})+}_{g})^{*}$
$\displaystyle\times\chi^{\alpha_{r^{\prime}}({\bm{k}}^{p}_{i^{\prime}})+}_{g}e^{-i\phi_{g}p_{g}({\bm{k}}^{p+1}_{i}-{\bm{k}}^{p}_{i^{\prime}})\cdot\bm{a}_{g}}$
(72)

for $D^{p}_{i^{\prime}}\in\partial D^{p+1}_{i}$ and
$W^{\alpha({\bm{k}}^{p}_{i^{\prime}})}_{\Gamma}=W^{\alpha({\bm{k}}^{p+1}_{i})}_{\Gamma}=1$,
and $[\delta^{\Gamma}_{p}]_{ir,i^{\prime}r^{\prime}}=0$ otherwise. Note that
$\delta^{\Gamma}_{p}$ depends on choices of irreps
$\alpha({\bm{k}}^{p}_{i})+$.

For an irrep $\alpha({\bm{k}})$ of $G^{0}_{\bm{k}}$ at ${\bm{k}}\in X$ and
$h\in G$, we introduce the mapped irrep $h[\alpha({\bm{k}})]$ by $h$, which is
an irrep of $G^{0}_{\phi_{h}p_{h}{\bm{k}}}=hG^{0}_{\bm{k}}h^{-1}$ at
$\phi_{h}p_{h}{\bm{k}}\in X$ whose character is

$\displaystyle\chi^{h[\alpha({\bm{k}})]}_{g\in
hG^{0}_{\bm{k}}h^{-1}}=\frac{z_{g,h}({\bm{k}})}{z_{h,h^{-1}gh}({\bm{k}})}\chi^{\alpha({\bm{k}})}_{h^{-1}gh}.$
(73)

When $W^{\alpha({\bm{k}})}_{\Gamma}=1$, the mapped irrep
$h[\alpha({\bm{k}})+]$ is defined in the same way. Note that the two irreps
$h[\alpha({\bm{k}})+]$ and $h[\alpha({\bm{k}})]+$ of the group
$h(G^{0}_{{\bm{k}}}\coprod\gamma G^{0}_{\bm{k}})h^{-1}$ may not be unitary
equivalent to each other. We define the sign $s_{h}\in\\{\pm 1\\}$ as
$s_{h}=1$ if $h[\alpha({\bm{k}})+]$ is unitary equivalent to
$h[\alpha({\bm{k}})]+$ and $s_{h}=-1$ else. From (46), $s_{h}$ can be computed
from

$\displaystyle s_{h}=\frac{1}{|G^{0}_{\bm{k}}|}\sum_{g\in h(\gamma
G^{0}_{\bm{k}})h^{-1}}(\chi^{h[\alpha({\bm{k}})]+}_{g})^{*}\chi^{h[\alpha({\bm{k}})+]}_{g}.$
(74)

The sign $s_{h}$ is the relative sign between the two chiral operators

$\displaystyle
u_{h}({\bm{k}})(\Gamma_{\alpha({\bm{k}})})^{\phi_{h}}u_{h}({\bm{k}})^{-1}=s_{h}\Gamma_{h[\alpha({\bm{k}})]}.$
(75)

###  IV.2 $E_{1}$ page

We will define two integer sublattices $\tilde{E}_{1}^{p,-n}$ and ${\cal
P}_{1}^{p,-n}$ of $E_{0}^{p}$ such that $\tilde{E}_{1}^{p,-n}/{\cal
P}_{1}^{p,-n}=E_{1}^{p,-n}$.

Based on (57), $E_{1}^{p,-n}$ is generated by an orbit of $n$-dimensional
massive Dirac Hamiltonians

$\displaystyle
H_{{\bm{k}}^{p}_{i}}(\tilde{\bm{k}})=\sum_{\mu=1}^{n}\tilde{k}_{\mu}\gamma_{\mu}+m\gamma_{0},\quad\tilde{\bm{k}}\in{{\bm{k}}^{p}_{i}}\times\tilde{S}^{n}$
(76)

on $p$-cells $D^{p}_{i}$. Here, ${\bm{k}}^{p}_{i}\in D^{p}_{i}$ is a
representative point within the $p$-cell $D^{p}_{i}$, and $\tilde{S}^{n}$ is
the virtual $n$-sphere on which the little group $G_{{\bm{k}}^{p}_{i}}$ acts
trivially. The symmetry implies that

$\displaystyle
H_{\phi_{g}p_{g}{\bm{k}}^{p}_{i}}(\tilde{\bm{k}})=c_{g}u_{g}({\bm{k}}^{p}_{i})H_{{\bm{k}}^{p}_{i}}(\tilde{\bm{k}})^{\phi_{g}}u_{g}({\bm{k}}^{p}_{i})^{-1},$
$\displaystyle\tilde{\bm{k}}\in\\{\phi_{g}p_{g}{\bm{k}}^{p}_{i}\\}\times\tilde{S}^{n},$
(77)

for the $p$-cells in the same orbit. For a given irrep
$\alpha({\bm{k}}^{p}_{i})$ of $p$-cell $D^{p}_{i}$, the classification of the
mass term $m\gamma_{0}$ is determined according to TABLE 1. In Table 1, the
matrix size of the generating Dirac Hamiltonian is shown on the right of the
parentheses. For example, $(\mathbb{Z},4)$ indicates that the classification
is $\mathbb{Z}$, and the generator $1\in\mathbb{Z}$ is represented by a
massive Dirac Hamiltonian whose matrix size is four.

####  IV.2.1 $\mathbb{Z}$ classification

For each irrep of the little group for each orbit, we construct a vector
$\vec{a}^{p,-n}\in E_{0}^{p}$ consisting of the matrix dimensions of the
generating Dirac Hamiltonians with the relative signs of $\mathbb{Z}$ numbers.
Pick a representative $p$-cell $D^{p}_{i}$ of an orbit of equivalent $p$-cells
in ${\cal C}_{p}$. For an irrep $\alpha_{r}({\bm{k}}^{p}_{i})$, if the
classification of degree $n$ is $\mathbb{Z}$, $a^{p,-n}_{ir}$ denotes the
matrix size of the generator Dirac Hamiltonian listed in Table 1. For other
equivalent irreps $h[\alpha({\bm{k}}^{p}_{i})]$ at
$\phi_{h}p_{h}{\bm{k}}^{p}_{i}\in X$, we should implement the relative sign of
$\mathbb{Z}$ invariant from the relation (77).

For even $n$, the $\mathbb{Z}$ invariant is the Chern number

$\displaystyle
ch_{n/2}=\frac{1}{(n/2)!}\left(\frac{i}{2\pi}\right)^{n/2}\int_{\tilde{S}^{n}}{\rm
tr\,}[{\cal F}^{n/2}],$ (78)

where ${\cal F}$ is the Berry connection of the occupied state of the
Hamiltonian $H(\tilde{{\bm{k}}})$, and integral is taken over the virtual
$n$-sphere $\tilde{S}^{n}$. For $n=0$, we define $ch_{0}$ as the number of
occupied states minus the number of unoccupied states. Since the Chern number
of the unoccupied band is $-ch_{n/2}$, we have the factor $c_{h}$. Moreover,
since the Berry curvature ${\cal F}$ changes its sign if $h$ is antiunitary,
we have the factor $\phi_{h}$ when $n\in 4\mathbb{Z}+2$.

For odd $n$, the $\mathbb{Z}$ invariant is the winding number

$\displaystyle w_{n}$ $\displaystyle=\frac{((n+1)/2)!}{(2\pi
i)^{(n+1)/2}(n+1)!}$ $\displaystyle\qquad\int_{\tilde{S}^{n}}{\rm
tr\,}[(H^{-1}dH)^{n}\Gamma_{\alpha({\bm{k}}^{p}_{i})}],$ (79)

where $\Gamma_{\alpha({\bm{k}}^{p}_{i})}$ is the chiral operator, whose sign
depends on the choice of $\alpha({\bm{k}}^{p}_{i})+$, and integral is taken
over the virtual $n$-sphere $\tilde{S}^{n}$. From the sign change between the
chiral operators, we have the factor $s_{h}$. Moreover, since the winding
number (79) includes the imaginary unit, we have the factor $\phi_{h}$ when
$n\in 4\mathbb{Z}+1$.

Let us introduce $h(i)$ and $h(r)$ so that $D^{p}_{h(i)}=h(D^{p}_{i})$ and
$\alpha_{h(r)}((\phi_{j}p_{h}{\bm{k}})^{p}_{h(i)})=h[\alpha_{r}({\bm{k}}^{p}_{i})]$.
Then, for other components, we set

$\displaystyle
a^{p,-n}_{h(i)h(r)}=a^{p,-n}_{ir}\times\left\\{\begin{array}[]{ll}c_{h}&(n=0),\\\
s_{h}\phi_{h}&(n=1),\\\ \phi_{h}c_{h}&(n=2),\\\ s_{h}&(n=3),\\\
c_{h}&(n=4),\\\ s_{h}\phi_{h}&(n=5),\\\ \phi_{h}c_{h}&(n=6),\\\
s_{h}&(n=7).\\\ \end{array}\right.$ (88)

The last factor in (88) is also shown in Table 1. Constructing the vectors
$\vec{a}^{p,-n}_{1},\vec{a}^{p,-n}_{2},\dots,$ for all inequivalent irreps and
inequivalent orbits, we have the sublattice

$\displaystyle
E_{1\mathbb{Z}}^{p,-n}:=\braket{\vec{a}^{p,-n}_{1},\vec{a}^{p,-n}_{2},\dots}\subset
E_{0}^{p}.$ (89)

We note that since the $p$-cells in ${\cal C}_{p}$ are oriented symmetrically,
there is no sign change due to the mismatch of orientations.

####  IV.2.2 $\mathbb{Z}_{2}$ classification

No sign difference exists in the $\mathbb{Z}_{2}$ number. For each irrep of
each orbit, we construct a vector $\vec{b}^{p,-n}\in E_{0}^{p}$ consisting of
matrix sizes of generator Dirac Hamiltonians as follows. Pick a representative
$p$-cell $D^{p}_{i}$ of an orbit of equivalent $p$-cells in ${\cal C}_{p}$.
For an irrep $\alpha_{r}({\bm{k}}^{p}_{i})$, if the classification of degree
$n$ is $\mathbb{Z}_{2}$, we define $b^{p,-n}_{ir}$ by the matrix size of the
generator Dirac Hamiltonian listed in Table 1, i.e., $b^{p,-n}_{ir}={\rm
dim\,}(H)$. For other equivalent irreps $h[\alpha({\bm{k}}^{p}_{i})]$ at
$\phi_{h}p_{h}{\bm{k}}\in X$, $b^{p,-n}_{h(i)h(r)}=b^{p,-n}_{ir}$.
Constructing the vectors $\vec{b}^{p,-n}_{1},\vec{b}^{p,-n}_{2},\dots,$ for
all inequivalent irreps and inequivalent orbits, we have

$\displaystyle
E_{1\mathbb{Z}_{2}}^{p,-n}:=\braket{\vec{b}^{p,-n}_{1},\vec{b}^{p,-n}_{2},\dots}\subset
E_{0}^{p}$ (90)

and ${\cal P}_{1}^{p,-n}:=\\{2x\in E_{0}^{p}|x\in
E_{1\mathbb{Z}_{2}}^{p,-n}\\}$.

The group $E_{1}^{p,-n}$ is represented as the quotient group

$\displaystyle E_{1}^{p,-n}=(E_{1\mathbb{Z}}^{p,-n}\oplus
E_{1\mathbb{Z}_{2}}^{p,-n})/{\cal P}^{p,-n}_{1}.$ (91)

Note that by construction $E_{1\mathbb{Z}}^{p,-n}\cap
E_{1\mathbb{Z}_{2}}^{p,-n}=\\{\bm{0}\\}$. We write
$\tilde{E}_{1}^{p,-n}:=E_{1\mathbb{Z}}^{p,-n}\oplus
E_{1\mathbb{Z}_{2}}^{p,-n}$.

###  IV.3 The first differential $d_{1}$

The homomorphism $d_{1}^{p,-n}$ is given by the expansion coefficients

$\displaystyle
d_{1}^{p,-n}(\vec{a}^{p,-n}_{\lambda})=\sum_{\kappa}\vec{a}^{p+1,-n}_{\kappa}[M^{p,-n}_{\mathbb{Z}\mathbb{Z}}]_{\kappa\lambda}$
$\displaystyle\qquad\qquad+\sum_{\kappa}\vec{b}^{p+1,-n}_{\kappa}[M^{p,-n}_{\mathbb{Z}\mathbb{Z}_{2}}]_{\kappa\lambda},$
(92) $\displaystyle
d_{1}^{p,-n}(\vec{b}^{p,-n}_{\lambda})=\sum_{\kappa}\vec{b}^{p+1,-n}_{\kappa}[M^{p,-n}_{\mathbb{Z}_{2}\mathbb{Z}_{2}}]_{\kappa\lambda},$
(93)

where $[M^{p,-n}_{\mathbb{Z}\mathbb{Z}}]_{\kappa\lambda}\in\mathbb{Z}$ and
$[M^{p,-n}_{\mathbb{Z}\mathbb{Z}_{2}}]_{\kappa\lambda},[M^{p,-n}_{\mathbb{Z}_{2}\mathbb{Z}_{2}}]_{\kappa\lambda}\in\\{0,1\\}$.
A vector $\vec{v}=(v_{ir})\in E_{1}^{p,-n}$ represents a set of massive Dirac
Hamiltonians over the $n$-spheres $\\{{\bm{k}}^{p}_{i}\\}\times\tilde{S}^{n}$
with $\mathbb{Z}$ or $\mathbb{Z}_{2}$ invariants specified by $v_{ir}$. This
set of the massive Dirac Hamiltonians over
$\\{{\bm{k}}^{p}_{i}\\}\times\tilde{S}^{n}$ may be incompatible on adjacent
$(p+1)$-cells whose obstruction is given by the vector
$d_{1}^{p,-n}(\vec{v})\in E_{1}^{p+1,-n}$.

The expansion coefficients are computed as follows. We denote the matrices
consisting of generators of $E^{p,-n}_{1\mathbb{Z}}$ and
$E_{1\mathbb{Z}_{2}}^{p,-n}$ by

$\displaystyle A_{1}^{p,-n}:=(\vec{a}^{p,-n}_{1},\vec{a}^{p,-n}_{2},\dots),$
(94) $\displaystyle
B_{1}^{p,-n}:=(\vec{b}^{p,-n}_{1},\vec{b}^{p,-n}_{2},\dots).$ (95)

We claim that $\delta_{p}(E_{1\mathbb{Z}}^{p,-n})\subset
E_{1\mathbb{Z}}^{p+1,-n}$ for even $n$: A $\mathbb{Z}$-valued vector
$\vec{a}=(a_{ir})\in E^{p,-n}_{1\mathbb{Z}}$ represents the set of massive
Dirac Hamiltonians with $\mathbb{Z}$ numbers specified by $a_{ir}$ over the
$p$-cells $D^{p}_{i}$. The vector $\delta_{p}\vec{a}\in E_{0}^{p+1}$
represents the $\mathbb{Z}$-valued obstruction to glue the massive Dirac
Hamiltonians on adjacent $(p+1)$-cells. Suppose that an irrep
$\alpha_{r^{\prime}}({\bm{k}}^{p+1}_{i^{\prime}})$ at a point of
${\bm{k}}^{p+1}_{i^{\prime}}$ of a $(p+1)$-cell is classified as
$\mathbb{Z}_{2}$. The relation $d_{1}^{p,-n}\vec{a}\in E_{1}^{p+1,-n}$ implies
that the total $\mathbb{Z}$ number of the irrep
$\alpha_{r^{\prime}}({\bm{k}}^{p+1}_{i^{\prime}})$, the coefficient
$[\delta_{p}\vec{a}]_{i^{\prime}r^{\prime}}$ must vanish. For odd $n$, the
same relation $\delta_{p}^{\Gamma}(E_{1\mathbb{Z}}^{p,-n})\subset
E_{1\mathbb{Z}}^{p+1,-n}$ holds true. Therefore, the expansion coefficient
$M^{p,-n}_{\mathbb{Z}\mathbb{Z}}$ is given as

$\displaystyle M^{p,-n}_{\mathbb{Z}\mathbb{Z}}$
$\displaystyle=\left\\{\begin{array}[]{ll}(A_{1}^{p+1,-n})^{+}\delta_{p}A_{1}^{p,-n}&({\rm
for\ even\ }n),\\\ (A_{1}^{p+1,-n})^{+}\delta^{\Gamma}_{p}A_{1}^{p,-n}&({\rm
for\ odd\ }n).\end{array}\right.$ (98)

Here, $X^{+}$ is the pseudoinverse of the matrix $X$. (Note that the
pseudoinverse $X^{+}=X^{+}XX^{+}$ itself involves the projection $XX^{+}$ onto
${\rm Im\,}X$.) For the coefficient of $\mathbb{Z}_{2}$ groups, since the
$\mathbb{Z}_{2}$ number is given by the matrix size of Dirac Hamiltonians,
without considering either orientations or the sign of $\mathbb{Z}$ numbers,
the expansion coefficients are given as

$\displaystyle
M^{p,-n}_{\mathbb{Z}\mathbb{Z}_{2}}=(B_{1}^{p+1,-n})^{+}|\delta_{p}||A_{1}^{p,-n}|\quad{\rm\
mod\ }2,$ (99) $\displaystyle
M^{p,-n}_{\mathbb{Z}_{2}\mathbb{Z}_{2}}=(B_{1}^{p+1,-n})^{+}|\delta_{p}|B_{1}^{p,-n}\quad{\rm\
mod\ }2.$ (100)

Here, $|X|$ is the matrix whose component is $|X_{ij}|$.

Introduce an integer lift

$\displaystyle\tilde{d}_{1}^{p,-n}:\tilde{E}_{1}^{p,-n}\to\tilde{E}_{1}^{p+1,-n}.$
(101)

By considering $M^{p,-n}_{\mathbb{Z}\mathbb{Z}_{2}}$ and
$M^{p,-n}_{\mathbb{Z}_{2}\mathbb{Z}_{2}}$ as $\mathbb{Z}$-valued matrices,
such a lift is given. The group $E_{2}^{p,-n}$ is computed as the quotient of
two integer sublattices of $E_{0}^{p}$:

$\displaystyle E_{2}^{p,-n}=\frac{{\rm Ker\,}(\tilde{d}_{1}^{p,-n}\oplus{\rm
Id}_{{\cal P}_{1}^{p+1,-n}})|_{\tilde{E}_{1}^{p,-n}}}{{\rm
Im\,}(\tilde{d}_{1}^{p-1,-n})+{\cal P}_{1}^{p,-n}}.$ (102)

Here, $\tilde{d}_{1}^{p,-n}\oplus{\rm Id}_{{\cal
P}_{1}^{p+1,-n}}(\vec{u},\vec{v})=\tilde{d}_{1}^{p,-n}(\vec{u})+\vec{v}$, and
${\rm Ker\,}(\tilde{d}_{1}^{p,-n}\oplus{\rm Id}_{{\cal
P}_{1}^{p+1,-n}})|_{\tilde{E}_{1}^{p,-n}}$ means the restriction to
$\tilde{E}_{1}^{p,-n}$. See Appendix C for a derivation of (102).

##  V Real-space AHSS

We present a method for calculating the first differential $d^{1}$ of the
real-space AHSS [28]. In this section, $g,h,\dots$ refer to elements of the
group ${\cal G}$. We denote the ${\cal G}$-action on the real space by
${\bm{x}}\mapsto g({\bm{x}})=p_{g}{\bm{x}}+\bm{t}_{g}$ for $g\in{\cal G}$.
This formulation shares similarities with the momentum-space AHSS, so our
primary focus will be on the differences.

The real-space AHSS is based on the concept of “crystalline topological
liquid” [49]: We assume the scale of lattice translations is much larger than
the microscopic scale. A topological phase protected by a MSG is represented
as a ”patchwork” of the set of topological phases localized on $p$-cells. With
this perspective, we denote a Hamiltonian near the position
${\bm{x}}\in\mathbb{E}^{3}$ by a single variable as $H({\bm{x}})$. The
symmetry constraint on $H({\bm{x}})$ is written as

$\displaystyle u_{g}H({\bm{x}})^{\phi_{g}}u_{g}^{-1}=c_{g}H(g({\bm{x}})),$
(103) $\displaystyle u_{g}u_{h}^{\phi_{g}}=z^{\rm int}_{g,h}u_{gh},\quad
g,h\in{\cal G}.$ (104)

We shall write the $K$-theory classification of Hamiltonians $H({\bm{x}})$
with ${\cal G}$ symmetry by ${}^{\phi}K^{\cal G}_{(z^{\rm
int},c)-n}(\mathbb{R}^{3})$. Here, $n\in\mathbb{Z}$ is the degree of the
$K$-homology group. The physical meaning of the degree $n$ is opposite to the
$K$-cohomology group ${}^{\phi}K_{G}^{(z,c)-n}(X)$: For $n=1,0,-1$, the group
${}^{\phi}K^{\cal G}_{(z^{\rm int},c)-n}(\mathbb{R}^{3})$ represents the
classification of gapless Hamiltonians, gapped Hamiltonians, and one-parameter
families gapped Hamiltonians (adiabatic pumps, says), respectively. It is
expected that

$\displaystyle{}^{\phi}K^{\cal G}_{(z^{\rm
int},c)-n}(\mathbb{R}^{3})\cong{}^{\phi}K_{G}^{(z,c)+n}(X)$ (105)

since the two $K$-groups classify the physically same systems, and the
isomorphism (105) was proved in [45].

In the following, the superscript of the factor system $z^{\rm int}$ will be
omitted and simply written as $z$.

###  V.1 Preliminary

For a cell decomposition $X_{0}\subset X_{1}\subset X_{2}\subset
X_{3}=\mathbb{R}^{3}$ introduced in Sec. III.2, the $E^{1}$-page
$E^{1}_{p,-n}$ is defined as the relative $K$-homology group

$\displaystyle E^{1}_{p,-n}:={}^{\phi}K_{(z,c)+p-n}^{\cal G}(X_{p},X_{p-1}).$
(106)

We denote the label set of orbits of $p$-cells by $I^{p}_{\rm orb}$. We have

$\displaystyle E^{1}_{p,-n}\cong{}^{\phi}K^{\cal G}_{(z,c)+p-n}(\coprod_{{\sf
a}\in I^{p}_{\rm orb}}{\cal G}/{\cal G}_{D^{p}_{\sf a}}\times D^{p}_{\sf a},$
$\displaystyle\hskip 50.0pt\coprod_{{\sf a}\in I^{p}_{\rm orb}}{\cal G}/{\cal
G}_{D^{p}_{\sf a}}\times\partial D^{p}_{\sf a})$
$\displaystyle\cong\bigoplus_{{\sf a}\in I^{p}_{\rm orb}}{}^{\phi|_{D^{p}_{\sf
a}}}K_{(z|_{D^{p}_{\sf a}},c|_{D^{p}_{\sf a}})+p-n}^{{\cal G}_{D^{p}_{\sf
a}}}(D^{p}_{\sf a},\partial D^{p}_{\sf a}).$ (107)

Here, $D^{p}_{\sf a}$ is a representative of the orbit ${\sf a}\in I^{p}_{\rm
orb}$, ${\cal G}_{D^{p}_{\sf a}}=\\{g\in{\cal G}|g({\bm{x}})={\bm{x}}{\rm\
for\ }{\bm{x}}\in D^{p}_{\sf a}\\}$ is the little group of the $p$-cell
$D^{p}_{\sf a}$, and $\phi|_{D^{p}_{\sf a}},c|_{D^{p}_{\sf a}},z|_{D^{p}_{\sf
a}}$ are symmetry data restricted in the $p$-cell $D^{p}_{\sf a}$. Since the
little group ${\cal G}_{D^{p}_{\sf a}}$ fixes the $p$-cell $D^{p}_{\sf a}$
pointwise,

$\displaystyle E^{1}_{p,-n}$ $\displaystyle\cong\bigoplus_{{\sf a}\in
I^{p}_{\rm orb}}{}^{\phi|_{D^{p}_{\sf a}}}\tilde{K}^{(z|_{D^{p}_{\sf
a}},c|_{D^{p}_{\sf a}})+p-n}_{{\cal G}_{D^{p}_{\sf a}}}(D^{p}_{\sf a}/\partial
D^{p}_{\sf a})$ $\displaystyle\cong\bigoplus_{{\sf a}\in I^{p}_{\rm
orb}}{}^{\phi|_{{\bm{x}}^{p}_{\sf a}}}K^{(z|_{{\bm{x}}^{p}_{\sf
a}},c|_{{\bm{x}}^{p}_{\sf a}})-n}_{{\cal G}_{{\bm{x}}^{p}_{\sf
a}}}(\\{{\bm{x}}^{p}_{\sf a}\\}).$ (108)

In the same way as the momentum-space AHSS, to represent the $K$-group with
degree $-n$, we use the suspension isomorphism

$\displaystyle{}^{\phi|_{{\bm{x}}^{p}_{\sf a}}}K_{(z|_{{\bm{x}}^{p}_{\sf
a}},c|_{{\bm{x}}^{p}_{\sf a}})-n}^{{\cal G}_{{\bm{x}}^{p}_{\sf
a}}}(\\{{\bm{x}}^{p}_{\sf a}\\})$
$\displaystyle\cong{}^{\phi|_{{\bm{x}}^{p}_{\sf
a}}}\tilde{K}_{(z|_{{\bm{x}}^{p}_{\sf a}},c|_{{\bm{x}}^{p}_{\sf a}})}^{{\cal
G}_{{\bm{x}}^{p}_{\sf a}}}(\\{{\bm{x}}^{p}_{\sf a}\\}\times\tilde{S}^{n}),$
(109)

where the little group ${\cal G}_{{\bm{x}}^{p}_{\sf a}}$ trivially acts on the
sphere $\tilde{S}^{n}$. To distinguish it from real space, we use the symbol
of the virtual sphere as $\tilde{S}^{n}$.

The first differential $d^{1}_{p,-n}$ is defined as the composition of the
homomorphisms below

$\displaystyle d^{1}_{p,-n}:{}^{\phi}K^{\cal G}_{(z,c)+p-n}(X_{p},X_{p-1})$
$\displaystyle\to{}^{\phi}K^{\cal G}_{(z,c)+p-n-1}(X_{p-1})$
$\displaystyle\to{}^{\phi}K^{\cal
G}_{(z,c)+p-n-1}(X_{p-1},X_{p-2})=E^{1}_{p-1,-n}.$ (110)

Here, the first homomorphism is the boundary map (physically, the bulk-
boundary correspondence) and the second one is from the inclusion
$X_{p-1}\to(X_{p-1},X_{p-2})$. The relation

$\displaystyle d_{1}^{p-1,-n}\circ d_{1}^{p,-n}=0$ (111)

holds and the the $E_{2}$-page is defined as

$\displaystyle E_{2}^{p,-n}:={\rm Ker\,}d_{1}^{p,-n}/{\rm
Im\,}d_{1}^{p+1,-n}.$ (112)

(For $p=3$, $E_{2}^{3.-n}={\rm Ker\,}d_{1}^{3,-n}$.) The physical meaning of
$d^{1}_{p,-n}$ is to measure how much of the boundary gapless state remains in
the adjacent $(p-1)$-cells for a gapped Hamiltonian inside $p$-cells.

Figure 2: A cell decomposition of real space for MSG $P2_{1}1^{\prime}$. The
figure shows the fundamental domain (3-cell). The 0-, 1-, and 2-cells are
shown with black, red, and blue Arabic numerals, respectively. The coordinates
of 0-cells are $(-\frac{1}{2},-\frac{1}{4},-\frac{1}{2})$,
$(-\frac{1}{2},\frac{1}{4},-\frac{1}{2})$,
$(\frac{1}{2},-\frac{1}{4},-\frac{1}{2})$,
$(\frac{1}{2},\frac{1}{4},-\frac{1}{2})$,
$(-\frac{1}{2},-\frac{1}{4},\frac{1}{2})$,
$(-\frac{1}{2},\frac{1}{4},\frac{1}{2})$,
$(\frac{1}{2},-\frac{1}{4},\frac{1}{2})$,
$(\frac{1}{2},\frac{1}{4},\frac{1}{2})$ in order.

It turns out that we only need to consider the $p$-cells intersecting the
closure of the fundamental domain (asymmetric unit). We introduce the integer
lattices $E^{0}_{p}$ for $p=0,1,2,3$ as follows. Let $D^{3}$ be the
fundamental domain. We define the set of relevant $p$-cells ${\cal C}_{p}$ as
before. (For example, see Fig. 2 for a choice of fundamental domain for the
MSG $P2_{1}1^{\prime}$ together with the boundary 0-, 1-, and 2-cells.) For
each $p$-cell, we pick a representative point ${\bm{x}}^{p}_{i}\in D^{p}_{i}$
for each $p$-cell. For each ${\bm{x}}$, introduce the left coset decomposition

$\displaystyle{\cal G}_{{\bm{x}}}={\cal G}_{{\bm{x}}}^{0}\coprod{\sf t}{\cal
G}_{\bm{x}}^{0}\coprod{\sf c}{\cal G}_{\bm{x}}^{0}\coprod\gamma{\cal
G}_{\bm{x}}^{0},$ (113)

where ${\cal G}_{\bm{x}}^{0}={\rm Ker\,}\phi\cap{\rm Ker\,}c\cap{\cal
G}_{\bm{x}}$, and

$\displaystyle{\sf t}\in\\{g\in{\cal G}_{\bm{x}}|-\phi_{g}=c_{g}=1\\},$ (114)
$\displaystyle{\sf c}\in\\{g\in{\cal G}_{\bm{x}}|-\phi_{g}=-c_{g}=1\\},$ (115)
$\displaystyle\gamma\in\\{g\in{\cal G}_{\bm{x}}|\phi_{g}=-c_{g}=1\\},$ (116)

are representatives. In general, ${\sf t},{\sf c}$, and $\gamma$ depend on the
${\bm{x}}\in\mathbb{R}^{3}$, but ${\bm{x}}$ is omitted unless misunderstanding
arises. We denote by
$\alpha_{1}({\bm{x}}),\dots,\alpha_{N_{\bm{x}}}({\bm{x}})$ the set of irreps
of ${\cal G}_{\bm{x}}^{0}$ with the factor system $z_{g,h}$. The lattice
$E^{0}_{p}$ is defined as the $\mathbb{Z}$-module generated by irreps inside
the relevant $p$-cells

$\displaystyle E_{0}^{p}:=\Braket{\bigcup_{D^{p}_{i}\in{\cal
C}_{p}}\bigcup_{r=1}^{N_{{\bm{x}}^{p}_{i}}}\alpha_{r}({\bm{x}}^{p}_{i})}.$
(117)

Denote by $\chi^{\alpha_{r}({\bm{x}}^{p}_{i})}_{g\in{\cal
G}_{{\bm{x}}^{p}_{i}}}$ the irreducible character of the irrep
$\alpha_{r}({\bm{x}}^{p}_{i})$. For each irrep $\alpha({\bm{x}})$ at
${\bm{x}}$, we identify the AZ class by the Wigner criteria

$\displaystyle W^{\alpha({\bm{x}})}_{T}:=\frac{1}{|{\cal
G}^{0}_{\bm{x}}|}\sum_{g\in{\sf t}{\cal
G}_{\bm{x}}^{0}}z_{g,g}\chi^{\alpha({\bm{x}})}_{g^{2}}\in\\{\pm 1,0\\},$ (118)
$\displaystyle W^{\alpha({\bm{x}})}_{C}:=\frac{1}{|{\cal
G}^{0}_{\bm{x}}|}\sum_{g\in{\sf
c}G_{\bm{x}}^{0}}z_{g,g}\chi^{\alpha({\bm{x}})}_{g^{2}}\in\\{\pm 1,0\\},$
(119) $\displaystyle W^{\alpha({\bm{x}})}_{\Gamma}:=\frac{1}{|{\cal
G}^{0}_{\bm{x}}|}\sum_{g\in
G_{\bm{x}}^{0}}(\chi^{\alpha({\bm{x}})}_{g})^{*}\frac{z_{g,\gamma}}{z_{\gamma,\gamma^{-1}g\gamma}}\chi^{\alpha({\bm{x}})}_{\gamma^{-1}g\gamma}$
$\displaystyle\hskip 50.0pt\in\\{1,0\\}.$ (120)

For each pattern of
$(W^{\alpha({\bm{x}})}_{T},W^{\alpha({\bm{x}})}_{C},W^{\alpha({\bm{x}})}_{\Gamma})$,
the AZ class and one of the $\mathbb{Z}$, $\mathbb{Z}_{2}$, or trivial
classification are listed in Table 2. For each irrep $\alpha({\bm{x}})$, if
$W^{\alpha({\bm{x}})}_{\Gamma}=1$, we pick an irrep $\alpha({\bm{x}})+$ of
${\cal G}^{0}_{\bm{x}}\coprod\gamma{\cal G}^{0}_{\bm{x}}$ so that
$\chi^{\alpha({\bm{x}})+}_{g\in{\cal
G}^{0}_{\bm{x}}}=\chi^{\alpha({\bm{x}})}_{g}$.

We will define the homomorphisms $\delta_{p},\delta^{\Gamma}_{p}:E_{0}^{p}\to
E_{0}^{p+1}$ in the same way as before. The integer $n_{i,i^{\prime}}\in\\{\pm
1,0\\}$ is defined as $n_{i,i^{\prime}}=1$ ($n_{i,i^{\prime}}=-1$) if
$D^{p}_{i^{\prime}}\in\partial D^{p+1}_{i}$ and the orientation of
$D^{p}_{i^{\prime}}$ agrees (disagrees, resp.), and $n_{i,i^{\prime}}=0$ else.
We define $[\delta_{p}]_{ir,i^{\prime}r^{\prime}}=0$ for
$D^{p}_{i^{\prime}}\notin\partial D^{p+1}_{i}$ and

$\displaystyle[\delta_{p}]_{ir,i^{\prime}r^{\prime}}$
$\displaystyle:=n_{i,i^{\prime}}\times\frac{1}{|{\cal
G}^{0}_{{\bm{x}}^{p+1}_{i}}|}$ $\displaystyle\sum_{g\in{\cal
G}^{0}_{{\bm{x}}^{p+1}_{i}}}(\chi^{\alpha_{r}({\bm{x}}^{p+1}_{i})}_{g})^{*}\chi^{\alpha_{r^{\prime}}({\bm{x}}^{p}_{i^{\prime}})}_{g}$
(121)

for $D^{p}_{i^{\prime}}\in\partial D^{p+1}_{i}$. We define
$\delta_{p}^{\Gamma}$ by

$\displaystyle[\delta^{\Gamma}_{p}]_{ir,i^{\prime}r^{\prime}}$
$\displaystyle:=n_{i,i^{\prime}}\times\frac{1}{|{\cal
G}^{0}_{{\bm{x}}^{p+1}_{i}}|}$ $\displaystyle\sum_{g\in\gamma{\cal
G}^{0}_{{\bm{x}}^{p+1}_{i}}}(\chi^{\alpha_{r}({\bm{x}}^{p+1}_{i})+}_{g})^{*}\chi^{\alpha_{r^{\prime}}({\bm{x}}^{p}_{i^{\prime}})+}_{g}$
(122)

when $D^{p}_{i^{\prime}}\in\partial D^{p+1}_{i}$ and
$W^{\alpha({\bm{k}}^{p}_{i^{\prime}})}_{\Gamma}=W^{\alpha({\bm{k}}^{p+1}_{i})}_{\Gamma}=1$
and $[\delta^{\Gamma}_{p}]_{ir,i^{\prime}r^{\prime}}=0$ else.

For an irrep $\alpha({\bm{x}})$ of ${\cal G}^{0}_{\bm{x}}$ at ${\bm{x}}$ and
$h\in{\cal G}$, we introduce the mapped irrep $h[\alpha({\bm{x}})]$ by $h$,
which is an irrep of ${\cal G}^{0}_{h({\bm{x}})}=h{\cal G}^{0}_{\bm{x}}h^{-1}$
at $p_{h}{\bm{x}}+\bm{t}_{h}\in X$ whose character is

$\displaystyle\chi^{h[\alpha({\bm{x}})]}_{g\in h{\cal
G}^{0}_{\bm{x}}h^{-1}}=\frac{z_{g,h}}{z_{h,h^{-1}gh}}\chi^{\alpha({\bm{x}})}_{h^{-1}gh}.$
(123)

When $W^{\alpha({\bm{x}})}_{\Gamma}=1$, the mapped irrep
$h[\alpha({\bm{x}})+]$ is defined similarly. We define the sign
$s_{h}\in\\{\pm 1\\}$ as before.

$\displaystyle s_{h}=\frac{1}{|{\cal G}^{0}_{\bm{x}}|}\sum_{g\in h(\gamma{\cal
G}^{0}_{\bm{x}})h^{-1}}(\chi^{h[\alpha({\bm{x}})]+}_{g})^{*}\chi^{h[\alpha({\bm{x}})+]}_{g},$
(124)

which measures the relative sign between the two chiral operators at
$h({\bm{x}})$ as in

$\displaystyle
u_{h}(\Gamma_{\alpha({\bm{x}})})^{\phi_{h}}u_{h}=s_{h}\Gamma_{h[\alpha({\bm{x}})]}.$
(125)

The new ingredient specific to the real-space AHSS is the homomorphisms
$\delta_{p}^{T}$ and $\delta_{p}^{\Gamma,T}$, which are needed in the MSGs of
types III and IV. We define

$\displaystyle[\delta^{T}_{p}]_{ir,i^{\prime}r^{\prime}}$
$\displaystyle:=n_{i,i^{\prime}}\times\frac{1}{|{\cal
G}^{0}_{{\bm{x}}^{p+1}_{i}}|}$ $\displaystyle\sum_{g\in{\cal
G}^{0}_{{\bm{x}}^{p+1}_{i}}}(\chi^{\alpha_{r}({\bm{x}}^{p+1}_{i})}_{g})^{*}\chi^{{\sf
t}[\alpha_{r^{\prime}}({\bm{x}}^{p}_{i^{\prime}})]}_{g}$ (126)

only if the following conditions are fulfilled, and $0$ otherwise:
$D^{p}_{i^{\prime}}\in\partial D^{p+1}_{i}$, ${\sf t}{\cal
G}^{0}_{{\bm{x}}^{p+1}_{i}}=\emptyset$, and ${\sf t}{\cal
G}^{0}_{{\bm{x}}^{p}_{i^{\prime}}}\neq\emptyset$. Similarly, we define

$\displaystyle[\delta^{\Gamma,T}_{p}]_{ir,i^{\prime}r^{\prime}}$
$\displaystyle:=n_{i,i^{\prime}}\times\frac{1}{|{\cal
G}^{0}_{{\bm{x}}^{p+1}_{i}}|}$ $\displaystyle\sum_{g\in\gamma{\cal
G}^{0}_{{\bm{x}}^{p+1}_{i}}}(\chi^{\alpha_{r}({\bm{x}}^{p+1}_{i})+}_{g})^{*}\chi^{{\sf
t}[\alpha_{r^{\prime}}({\bm{x}}^{p}_{i^{\prime}})+]}_{g}$ (127)

only if the following conditions are fulfilled, and $0$ otherwise:
$D^{p}_{i^{\prime}}\in\partial D^{p+1}_{i}$, $\gamma{\cal
G}^{0}_{{\bm{x}}^{p}_{i^{\prime}}}\supset\gamma{\cal
G}^{0}_{{\bm{x}}^{p+1}_{i}}\neq\emptyset$, ${\sf t}{\cal
G}^{0}_{{\bm{x}}^{p+1}_{i}}=\emptyset$, and ${\sf t}{\cal
G}^{0}_{{\bm{x}}^{p}_{i^{\prime}}}\neq\emptyset$.

###  V.2 $E^{1}$ page

In the view of (109), $E^{1}_{p,-n}$ is generated by an orbit of
$n$-dimensional massive Dirac Hamiltonians over the virtual $n$-sphere labeled
by $p$-cells as in

$\displaystyle
H_{{\bm{x}}^{p}_{i}}(\tilde{\bm{x}})=\sum_{\mu=1}^{n}-i\tilde{\partial}_{\mu}\gamma_{\mu}+m\gamma_{0},$
$\displaystyle\tilde{\bm{x}}\in\\{{\bm{x}}^{p}_{i}\\}\times\tilde{S}^{n}.$
(128)

Here, $\tilde{\partial}_{\mu}=\frac{\partial}{\partial\tilde{x}_{\mu}}$. The
Hamiltonians for equivalent $p$-cells are related as

$\displaystyle
H_{g({\bm{x}}^{p}_{i})}(\tilde{\bm{x}})=c_{g}u_{g}H_{{\bm{x}}^{p}_{i}}(\tilde{\bm{x}})^{\phi_{g}}u_{g}^{-1},$
$\displaystyle\tilde{\bm{x}}\in\\{g({\bm{x}}^{p}_{i})\\}\times\tilde{S}^{n},$
(129)

for $g\in{\cal G}$. For a given irrep $\alpha({\bm{x}}^{p}_{i})$ of $p$-cell
$D^{p}_{i}$, the classification of the mass term $m\gamma_{0}$ is determined
according to Table2, which is the same as the periodic table of topological
insulators and superconductors [2, 3]. In Table 2, the matrix size of the
generating Dirac Hamiltonian is shown on the right of the parentheses.

$\begin{array}[]{c|ccc|cccccccccccc}&W^{\alpha}_{T}&W^{\alpha}_{C}&W^{\alpha}_{\Gamma}&n=0&n=1&n=2&n=3&n=4&n=5&n=6&n=7\\\
\hline\cr{\rm
A}&0&0&0&(\mathbb{Z},1)&0&(\mathbb{Z},2)&0&(\mathbb{Z},4)&0&(\mathbb{Z},8)&0\\\
{\rm
AIII}&0&0&1&0&(\mathbb{Z},2)&0&(\mathbb{Z},4)&0&(\mathbb{Z},8)&0&(\mathbb{Z},16)\\\
\hline\cr{\rm
AI}&1&0&0&(\mathbb{Z},1)&0&0&0&(\mathbb{Z},8)&0&(\mathbb{Z}_{2},16)&(\mathbb{Z}_{2},16)\\\
{\rm
BDI}&1&1&1&(\mathbb{Z}_{2},2)&(\mathbb{Z},2)&0&0&0&(\mathbb{Z},16)&0&(\mathbb{Z}_{2},32)\\\
{\rm
D}&0&1&0&(\mathbb{Z}_{2},2)&(\mathbb{Z}_{2},2)&(\mathbb{Z},2)&0&0&0&(\mathbb{Z},16)&0\\\
{\rm
DIII}&-1&1&1&0&(\mathbb{Z}_{2},4)&(\mathbb{Z}_{2},4)&(\mathbb{Z},4)&0&0&0&(\mathbb{Z},32)\\\
{\rm
AII}&-1&0&0&(\mathbb{Z},2)&0&(\mathbb{Z}_{2},4)&(\mathbb{Z}_{2},4)&(\mathbb{Z},4)&0&0&0\\\
{\rm
CII}&-1&-1&1&0&(\mathbb{Z},4)&0&(\mathbb{Z}_{2},8)&(\mathbb{Z}_{2},8)&(\mathbb{Z},8)&0&0\\\
{\rm
C}&0&-1&0&0&0&(\mathbb{Z},4)&0&(\mathbb{Z}_{2},8)&(\mathbb{Z}_{2},8)&(\mathbb{Z},8)&0\\\
{\rm
CI}&1&-1&1&0&0&0&(\mathbb{Z},8)&0&(\mathbb{Z}_{2},16)&(\mathbb{Z}_{2},16)&(\mathbb{Z},16)\\\
\hline\cr{\rm mapped\
rep.}&&&&c_{h}&s_{h}&\phi_{h}c_{h}&s_{h}\phi_{h}&c_{h}&s_{h}&\phi_{h}c_{h}&s_{h}\phi_{h}\\\
\end{array}$ Table 2: Wignar criteria, AZ classes, and classification of
gapped Hamiltonian over the real-space $n$-sphere. In the table,
$0,\mathbb{Z}$ and $\mathbb{Z}_{2}$ indicate the classification, and when the
classification is nontrivial, the matrix size of the generator Dirac
Hamiltonian is shown together. The last row shows the sign change of the
$\mathbb{Z}$ topological invariant under the action of the group element $h$.

####  V.2.1 $\mathbb{Z}$ classification

For each irrep of each orbit, we construct a vector $\vec{a}^{p,-n}\in
E^{0}_{p}$. Pick a representative $p$-cell $D^{p}_{i}$ of an orbit of
equivalent $p$-cells in ${\cal C}_{p}$. For an irrep
$\alpha_{r}({\bm{x}}^{p}_{i})$, if the classification of degree $-n$ is
$\mathbb{Z}$, we set $a^{p,-n}_{ir}$ to be the matrix size of generator Dirac
Hamiltonian listed in Table 2. The components for other equivalent irreps
$h[\alpha({\bm{x}}^{p}_{i})]$ are determined by the relation (129).

For even $n$, the $\mathbb{Z}$ invariant is the Chern number (78). Since the
Chern number of the unoccupied band is $-ch_{n/2}$, we have the factor
$c_{h}$. Moreover, since the Berry curvature ${\cal F}$ changes its sign if
$h$ is antiunitary, we have the factor $\phi_{h}$ when $n\in 4\mathbb{Z}+2$.
No sign change arises from the flip of momenta ${\bm{k}}\to-{\bm{k}}$.

For odd $n$, the $\mathbb{Z}$ invariant is the winding number (79) with the
chiral operator $\Gamma_{\alpha({\bm{x}}^{p}_{i})}$. We have the factor
$s_{h}$ for all $n\in 2\mathbb{Z}+1$ and $\phi_{h}$ for $n\in 4\mathbb{Z}+1$
as before. Moreover, from the flip of momenta ${\bm{k}}\to-{\bm{k}}$, we have
the additional factor $\phi_{h}$ for all $n\in 2\mathbb{Z}+1$.

Let us introduce $h(i)$ and $h(r)$ as before, i.e.,
$D^{p}_{h(i)}=h(D^{p}_{i})$ and
$\alpha_{h(r)}(h({\bm{x}}^{p}_{h(i)}))=h[\alpha_{r}({\bm{x}}^{p}_{i})]$. In
sum, we have

$\displaystyle
a^{p,-n}_{h(i)h(r)}=a^{p,-n}_{ir}\times\left\\{\begin{array}[]{ll}c_{h}&(n=0),\\\
s_{h}&(n=1),\\\ \phi_{h}c_{h}&(n=2),\\\ s_{h}\phi_{h}&(n=3),\\\
c_{h}&(n=4),\\\ s_{h}&(n=5),\\\ \phi_{h}c_{h}&(n=6),\\\
s_{h}\phi_{h}&(n=7).\\\ \end{array}\right.$ (138)

The last factor in (138) is also shown in Table 2. Constructing the vectors
$\vec{a}^{p,-n}_{1},\vec{a}^{p,-n}_{2},\dots,$ for all inequivalent irreps and
inequivalent orbits, we have the sublattice

$\displaystyle
E^{1\mathbb{Z}}_{p,-n}:=\braket{\vec{a}^{p,-n}_{1},\vec{a}^{p,-n}_{2},\dots}\subset
E^{0}_{p}.$ (139)

####  V.2.2 $\mathbb{Z}_{2}$ classification

For an irrep $\alpha_{r}({\bm{x}}^{p}_{i})$, if the classification of degree
$-n$ is $\mathbb{Z}_{2}$, set $b^{p,-n}_{ir}={\rm dim\,}(H)$, the matrix
dimension listed in Table 2. For other equivalent irreps
$h[\alpha({\bm{x}}^{p}_{i})]$ at $h({\bm{x}})\in X$, set
$b^{p,-n}_{h(i)h(r)}=b^{p,-n}_{ir}$. Constructing the vectors
$\vec{b}^{p,-n}_{1},\vec{b}^{p,-n}_{2},\dots,$ for all inequivalent irreps and
inequivalent orbits, we have

$\displaystyle
E^{1\mathbb{Z}_{2}}_{p,-n}:=\braket{\vec{b}^{p,-n}_{1},\vec{b}^{p,-n}_{2},\dots}\subset
E^{0}_{p}$ (140)

and ${\cal P}^{1}_{p,-n}=\\{2x\in E^{0}_{p}|x\in
E^{1\mathbb{Z}_{2}}_{p,-n}\\}$.

The group $E^{1}_{p,-n}$ is represented as the quotient group

$\displaystyle E^{1}_{p,-n}=(E^{1\mathbb{Z}}_{p,-n}\oplus
E^{1\mathbb{Z}_{2}}_{p,-n})/{\cal P}_{p,-n}^{1}.$ (141)

By construction, $E^{1\mathbb{Z}}_{p,-n}\cap
E^{1\mathbb{Z}_{2}}_{p,-n}=\\{\bm{0}\\}$. We write
$\tilde{E}^{1}_{p,-n}=E^{1\mathbb{Z}}_{p,-n}\oplus
E^{1\mathbb{Z}_{2}}_{p,-n}$.

###  V.3 The first differential $d^{1}$

Let us focus on the ${\sf a}$-th and ${\sf b}$-th orbits of $p$\- and
$(p-1)$-cells, respectively. Let $D^{p}_{\sf a}$ and $D^{p-1}_{\sf b}$ be
representative of each orbit. The small block of the first differential is

$\displaystyle[d^{1}_{p,-n}]_{\sf ab}:{}^{\phi}K^{{\cal G}}_{(z,c)+p-n}({\cal
G}/{\cal G}_{D^{p}_{\sf a}}\times D^{p}_{\sf a},{\cal G}/{\cal G}_{D^{p}_{\sf
a}}\times\partial D^{p}_{\sf a})$ $\displaystyle\to{}^{\phi}K^{{\cal
G}}_{(z,c)+(p-1)-n}({\cal G}/{\cal G}_{D^{p-1}_{\sf b}}\times D^{p-1}_{\sf
b},{\cal G}/{\cal G}_{D^{p-1}_{\sf b}}\times\partial D^{p-1}_{\sf b}).$ (142)

This is done by “expanding” the orbit of Hamiltonians over the $p$-cells
${\cal G}({\cal G}/{\cal G}_{D^{p}_{\sf a}}\times D^{p}_{\sf a})$ by the orbit
of Hamiltonians over the $(p-1)$-cells ${\cal G}({\cal G}/{\cal
G}_{D^{p-1}_{\sf b}}\times D^{p-1}_{\sf b})$. 111 Note that, as a matter of
mathematical fact, for a subgroup $H\subset G$, the group $G$ naturally acts
on the set of left cosets $G/H$ of $H$ in $G$: Let
$G/H=\prod_{\sigma}g_{\sigma}H$ be a complete set of left coset
representatives. For $k\in G$, we have a unique representative $g_{k(\sigma)}$
such that $kg_{\sigma}=g_{k(\sigma)}h$, with $h\in H$. The map $\sigma\mapsto
k(\sigma)$ is well-defined, as for $h^{\prime}\in H$,
$kg_{\sigma}h=g_{k(\sigma)}hh^{\prime}\in g_{k(\sigma)}H$. The orbit of the
action $G$ on $G/H$ in this sense is denoted by $G(G/H)$. There, each boundary
of the $p$-cell $D^{p}_{\sf a}$, $D^{p-1}_{i}\in\partial D^{p}_{\sf a}$ say,
contribute to $[d^{1}_{p,-n}]_{\sf ab}$ if the $(p-1)$-cell $D^{p-1}_{i}$ is
in the $b$-th orbit ${\cal G}({\cal G}/{\cal G}_{D^{p-1}_{\sf b}}\times
D^{p-1}_{\sf b})$. To implement this, we write the boundary of $D^{p}_{\sf a}$
as the sum of orbit of $(p-1)$-cells as in

$\displaystyle\partial D^{p}_{\sf a}$ $\displaystyle=\sum_{{\sf b}\in
I^{p-1}_{\rm orb}}\sum_{h\in{\cal G},h(D^{p-1}_{\sf b})\in\partial D^{p}_{\sf
a}}n_{{\sf a},h({\sf b})}h(D^{p-1}_{\sf b}).$ (143)

See Fig. 3. It is important to note that since PHS is an internal symmetry the
sum of $h$ can be limited to $h\in{\cal G}^{0}\coprod{\sf t}{\cal G}^{0}$ for
the symmetry classes discussed in this paper.

Figure 3: The relationship between the boundary $(p-1)$-cells of the
representative $p$-cell $D^{p}_{\sf a}$ and the ${\sf b}$th orbit of
$(p-1)$-cells.

Fix $h\in{\cal G}^{0}\coprod{\sf t}{\cal G}^{0}$ such that $h(D^{p-1}_{\sf
b})\in\partial D^{p}_{\sf a}$. For an irrep $\alpha_{r}({\bm{x}}^{p}_{\sf a})$
of ${\cal G}^{0}_{{\bm{x}}^{p}_{\sf a}}$, the mapped irrep
$h^{-1}[\alpha_{r}({\bm{x}}^{p}_{\sf
a})]=\alpha_{h^{-1}(r)}(h^{-1}({\bm{x}}^{p}_{\sf a}))$ is the irrep at
$h^{-1}({\bm{x}}^{p}_{\sf a})$. Let $\\{g_{\sigma}\\}_{\sigma=1,\dots,|{\cal
G}_{{\bm{x}}^{p-1}_{\sf b}}|/|{\cal G}_{{\bm{x}}^{p}_{\sf a}}|}$ be a complete
set of the left coset representatives of ${\cal G}_{h^{-1}({\bm{x}}^{p}_{\sf
a})}$ in ${\cal G}_{{\bm{x}}^{p-1}_{\sf b}}$, say

$\displaystyle{\cal G}_{{\bm{x}}^{p-1}_{\sf b}}=\prod_{\sigma=1}^{|{\cal
G}_{{\bm{x}}^{p-1}_{\sf b}}|/|{\cal G}_{{\bm{x}}^{p}_{\sf a}}|}g_{\sigma}{\cal
G}_{h^{-1}({\bm{x}}^{p}_{\sf a})}.$ (144)

Here, $g_{\sigma}$ takes an element in the group according to the following
cases: $g_{\sigma}\in{\cal G}^{0}\coprod{\sf t}{\cal G}^{0}$ if ${\sf t}{\cal
G}_{{\bm{x}}^{p-1}_{\sf b}}\neq\emptyset$ and ${\sf t}{\cal
G}_{{\bm{x}}^{p}_{\sf a}}=\emptyset$, and $s_{\sigma}\in{\cal G}^{0}$
otherwise. The $p$-cells

$\displaystyle(g_{\sigma}h^{-1})(D^{p}_{\sf a}),\quad\sigma=1,\dots,|{\cal
G}_{{\bm{x}}^{p-1}_{\sf b}}|/|{\cal G}_{{\bm{x}}^{p}_{\sf a}}|,$ (145)

adjacent to the representative $(p-1)$-cell $D^{p-1}_{\sf b}$. We introduce
the “induced representation” of $h^{-1}[\alpha_{r}({\bm{x}}^{p}_{\sf a})]$ by
$g_{\sigma}$s, which is a representation of ${\cal G}_{{\bm{x}}^{p-1}_{\sf
b}}$, as in

$\displaystyle{\rm Ind}_{\\{g_{\sigma}\\}}(h^{-1}[\alpha_{r}({\bm{x}}^{p}_{\sf
a})])$ $\displaystyle:=\bigoplus_{\sigma=1}^{|{\cal G}_{{\bm{x}}^{p-1}_{\sf
b}}|/|{\cal G}_{{\bm{x}}^{p}_{\sf
a}}|}(g_{\sigma}h^{-1})[\alpha_{r}({\bm{x}}^{p}_{\sf a})],$ (146)

whose character is

$\displaystyle\chi^{{\rm
Ind}_{\\{g_{\sigma}\\}}(h^{-1}[\alpha_{r}({\bm{x}}^{p}_{\sf a})])}_{g\in{\cal
G}^{0}_{{\bm{x}}^{p-1}_{\sf b}}}$ $\displaystyle=\sum_{\sigma=1}^{|{\cal
G}_{{\bm{x}}^{p-1}_{\sf b}}|/|{\cal G}_{{\bm{x}}^{p}_{\sf
a}}|}\delta_{g\in{\cal G}^{0}_{(g_{\sigma}h^{-1})({\bm{x}}^{p}_{\sf
a})}}\chi^{(g_{\sigma}h^{-1})[\alpha_{r}({\bm{x}}^{p}_{\sf a})]}_{g}.$ (147)

Note that when $\\{g_{\sigma}\\}_{\sigma=1,\dots,|{\cal
G}_{{\bm{x}}^{p-1}_{\sf b}}|/|{\cal G}_{{\bm{x}}^{p}_{\sf a}}|}\subset{\cal
G}^{0}$, this is the standard induced representation ${\rm Ind}_{{\cal
G}^{0}_{h^{-1}({\bm{x}}^{p}_{\sf a})}}^{{\cal G}^{0}_{{\bm{x}}^{p-1}_{\sf
b}}}(h^{-1}[\alpha_{r}({\bm{x}}^{p}_{\sf a})])$. We expand the representation
${\rm Ind}_{\\{g_{\sigma}\\}}(h^{-1}[\alpha_{r}({\bm{x}}^{p}_{\sf a})])$ by
the irreps $\beta_{r^{\prime}}({\bm{x}}^{p-1}_{\sf b})$ at
${\bm{x}}^{p-1}_{\sf b}$. Let us denote the inner product between two reps
$\alpha,\beta$ of $G_{0}$ by
$\braket{\alpha,\beta}=\frac{1}{|G_{0}|}\sum_{g\in
G_{0}}(\chi^{\alpha}_{g})^{*}\chi^{\beta}_{g}\in\mathbb{Z}_{\geq 0}$. Then,
the expansion coefficient is given by

$\displaystyle\Braket{\beta_{r^{\prime}}({\bm{x}}^{p-1}_{\sf b}),{\rm
Ind}_{\\{g_{\sigma}\\}}(h^{-1}[\alpha_{r}({\bm{x}}^{p}_{\sf a})])}_{{\cal
G}^{0}_{{\bm{x}}^{p-1}_{\sf b}}}$ $\displaystyle=\sum_{\sigma=1}^{|{\cal
G}_{{\bm{x}}^{p-1}_{\sf b}}|/|{\cal G}_{{\bm{x}}^{p}_{\sf a}}|}\frac{1}{|{\cal
G}^{0}_{{\bm{x}}^{p-1}_{\sf b}}|}\sum_{g\in{\cal G}^{0}_{{\bm{x}}^{p-1}_{\sf
b}}}(\chi^{\beta_{r^{\prime}}({\bm{x}}^{p-1}_{\sf
b})}_{g})^{*}\delta_{g\in{\cal G}^{0}_{(g_{\sigma}h^{-1})({\bm{x}}^{p}_{\sf
a})}}\chi^{(g_{\sigma}h^{-1})[\alpha_{r}({\bm{x}}^{p}_{\sf a})]}_{g}$
$\displaystyle=\sum_{\sigma=1}^{|{\cal G}_{{\bm{x}}^{p-1}_{\sf b}}|/|{\cal
G}_{{\bm{x}}^{p}_{\sf a}}|}\frac{1}{|{\cal G}^{0}_{{\bm{x}}^{p-1}_{\sf
b}}|}\sum_{g\in{\cal G}^{0}_{(g_{\sigma}h^{-1})({\bm{x}}^{p}_{\sf
a})}}(\chi^{\beta_{r^{\prime}}({\bm{x}}^{p-1}_{\sf
b})}_{g})^{*}\chi^{(g_{\sigma}h^{-1})[\alpha_{r}({\bm{x}}^{p}_{\sf a})]}_{g}$
$\displaystyle=\frac{|{\cal G}^{0}_{{\bm{x}}^{p}_{\sf a}}|}{|{\cal
G}^{0}_{{\bm{x}}^{p-1}_{\sf b}}|}\sum_{\sigma=1}^{|{\cal
G}_{{\bm{x}}^{p-1}_{\sf b}}|/|{\cal G}_{{\bm{x}}^{p}_{\sf
a}}|}\Braket{\beta_{r^{\prime}}({\bm{x}}^{p-1}_{\sf b})}{{}_{{\cal
G}^{0}_{(g_{\sigma}h^{-1})({\bm{x}}^{p}_{\sf
a})}},(g_{\sigma}h^{-1})[\alpha_{r}({\bm{x}}^{p}_{\sf a})]}_{{\cal
G}^{0}_{(g_{\sigma}h^{-1})({\bm{x}}^{p}_{\sf a})}}$
$\displaystyle=\frac{|{\cal G}^{0}_{{\bm{x}}^{p}_{\sf a}}|}{|{\cal
G}^{0}_{{\bm{x}}^{p-1}_{\sf b}}|}\sum_{\sigma=1}^{|{\cal
G}_{{\bm{x}}^{p-1}_{\sf b}}|/|{\cal G}_{{\bm{x}}^{p}_{\sf
a}}|}\Braket{(hg_{\sigma}^{-1})[\beta_{r^{\prime}}({\bm{x}}^{p-1}_{\sf
b})]}{{}_{{\cal G}^{0}_{{\bm{x}}^{p}_{\sf a}}},\alpha_{r}({\bm{x}}^{p}_{\sf
a})}_{{\cal G}^{0}_{{\bm{x}}^{p}_{\sf a}}}.$ (148)

Because $g_{\sigma}\in{\cal G}^{0}_{{\bm{x}}^{p-1}_{\sf b}}\coprod{\sf t}{\cal
G}^{0}_{{\bm{x}}^{p-1}_{\sf b}}$, the equivalent class of the mapped irrep
$g_{\sigma}^{-1}[\beta_{r^{\prime}}({\bm{x}}^{p-1}_{\sf b})]$ depends only on
which $g_{\sigma}$ belongs to ${\cal G}^{0}_{{\bm{x}}^{p-1}_{\sf b}}$ or ${\sf
t}{\cal G}^{0}_{{\bm{x}}^{p-1}_{\sf b}}$. Namely,
$g_{\sigma}^{-1}[\beta_{r^{\prime}}({\bm{x}}^{p-1}_{\sf
b})]\sim\beta_{r^{\prime}}({\bm{x}}^{p-1}_{\sf b})$ for $g_{\sigma}\in{\cal
G}^{0}_{{\bm{x}}^{p-1}_{\sf b}}$ and
$g_{\sigma}^{-1}[\beta_{r^{\prime}}({\bm{x}}^{p-1}_{\sf b})]\sim{\sf
t}[\beta_{r^{\prime}}({\bm{x}}^{p-1}_{\sf b})]$ for $g_{\sigma}\in{\sf t}{\cal
G}^{0}_{{\bm{x}}^{p-1}_{\sf b}}$. Therefore,

$\displaystyle\Braket{\beta_{r^{\prime}}({\bm{x}}^{p-1}_{\sf b}),{\rm
Ind}_{\\{g_{\sigma}\\}}(h^{-1}[\alpha_{r}({\bm{x}}^{p}_{\sf a})])}_{{\cal
G}^{0}_{{\bm{x}}^{p-1}_{\sf b}}}$
$\displaystyle=\left\\{\begin{array}[]{ll}\Braket{\left\\{h[\beta_{r^{\prime}}({\bm{x}}^{p-1}_{\sf
b})]\oplus(h{\sf t})[\beta_{r^{\prime}}({\bm{x}}^{p-1}_{\sf
b})]\right\\}}{{}_{{\cal G}^{0}_{{\bm{x}}^{p}_{\sf
a}}},\alpha_{r}({\bm{x}}^{p}_{\sf a})}_{{\cal G}^{0}_{{\bm{x}}^{p}_{\sf
a}}}&({\sf t}{\cal G}_{{\bm{x}}^{p-1}_{\sf b}}\neq\emptyset,{\sf t}{\cal
G}_{{\bm{x}}^{p}_{\sf a}}=\emptyset),\\\
\Braket{h[\beta_{r^{\prime}}({\bm{x}}^{p-1}_{\sf b})]}{{}_{{\cal
G}^{0}_{{\bm{x}}^{p}_{\sf a}}},\alpha_{r}({\bm{x}}^{p}_{\sf a})}_{{\cal
G}^{0}_{{\bm{x}}^{p}_{\sf a}}}&({\rm else}).\\\ \end{array}\right.$ (151)

Since (151) is a slight extension of the Frobenius reciprocity, we will refer
to it as the Frobenius reciprocity as well. Using the formula (151) and the
connectivity (143) of representative $p$-cells, we find that the irrep
$\alpha_{r}({\bm{x}}^{p}_{\sf a})$ at the $p$-cell $D^{p}_{\sf a}$ contributes
to the irrep $\beta_{r^{\prime}}({\bm{x}}^{p-1}_{\sf b})$ at the $(p-1)$-cell
$D^{p-1}_{\sf b}$ by the integer

$\displaystyle\sum_{h\in{\cal G},h(D^{p-1}_{\sf b})\in\partial D^{p}_{\sf
a}}[\delta_{p-1}+\delta^{T}_{p-1}]_{{\sf a}r,h({\sf b})r^{\prime}}.$ (152)

(Note that (V.1) and (V.1) involve the connectivity $n_{i,i^{\prime}}$.) Here,
we have used $h{\sf t}[\beta_{r^{\prime}}({\bm{x}}^{p-1}_{\sf b})]\sim({\sf
t}^{\prime}h)[\beta_{r^{\prime}}({\bm{x}}^{p-1}_{\sf b})]$ with ${\sf
t}^{\prime}\in{\sf t}{\cal G}^{0}_{h({\bm{x}}^{p-1}_{\sf b})}$. Since the
$(p-1)$-cell $h(D^{p-1}_{\sf b})=(h{\sf t})(D^{p-1}_{\sf b})$ is a boundary of
$D^{p}_{\sf a}$, all the $p$\- and $(p-1)$-cells in the expression (151) are
in the relevant $p$-cells ${\cal C}_{p}$, meaning that the first differential
$d^{1}_{p,-n}$ can be computed in the lattice $E^{p}_{0}$.

In the same way, for a pair of irreps $\alpha_{r}({\bm{x}}^{p}_{\sf a})$ and
$\beta_{r^{\prime}}({\bm{x}}^{p}_{\sf a})$ with the values
$W_{\Gamma}^{\alpha_{r}({\bm{x}}^{p}_{\sf
a})}=W_{\Gamma}^{\beta_{r^{\prime}}({\bm{x}}^{p-1}_{\sf b})}=1$ so that the
winding number is well-defined for odd $n$, we have

$\displaystyle\Braket{\beta_{r^{\prime}}({\bm{x}}^{p-1}_{\sf b})+,{\rm
Ind}_{\\{g_{\sigma}\\}}(h^{-1}[\alpha_{r}({\bm{x}}^{p}_{\sf a})+])}_{{\cal
G}^{0}_{{\bm{x}}^{p-1}_{\sf b}}\coprod\gamma{\cal G}^{0}_{{\bm{x}}^{p-1}_{\sf
b}}}$
$\displaystyle=\left\\{\begin{array}[]{ll}\Braket{\left\\{h[\beta_{r^{\prime}}({\bm{x}}^{p-1}_{\sf
b})+]\oplus(h{\sf t})[\beta_{r^{\prime}}({\bm{x}}^{p-1}_{\sf
b})+]\right\\}}{{}_{{\cal G}^{0}_{{\bm{x}}^{p-1}_{\sf b}}\coprod\gamma{\cal
G}^{0}_{{\bm{x}}^{p-1}_{\sf b}}},\alpha_{r}({\bm{x}}^{p}_{\sf a})+}_{{\cal
G}^{0}_{{\bm{x}}^{p-1}_{\sf b}}\coprod\gamma{\cal G}^{0}_{{\bm{x}}^{p-1}_{\sf
b}}}&({\sf t}{\cal G}_{{\bm{x}}^{p-1}_{\sf b}}\neq\emptyset,{\sf t}{\cal
G}_{{\bm{x}}^{p}_{\sf a}}=\emptyset),\\\
\Braket{h[\beta_{r^{\prime}}({\bm{x}}^{p-1}_{\sf b})+]}{{}_{{\cal
G}^{0}_{{\bm{x}}^{p-1}_{\sf b}}\coprod\gamma{\cal G}^{0}_{{\bm{x}}^{p-1}_{\sf
b}}},\alpha_{r}({\bm{x}}^{p}_{\sf a})+}_{{\cal G}^{0}_{{\bm{x}}^{p-1}_{\sf
b}}\coprod\gamma{\cal G}^{0}_{{\bm{x}}^{p-1}_{\sf b}}}&({\rm else}),\\\
\end{array}\right.$ (155)

and restricting the summation to elements of chiral type and subtracting
(151), we obtain

$\displaystyle\Braket{\beta_{r^{\prime}}({\bm{x}}^{p-1}_{\sf b})+,{\rm
Ind}_{\\{g_{\sigma}\\}}(h^{-1}[\alpha_{r}({\bm{x}}^{p}_{\sf
a})+])}_{\gamma{\cal G}^{0}_{{\bm{x}}^{p-1}_{\sf b}}}$
$\displaystyle=\left\\{\begin{array}[]{ll}\Braket{\left\\{h[\beta_{r^{\prime}}({\bm{x}}^{p-1}_{\sf
b})+]\oplus(h{\sf t})[\beta_{r^{\prime}}({\bm{x}}^{p-1}_{\sf
b})+]\right\\}}{{}_{{\cal G}^{0}_{{\bm{x}}^{p-1}_{\sf b}}\coprod\gamma{\cal
G}^{0}_{{\bm{x}}^{p-1}_{\sf b}}},\alpha_{r}({\bm{x}}^{p}_{\sf
a})+}_{\gamma{\cal G}^{0}_{{\bm{x}}^{p-1}_{\sf b}}}&({\sf t}{\cal
G}_{{\bm{x}}^{p-1}_{\sf b}}\neq\emptyset,{\sf t}{\cal G}_{{\bm{x}}^{p}_{\sf
a}}=\emptyset),\\\ \Braket{h[\beta_{r^{\prime}}({\bm{x}}^{p-1}_{\sf
b})+]}{{}_{{\cal G}^{0}_{{\bm{x}}^{p-1}_{\sf b}}\coprod\gamma{\cal
G}^{0}_{{\bm{x}}^{p-1}_{\sf b}}},\alpha_{r}({\bm{x}}^{p}_{\sf
a})+}_{\gamma{\cal G}^{0}_{{\bm{x}}^{p-1}_{\sf b}}}&({\rm else}).\\\
\end{array}\right.$ (158)

Here, we have introduced the notation $\braket{\alpha+,\beta+}_{\gamma
G_{0}}:=\frac{1}{|G_{0}|}\sum_{g\in\gamma
G_{0}}(\chi^{\alpha+}_{g})^{*}\chi^{\beta+}_{g}\in\mathbb{Z}$.

With the preparation above, we develop the formula to compute $d^{1}_{p,-n}$.
We want to compute the expansion coefficients

$\displaystyle
d^{1}_{p,-n}(\vec{a}^{p,-n}_{\lambda})=\sum_{\kappa}\vec{a}^{p-1,-n}_{\kappa}[M^{p,-n}_{\mathbb{Z}\mathbb{Z}}]_{\kappa\lambda}+\sum_{\kappa}\vec{b}^{p-1,-n}_{\kappa}[M^{p,-n}_{\mathbb{Z}\mathbb{Z}_{2}}]_{\kappa\lambda},$
(159) $\displaystyle
d^{1}_{p,-n}(\vec{b}^{p,-n}_{\lambda})=\sum_{\kappa}\vec{b}^{p-1,-n}_{\kappa}[M^{p,-n}_{\mathbb{Z}_{2}\mathbb{Z}_{2}}]_{\kappa\lambda},$
(160)

where $[M^{p,-n}_{\mathbb{Z}\mathbb{Z}}]_{\kappa\lambda}\in\mathbb{Z}$ and
$[M^{p,-n}_{\mathbb{Z}\mathbb{Z}_{2}}]_{\kappa\lambda},[M^{p,-n}_{\mathbb{Z}_{2}\mathbb{Z}_{2}}]_{\kappa\lambda}\in\\{0,1\\}$.
Let $\pi^{p}_{j}:E^{p}_{0}\to E^{p}_{0}$ the projection onto irreps at $j$-th
$p$-cell. Namely, for $\vec{v}=(v_{ir})\in E^{p}_{0}$,

$\displaystyle\pi^{p}_{j}(\vec{v})=(0,\dots,0,\underbrace{v_{j1},v_{j2},\dots}_{i=j},0,\dots,0)^{tr}.$
(161)

The $\lambda$-th basis vector
$\vec{a}_{\lambda}^{p,-n}=(a^{p,-n}_{\lambda,ir})$ of $E^{1\mathbb{Z}}_{p,-n}$
represents the set of the massive Dirac Hamiltonians
$H_{{\bm{x}}^{p}_{i}}^{\alpha_{r}({\bm{x}}^{p}_{i})}(\tilde{\bm{x}})$ of the
irrep $\alpha_{r}({\bm{x}}^{p}_{i})$ over the virtual $n$-sphere
$\tilde{\bm{x}}\in\\{{\bm{x}}^{p}_{i}\\}\times\tilde{S}^{n}$ with $\mathbb{Z}$
invariant of $a^{p,-n}_{\lambda,ir}$. We pick a representative $p$-cell
$D^{p}_{\sf a}$ from $\vec{a}_{\lambda}^{p,-n}$ such that
$a^{p,-n}_{\lambda,{\sf a}r}\neq 0$ and consider the projection onto the ${\sf
a}$-th $p$-cell

$\displaystyle\pi^{p}_{\sf a}(\vec{a}_{\lambda}^{p,-n})$
$\displaystyle=(0,\dots,0,\underbrace{a^{p,-n}_{\lambda,{\sf
a}1},a^{p,-n}_{\lambda,{\sf a}2},\dots}_{i={\sf a}},0,\dots,0)^{tr}.$ (162)

For the moment, $n=0$. From the Frobenius reciprocity (151), the vector

$\displaystyle\vec{w}=(\delta_{p-1}+\delta_{p-1}^{T})^{tr}\pi_{\sf
a}(\vec{a}_{\lambda}^{p,0})\in E^{p-1}_{0}$ (163)

represents how irreps $\alpha_{1}({\bm{x}}^{p}_{\sf a})^{\oplus
a^{p,0}_{\lambda,{\sf a}1}}\oplus\alpha_{2}({\bm{x}}^{p}_{\sf a})^{\oplus
a^{p,0}_{\lambda,{\sf a}2}}\oplus\cdots$ at the representative $p$-cell
$D^{p}_{\sf a}$ form the “induced representations” at adjacent $(p-1)$-cells
$D^{p-1}_{j}\in\partial D^{p}_{\sf a}$ and are expanded by the irreps at
$(p-1)$-cells $D^{p-1}_{j}$. Here,
$\delta_{p-1}^{tr},(\delta_{p-1}^{T})^{tr}:E^{p}_{0}\to E^{p-1}_{0}$ are the
transpose of $\delta_{p-1},\delta_{p-1}^{T}$. Since $\vec{w}$ contains only
the irreps at $(p-1)$-cells adjacent to $D^{p}_{\sf a}$,

$\displaystyle\vec{w}=\sum_{j\in{\cal C}_{p-1},D^{p-1}_{j}\in\partial
D^{p}_{\sf a}}\pi^{p-1}_{j}(\vec{w}).$ (164)

We compare the vector $\vec{w}$ with the basis
$\\{\vec{a}^{p-1,0}_{\kappa}\\}_{\kappa}$ of $E^{1\mathbb{Z}}_{p-1,0}$. In the
view of (152), each $(p-1)$-cells $D^{p-1}_{j}\in\partial D^{p}_{\sf a}$
contributes to the expansion coefficient
$[M^{\mathbb{Z}\mathbb{Z}}_{p,0}]_{\kappa\lambda}$ independently. We have

$\displaystyle[M^{\mathbb{Z}\mathbb{Z}}_{p,0}]_{\kappa\lambda}=\sum_{j\in{\cal
C}_{p-1},D^{p-1}_{j}\in\partial D^{p}_{\sf
a}}[\pi^{p-1}_{j}(\vec{a}^{p-1,0}_{\kappa})]^{+}\pi^{p-1}_{j}(\vec{w}).$ (165)

Here, $[\pi^{p-1}_{j}(\vec{a}^{p-1,0}_{\kappa})]^{+}$ is the pseudoinverse of
the vector $\pi^{p-1}_{j}(\vec{a}^{p-1,0}_{\kappa})$ as a ${\rm
dim\,}(E^{0}_{p-1})\times 1$ matrix. (Note that the pseudoinverse of the null
matrix is also the null matrix.) Eq.(165) can also be written as
$[M^{\mathbb{Z}\mathbb{Z}}_{p,0}]_{\kappa\lambda}=\sum_{j\in{\cal
C}_{p-1}}[\pi^{p-1}_{j}(\vec{a}^{p-1,0}_{\kappa})]^{+}\pi^{p-1}_{j}(\vec{w})$,
and $[M^{\mathbb{Z}\mathbb{Z}}_{p,0}]_{\kappa\lambda}$ should not depend on a
choice of representative $p$-cell $D^{p}_{\sf a}$, we get

$\displaystyle[M^{\mathbb{Z}\mathbb{Z}}_{p,0}]_{\kappa\lambda}=\frac{1}{{\cal
N}(\vec{a}^{p,0}_{\lambda})}\sum_{j\in{\cal
C}_{p-1}}[\pi^{p-1}_{j}(\vec{a}^{p-1,0}_{\kappa})]^{+}\pi^{p-1}_{j}\left((\delta_{p-1}+\delta_{p-1}^{T})^{tr}\vec{a}_{\lambda}^{p,0}\right).$
(166)

Here, ${\cal N}(\vec{v}):=|\\{i\in{\cal C}_{p}|v_{ir}\neq 0\\}|\in\mathbb{N}$
is the number of $p$-cells contained in $\vec{v}$. For other degrees $n$, we
have to implement the sign change (138) if the orbit $\\{h(D^{p}_{\sf
a})\\}_{h\in{\cal G}}$ includes a TRS-type element $h\in{\sf t}{\cal G}^{0}$.
We eventually get the formula

$\displaystyle[M^{\mathbb{Z}\mathbb{Z}}_{p,-n}]_{\kappa\lambda}=\frac{1}{{\cal
N}(\vec{a}^{p,-n}_{\lambda})}\sum_{j\in{\cal
C}_{p-1}}[\pi^{p-1}_{j}(\vec{a}^{p-1,-n}_{\kappa})]^{+}\cdot\left\\{\begin{array}[]{ll}\pi^{p-1}_{j}\left((\delta_{p-1}+\delta_{p-1}^{T})^{tr}\vec{a}_{\lambda}^{p,-n}\right)&(n\in
4\mathbb{Z}),\\\
\pi^{p-1}_{j}\left((\delta^{\Gamma}_{p-1}+\delta_{p-1}^{\Gamma,T})^{tr}\vec{a}_{\lambda}^{p,-n}\right)&(n\in
4\mathbb{Z}+1),\\\
\pi^{p-1}_{j}\left((\delta_{p-1}-\delta_{p-1}^{T})^{tr}\vec{a}_{\lambda}^{p,-n}\right)&(n\in
4\mathbb{Z}+2),\\\
\pi^{p-1}_{j}\left((\delta^{\Gamma}_{p-1}-\delta_{p-1}^{\Gamma,T})^{tr}\vec{a}_{\lambda}^{p,-n}\right)&(n\in
4\mathbb{Z}+3).\end{array}\right.$ (171)

For the coefficients
$[M^{p,-n}_{\mathbb{Z}\mathbb{Z}_{2}}]_{\kappa\lambda},[M^{p,-n}_{\mathbb{Z}_{2}\mathbb{Z}_{2}}]_{\kappa\lambda}$,
the derivation is similar. We neglect the orientation of $p$-cells and the
sign of components of vector. We have

$\displaystyle[M^{\mathbb{Z}\mathbb{Z}_{2}}_{p,-n}]_{\kappa\lambda}=\frac{1}{{\cal
N}(\vec{a}^{p,-n}_{\lambda})}\sum_{j\in{\cal
C}_{p-1}}[\pi^{p-1}_{j}(\vec{b}^{p-1,-n}_{\kappa})]^{+}\pi^{p-1}_{j}\left((|\delta_{p-1}|+|\delta_{p-1}^{T}|)^{tr}|\vec{a}_{\lambda}^{p,-n}|\right)\quad{\rm\
mod\ }2,$ (172)

and

$\displaystyle[M^{\mathbb{Z}_{2}\mathbb{Z}_{2}}_{p,-n}]_{\kappa\lambda}=\frac{1}{{\cal
N}(\vec{b}^{p,-n}_{\lambda})}\sum_{j\in{\cal
C}_{p-1}}[\pi^{p-1}_{j}(\vec{b}^{p-1,-n}_{\kappa})]^{+}\pi^{p-1}_{j}\left((|\delta_{p-1}|+|\delta_{p-1}^{T}|)^{tr}\vec{b}_{\lambda}^{p,-n}\right)\quad{\rm\
mod\ }2.$ (173)

In this way, we get the first differential $d^{1}_{p,-n}$. The computation of
$E^{2}$-page is the same as for the momentum-space AHSS. Introduce an integer
lift

$\displaystyle\tilde{d}^{1}_{p,-n}:\tilde{E}^{1}_{p,-n}\to\tilde{E}^{1}_{p-1,-n}$
(174)

of $d^{1}_{p,-n}$. Considering $M_{p,-n}^{\mathbb{Z}\mathbb{Z}_{2}}$ and
$M_{p,-n}^{\mathbb{Z}_{2}\mathbb{Z}_{2}}$ as $\mathbb{Z}$-valued matrices does
work for such a lift. Then, the group $E^{2}_{p,-n}$ is computed as the
quotient of two integer sublattices of $E^{0}_{p}$:

$\displaystyle E^{2}_{p,-n}=\frac{{\rm Ker\,}(\tilde{d}^{1}_{p,-n}\oplus{\rm
Id}_{{\cal P}^{1}_{p-1,-n}})|_{\tilde{E}^{1}_{p,-n}}}{{\rm
Im\,}(\tilde{d}^{1}_{p-1,-n})+{\cal P}^{1}_{p,-n}}.$ (175)

##  VI Constraint on $K$-group by $E_{2}$-pages

The $E_{2}$\- or $E^{2}$-page is sometimes sufficient to determine the
$K$-group as a $\mathbb{Z}$-module uniquely. However, in general, to obtain
the $K$-group from an $E_{2}$\- or $E^{2}$-page, one must compute higher-order
derivatives and then solve an extension problem. Nevertheless, one can
constrain the possible $K$-groups by comparing the momentum-space $E_{2}$-page
and real-space $E^{2}$-page, which we discuss in this section.

To illustrate our method, Tables 3 and 4 show the $E_{2}$\- and $E^{2}$-pages
for spinful class D superconductors with MSG $P2_{1}1^{\prime}$ and
$A$-representation of pairing symmetry, respectively. This kind of pair of
$E_{2}$\- and $E^{2}$-pages is the input for the following analysis.

$\begin{array}[]{l|cccc}n=0&0&0&0&0\\\
n=1&\mathbb{Z}&\mathbb{Z}_{2}\oplus\mathbb{Z}_{4}^{\oplus 3}&\mathbb{Z}&0\\\
n=2&0&\mathbb{Z}_{2}^{\oplus 3}&\mathbb{Z}_{2}&0\\\
n=3&0&\mathbb{Z}\oplus\mathbb{Z}_{2}^{\oplus 3}&\mathbb{Z}_{2}&\mathbb{Z}\\\
n=4&0&0&0&0\\\ n=5&\mathbb{Z}&0&\mathbb{Z}&0\\\ n=6&\mathbb{Z}_{2}^{\oplus
5}&0&\mathbb{Z}_{2}&0\\\ n=7&\mathbb{Z}_{2}^{\oplus
5}&\mathbb{Z}&\mathbb{Z}_{2}&\mathbb{Z}\\\ \hline\cr
E_{2}^{p,-n}&p=0&p=1&p=2&p=3\\\ \end{array}$ Table 3: The momentum space
$E_{2}$-page of spinful class D SC with MSG $P2_{1}1^{\prime}$ and $A$
representation of pairing symmetry. $\begin{array}[]{l|cccc}n=0&0&0&0&0\\\
n=1&\mathbb{Z}_{2}&\mathbb{Z}_{2}^{\oplus 3}&\mathbb{Z}_{2}^{\oplus
3}&\mathbb{Z}_{2}\\\ n=2&\mathbb{Z}_{2}&\mathbb{Z}_{2}^{\oplus
3}&\mathbb{Z}_{2}^{\oplus 3}&\mathbb{Z}_{2}\\\
n=3&\mathbb{Z}&\mathbb{Z}\oplus\mathbb{Z}_{2}^{\oplus
2}&\mathbb{Z}&\mathbb{Z}\\\ n=4&0&0&0&0\\\ n=5&0&0&0&0\\\ n=6&0&0&0&0\\\
n=7&\mathbb{Z}&\mathbb{Z}\oplus\mathbb{Z}_{2}^{\oplus
2}&\mathbb{Z}&\mathbb{Z}\\\ \hline\cr E^{2}_{p,-n}&p=0&p=1&p=2&p=3\\\
\end{array}$ Table 4: The real space $E^{2}$-page of spinful class D SC with
MSG $P2_{1}1^{\prime}$ and $A$ representation of pairing symmetry

###  VI.1 Higher differential

In the momentum-space AHSS, the differential $d_{r}$ and $E_{r}$-page for
$r\geq 1$ are defined successively as

$\displaystyle d_{r}^{p,-n}:E_{r}^{p,-n}\to E_{r}^{p+r,-n-r+1},$ (176)
$\displaystyle E_{r+1}^{p,-n}:={\rm Ker\,}d_{r}^{p,-n}/{\rm
Im\,}d_{r}^{p-r,-n+r-1}.$ (177)

Here, $d_{r}\circ d_{r}=0$ holds. Although the existence of higher-order
differentials $d_{r}$ for $r\geq 2$ is mathematically guaranteed, and there is
a physical picture of $d_{r}$ such as the creation of gapless points to
generic momenta associated with band inversion at a high-symmetry point, there
is no known formula to calculate $d_{r}$ automatically at this time. Recall
that we are interested in $K$-groups over a space in three dimensions. The
differentials $d_{r}$ for $r\geq 4$ are trivial, and the $E_{4}$-page is the
limit. In such cases, one of the composite $d_{r}^{p+r,-n-r+1}\circ
d_{r}^{p,-n}$ for $r=2,3$ is always trivial, and the composite
$d_{r}^{p+r,-n-r+1}\circ d_{r}^{p,-n}$ vanishes automatically. In other words,
any homomorphism $f\in{\rm Hom}_{\mathbb{Z}}(E_{r}^{p,-n},E_{r}^{p+r,-n-r+1})$
is a candidate for the true $d_{r}^{p,-n}$. Listing all homomorphisms
$f\in{\rm Hom}_{\mathbb{Z}}(E_{r}^{p,-n},E_{r}^{p+r,-n-r+1})$ for $r=2$ and
$r=3$, we obtain all candidates for $E_{4}$-pages.

The same is the case for the real-space AHSS. The differential $d^{r}$ and
$E^{r}$-page for $r\geq 1$ are defined successively as

$\displaystyle d^{r}_{p,-n}:E^{r}_{p,-n}\to E^{r}_{p-r,-n+r-1},$ (178)
$\displaystyle E^{r+1}_{p,-n}:={\rm Ker\,}d^{r}_{p,-n}/{\rm
Im\,}d^{r}_{p+r,-n-r+1},$ (179)

where $d^{r}\circ d^{r}=0$ holds. There is no known formula to compute the
higher-differential $d^{r}$ for $r\geq 2$. (Nevertheless, we should note that
Ref. [21] successfully computed $d^{2}$ for the time-reversal symmetric SCs
for spinful electrons for conventional pairing symmetry, based on the physical
picture that $d^{2}$ is equivalent to the inevitable vortex zero modes
enforced by crystalline symmetries.) The $E^{4}$-page is the limit, one of the
composite $d^{r}_{p-r,-n+r-1}\circ d^{r}_{p,-n}$ for $r=2,3$ is always
trivial, and the composite $d^{r}_{p-r,-n+r-1}\circ d^{r}_{p,-n}$ vanishes
automatically. Listing all homomorphisms $f\in{\rm
Hom}_{\mathbb{Z}}(E^{r}_{p,-n},E^{r}_{p-r,-n+r-1})$ for $r=2$ and $r=3$, we
get all candidates for $E^{4}$-pages.

Let us consider how to list all possible homomorphisms $f\in{\rm
Hom}_{\mathbb{Z}}(A,B)$ for a given pair of $\mathbb{Z}$-modules $A$ and $B$.
Precisely, we are only interested in the $\mathbb{Z}$-modules ${\rm Ker\,}f$
and ${\rm Coker\,}f$ per isomorphisms. (Different homomorphisms
$f_{1},f_{2}\in{\rm Hom}_{\mathbb{Z}}(A,B)$ may give the same kernel and
cokernel as $\mathbb{Z}$-modules.) To simplify the problem and maintain
physical relevance, we adopt the following working assumptions:

######  Assumption 1.

The rank of the $K$-group ${}^{\phi}K_{G}^{(z,c)-n}(X)$ is the same as the
rank of the $E_{2}$-page $\bigoplus_{p=0}^{3}E_{2}^{p,-n-p}$.

This property is known to hold for equivariant $K$-theory over any finite
proper $G$-complex [50]. Although no proof exists for twisted and equivariant
$K$-theory, we provide some reasons below for why this conjecture might be
correct. The higher differentials $d^{p,-n}_{r\geq 2}$ can be understood as
the band inversion at a high-symmetry $p$-cell followed by the creation of
gapless point inside adjacent $(p+r)$-cells [14]. For example, consider a
$C_{4}$-symmetric $0$-cell and a band inversion at the $0$-cell followed by
the creation of four gapless points in adjacent $2$-cells, as shown in Fig. 4
[a]. This is an example of nontrivial $d_{2}^{0,-n}(x)$ for $x\in
E_{2}^{0,-n}$. The gapless points in the 2-cells cannot be removed on 1-cells
by the $C_{4}$ symmetry. Let us consider the same band inversion twice as
shown in Fig. 4 [b], namely, the element $2d_{2}^{0,-n}(x)$. We have two
quartets of gapless points. Since each quartet can pass through 1-cells while
preserving $C_{4}$ symmetry, the two quartets can annihilate each other. This
implies that $2d_{2}^{0,-n}(x)=0$, that is, the image of $d_{2}^{0,-n}$ is a
torsion. We expect a similar picture to be true for any higher-differentials
$d_{r\geq 2}^{p,-n}$.

Figure 4:  [a] The higher differentials $d_{r\geq 2}$ can be understood as
band inversion followed by the creation of gapless points. [b] When the same
gapless point creation occurs twice, they can cancel each other out.

For the real-space AHSS, we assume the same one:

######  Assumption 2.

The rank of the $K$-group ${}^{\phi}K^{\cal G}_{(z^{\rm
int},c)-n}(\mathbb{E}^{3})$ is the same as the rank of the $E^{2}$-page
$\bigoplus_{p=0}^{3}E^{2}_{p,-n-p}$.

We leave a discussion of the validity of this assumption for SCs. As discussed
in [28], the origin of the second differential $d^{2}$ is a vortex zero mode
of topological SC with a unit Chern number. The two-layered SC with Chern
number 1 for each is equivalent to a single SC with Chern number 2 under the
stable equivalence, and there are no vortex zeros. This means that the image
of $d^{2}$ should be a 2-torsion.

These assumptions imply that higher-order differentials do not reduce the
rank, i.e., there is no $\mathbb{Z}$ in the image of $d_{r}$ and $d^{r}$ for
$r=2,3$. Thus, the homomorphisms to be considered have the following form

$\displaystyle f:\mathbb{Z}^{\oplus q}\oplus A\to B$ (180)

with $A$ and $B$ torsion $\mathbb{Z}$-modules. Let us denote
$f_{1}=f|_{\mathbb{Z}^{\oplus q}}$ and $f_{2}=f|_{A}$ so that $f=f_{1}\oplus
f_{2}$. ${\rm Ker\,}f$ and ${\rm Coker\,}f$ are computed as follows. The rank
of ${\rm Ker\,}f$ is the same as $\mathbb{Z}^{\oplus q}$, and the torsion part
of ${\rm Ker\,}f$ is ${\rm Ker\,}f_{2}$. (If $f_{1}(x)+f_{2}(a)=0$ and there
exists $k\in\mathbb{N}$ such that $k(x,a)=0$, then $x=0$ and $f_{2}(a)=0$.)
Therefore, ${\rm Ker\,}f$ is in a form $\mathbb{Z}^{\oplus q}\oplus{\rm
Ker\,}f_{2}$ and is determined by $f_{2}$ alone. Using the third-isomorphism
theorem, ${\rm Coker\,}f=B/({\rm Im\,}f_{1}+{\rm Im\,}f_{2})\cong(B/{\rm
Im\,}f_{2})/(({\rm Im\,}f_{1}+{\rm Im\,}f_{2})/{\rm Im\,}f_{2})$. The
corresponding theorem says there is a bijection between the set of
sub-$\mathbb{Z}$-modules of $B$ including ${\rm Im\,}f_{2}$ and the set of
sub-$\mathbb{Z}$-modules of $B/{\rm Im\,}f_{2}$. Here, the bijection is given
by ${\rm Im\,}f_{1}+{\rm Im\,}f_{2}\mapsto{\rm
Im\,}[f_{1}^{\prime}:\mathbb{Z}^{\oplus q}\to B/{\rm Im\,}f_{2}]$ where
$f_{1}^{\prime}$ is defined by $f_{1}^{\prime}(n):=f_{1}(n){\rm\ mod\ }{\rm
Im\,}f_{2}$. Therefore,

$\displaystyle{\rm Coker\,}f$ $\displaystyle\cong\frac{B/{\rm Im\,}f_{2}}{{\rm
Im\,}[f_{1}^{\prime}:\mathbb{Z}^{\oplus q}\to B/{\rm Im\,}f_{2}]}$
$\displaystyle={\rm Coker\,}f_{1}^{\prime}.$ (181)

From the observation above, to list all the possible pairs of ${\rm Ker\,}f$
and ${\rm Coker\,}f$, we do the following. First, we tabulate all possible
homomorphisms $f_{2}:A\to B$. Next, we tabulate all possible homomorphisms
$f_{1}^{\prime}:\mathbb{Z}^{\oplus q}\to B/{\rm Im\,}f_{2}$ for each $f_{2}$.

Computing the $E_{2}$ and $E^{2}$-pages explicitly for the symmetry classes
summarized in Sec. II, we find that the torsion sub-$\mathbb{Z}$-modules
appearing in the homomorphism $f_{2}:A\to B$ is in the following form

$\displaystyle f_{2}:\mathbb{Z}_{2}^{\oplus k}\oplus\mathbb{Z}_{4}^{\oplus
l}\to\mathbb{Z}_{2}^{\oplus m}\oplus\mathbb{Z}_{4}^{\oplus n}.$ (182)

Let us denote the matrix representation of $f_{2}$ as

$\displaystyle
M_{f_{2}}=\left[\begin{array}[]{cc}M^{\mathbb{Z}_{2}\to\mathbb{Z}_{2}}_{m\times
k}&M^{\mathbb{Z}_{4}\to\mathbb{Z}_{2}}_{l\times m}\\\
M^{\mathbb{Z}_{2}\to\mathbb{Z}_{4}}_{n\times
k}&M^{\mathbb{Z}_{4}\to\mathbb{Z}_{4}}_{l\times n}\\\ \end{array}\right].$
(185)

By the basis transformation, the diagonal blocks can be Smith normal forms

$\displaystyle M^{\mathbb{Z}_{2}\to\mathbb{Z}_{2}}_{m\times
k}=\begin{pmatrix}I_{r_{1}\times r_{1}}&O\\\ O&O\\\ \end{pmatrix},$
$\displaystyle 0\leq r_{1}\leq\min(m,k),$ (186) $\displaystyle
M^{\mathbb{Z}_{4}\to\mathbb{Z}_{4}}_{l\times n}=\begin{pmatrix}I_{r_{2}\times
r_{2}}&O&O\\\ O&2\times I_{r_{3}\times r_{3}}&O\\\ O&O&O\\\ \end{pmatrix},$
$\displaystyle 0\leq r_{2}+r_{3}\leq\min(l,n).$ (187)

For the off-diagonal ones $M^{\mathbb{Z}_{4}\to\mathbb{Z}_{2}}_{l\times m}$
and $M^{\mathbb{Z}_{2}\to\mathbb{Z}_{4}}_{n\times k}$, we consider all
possible $\mathbb{Z}_{2}$\- and $\mathbb{Z}_{4}$-valued matrices. For a given
matrix $M_{f_{2}}$, applying the method in Appendix C, we get ${\rm
Ker\,}f_{2}$ and ${\rm Coker\,}f_{2}$.

For a given $f_{2}$, the quotient group $B/{\rm Im\,}f_{2}$ is also in a form
$\mathbb{Z}_{2}^{\oplus s}\oplus\mathbb{Z}_{4}^{\oplus t}$. We tabulate all
homomorphisms $f_{1}^{\prime}:\mathbb{Z}^{\oplus q}\to\mathbb{Z}_{2}^{\oplus
s}\oplus\mathbb{Z}_{4}^{\oplus t}$, of which matrix expression is

$\displaystyle
M_{f_{1}^{\prime}}=\left[\begin{array}[]{cc}M^{\mathbb{Z}\to\mathbb{Z}_{2}}_{q\times
s}\\\ M^{\mathbb{Z}\to\mathbb{Z}_{4}}_{q\times t}\end{array}\right].$ (190)

By the basis transformation, $M^{\mathbb{Z}\to\mathbb{Z}_{2}}_{q\times s}$ can
be the Smith normal form, the same as (186). For
$M^{\mathbb{Z}\to\mathbb{Z}_{4}}_{q\times t}$, we consider all possible
$\mathbb{Z}_{4}$-valued matrices. Using the method in Appendix C, we get ${\rm
Coker\,}f_{1}^{\prime}$.

Following the procedure above, from a $E_{2}$-page of the momentum space , we
first compute possible $d_{2}^{p,-n}$ to get the $E_{3}$-page as
$E_{3}^{p,-n}={\rm Ker\,}d_{2}^{p,-n}$ and $E_{3}^{p+2,-n-1}={\rm
Coker\,}d_{2}^{p,-n}$ for $p=0,1$. For each candidate $E_{3}$-page, we compute
possible $d_{3}^{p,-n}$ to get $E_{4}^{0,-n}={\rm Ker\,}d_{3}^{0,-n}$ and
$E_{4}^{3,-n-2}={\rm Coker\,}d_{3}^{0,-n}$. Note that
$E_{4}^{p,-n}=E_{3}^{p,-n}$ for $p=1,2$. Eventually, we get the set of
candidates of $E_{4}$-pages, which we denote by ${\cal
S}_{k}=\\{E_{4,i}=\\{cE_{4,i}^{p,-n}\\}_{p=0,\dots,3,n=0,\dots,7}\\}_{i=1,\dots,|{\cal
S}_{k}|}$. Similarly, from a $E^{2}$-page, we compute possible $d^{3}_{p,-n}$
to get the candidates of $E^{3}$-pages by $E^{3}_{p,-n}={\rm
Ker\,}d^{2}_{p,-n}$ and $E^{3}_{p-2,-n+1}={\rm Coker\,}d^{2}_{p,-n}$ for
$p=2,3$, and for each $E^{3}$-page we compute possible $d^{3}_{3,-n}$ to get
$E^{4}_{3,-n}={\rm Ker\,}d^{3}_{3,-n}$ and $E^{4}_{0,-n+2}={\rm
Coker\,}d^{3}_{3,-n}$ with $E^{4}_{p,-n}=E^{3}_{p,-n}$ for $p=1,2$. We denote
the set of candidate $E^{4}$-pages by ${\cal
S}_{r}=\\{cE^{4,i}=\\{cE^{4,i}_{p,-n}\\}_{p=0,\dots,3,n=0,\dots,7}\\}_{i=1,\dots,|{\cal
S}_{r}|}$.

###  VI.2 Extension

The $E_{4}$-page approximates the momentum space $K$-group in the following
sense. We have a filtration

$\displaystyle 0=F_{4}K^{-n}\subset F_{3}K^{-n}\subset F_{2}K^{-n}$
$\displaystyle\subset F_{1}K^{-n}\subset
F_{0}K^{-n}={}^{\phi}K^{(z,c)-n}_{G}(X),$ (191)

where each quotient group is given as

$\displaystyle F_{p}K^{-n}/F_{p+1}K^{-n}\cong E_{4}^{p,-p-n}.$ (192)

The $K$-group is obtained by solving the extension problems as
$\mathbb{Z}$-module sequentially. $F_{3}K^{-n}=E_{4}^{3,-3-n}$. Then,
$F_{p}K^{-n}$ is given by an extension of $E_{4}^{p,-p-n}$ by $F_{p+1}K^{-n}$
for $p=2,1,0$ in order.

Similarly, the $E^{4}$-page approximates the real space $K$-group as

$\displaystyle 0=F_{-1}K_{-n}\subset F_{0}K_{-n}\subset F_{1}K_{-n}$
$\displaystyle\subset F_{2}K_{-n}\subset F_{3}K_{-n}={}^{\phi}K_{(z^{\rm
int},c)-n}^{{\cal G}}(\mathbb{R}^{3})$ (193)

with

$\displaystyle F_{p}K_{-n}/F_{p-1}K_{-n}\cong E^{4}_{p,-p-n}.$ (194)

We have $F_{0}K_{-n}=E^{4}_{0,-n}$, and $F_{p}K_{-n}$ is an extension of
$E^{4}_{p,-p-n}$ by $F_{p-1}K_{-n}$ for $p=1,2,3$ in order.

Given a pair of $\mathbb{Z}$-modules $A$ and $B$, equivalence classes of
$\mathbb{Z}$-module extensions of $A$ by $B$ are classified by ${\rm
Ext}_{\mathbb{Z}}^{1}(A,B)$ [51]. The group ${\rm Ext}^{1}_{\mathbb{Z}}(A,B)$
has the following properties

$\displaystyle{\rm
Ext}_{\mathbb{Z}}^{1}\Big{(}\bigoplus_{i}A_{i},\bigoplus_{j}B_{j}\Big{)}=\bigoplus_{i,j}{\rm
Ext}_{\mathbb{Z}}^{1}(A_{i},B_{j}),$ (195)

and

$\displaystyle{\rm
Ext}_{\mathbb{Z}}^{1}(\mathbb{Z}_{n},\mathbb{Z})=\mathbb{Z}_{n},$ (196)
$\displaystyle{\rm
Ext}_{\mathbb{Z}}^{1}(\mathbb{Z}_{n},\mathbb{Z}_{m})=\mathbb{Z}_{{\rm
gcd}(n,m)},$ (197) $\displaystyle{\rm
Ext}_{\mathbb{Z}}^{1}(\mathbb{Z},\mathbb{Z}_{n})={\rm
Ext}_{\mathbb{Z}}^{1}(\mathbb{Z},\mathbb{Z})=0.$ (198)

The free $\mathbb{Z}$-module $\mathbb{Z}$ is not extended. Hereafter we assume
$A$ is a torsion $\mathbb{Z}$-module. The abelian group structure of ${\rm
Ext}^{1}_{\mathbb{Z}}(A,B)$ is represented by the Baer sum of two extensions
$0\to B\to E_{i}\to A\to 0$ for $i=0,1$. (See Appendix D for the construction
of the Baer sum.) Thus, all extensions can be constructed from the extensions
corresponding to the generators of the $\mathbb{Z}$-module ${\rm
Ext}_{\mathbb{Z}}^{1}(A,B)$.

However, it is not feasible to compute extensions for all values of ${\rm
Ext}_{\mathbb{Z}}^{1}(A,B)$. For example, $|{\rm
Ext}_{\mathbb{Z}}^{1}(\mathbb{Z}_{2}^{\oplus n},\mathbb{Z}^{\oplus
m})|=2^{nm}$. Changing the basis of $A$ and $B$ induces an automorphism of
${\rm Ext}_{\mathbb{Z}}^{1}(A,B)$, but the $\mathbb{Z}$-module obtained as an
extension via this basis transformation does not change. For example, ${\rm
Ext}^{1}_{\mathbb{Z}}(\mathbb{Z}_{4},\mathbb{Z})=\mathbb{Z}_{4}$, and each
value of $\mathbb{Z}_{4}$ gives
$\mathbb{Z}\oplus\mathbb{Z}_{4},\mathbb{Z},\mathbb{Z}\oplus\mathbb{Z}_{2},\mathbb{Z}$
respectively. Here, the second and fourth extensions are related via the basis
transformation of $\mathbb{Z}_{4}$ (or $\mathbb{Z}$) multiplying the basis by
$-1$. Thus, if we are only interested in the $\mathbb{Z}$-module obtained as
extensions, we can contract the elements of ${\rm Ext}_{\mathbb{Z}}^{1}(A,B)$
via basis transformations. We will now formalize this contraction procedure.

We express an element of the group ${\rm Ext}_{\mathbb{Z}}^{1}(A,B)$ as a
matrix $M$, which we call an extension matrix, using the following procedure.
First, we decompose $A$ into its invariant factors as

$\displaystyle A=\mathbb{Z}_{p_{1}}^{\oplus
n_{1}}\oplus\cdots\oplus\mathbb{Z}_{p_{k}}^{\oplus n_{k}},\quad
p_{k}|p_{k-1}|\cdots|p_{1}.$ (199)

Here, $p|q$ denotes that $p$ divides $q$. We denote the basis of $A$ as

$\displaystyle
a^{(p_{1})}_{1},\dots,a^{(p_{1})}_{n_{1}},\dots,a^{(p_{k})}_{1},\dots,a^{(p_{k})}_{n_{k}}.$
(200)

Similarly, we decompose $B$ into its invariant factors as

$\displaystyle B=\mathbb{Z}^{\oplus m_{0}}\oplus\mathbb{Z}_{q_{1}}^{\oplus
m_{1}}\oplus\cdots\oplus\mathbb{Z}_{q_{l}}^{\oplus q_{l}},$ $\displaystyle
q_{l}|q_{l-1}|\cdots|q_{1}.$ (201)

We denote the basis of $B$ as

$\displaystyle{\cal B}$
$\displaystyle=(b^{(q_{0})}_{1},\dots,b^{(q_{0})}_{m_{0}},$
$\displaystyle\quad\quad
b^{(q_{1})}_{1},\dots,b^{(q_{1})}_{m_{1}},\dots,b^{(q_{l})}_{1},\dots,b^{(q_{l})}_{m_{l}}),$
(202)

where we formally wrote $q_{0}=\infty$. An element of ${\rm
Ext}^{1}_{\mathbb{Z}}(A,B)\supset{\rm Ext}^{1}_{\mathbb{Z}}(\mathbb{Z}_{p},B)$
represents how $``p"\in\mathbb{Z}_{p}$ is “carried-up” by an element of $B$.
We denote this correspondence as an extension matrix $M$ by

$\displaystyle f_{M}({\cal A})={\cal B}M,$ (203)

with the basis

$\displaystyle{\cal A}$
$\displaystyle=(p_{1}a^{(p_{1})}_{1},\dots,p_{1}a^{(p_{1})}_{n_{1}},$
$\displaystyle\quad\quad\dots,p_{k}a^{(p_{k})}_{1},\dots,p_{k}a^{(p_{k})}_{n_{k}}).$
(204)

Next, let’s consider possible basis transformations for ${\cal A}$ and ${\cal
B}$. For ${\cal B}$, any basis transformations that do not change the
structure of the $\mathbb{Z}$-module $B$ are allowed. Since it is allowed to
add linear combinations of basis elements $b^{(q_{j})}_{t}$ with $i\leq j$ to
a basis $b^{(q_{i})}_{s}$, we can use the following block lower triangular
unimodular matrix in the form

$\displaystyle V=\left[\begin{array}[]{ccccc}v_{00}&O&O&\cdots\\\
v_{10}&v_{11}&O\\\ v_{20}&v_{21}&v_{33}\\\ \vdots\\\ \end{array}\right]$ (209)

as a general basis transformation for ${\cal B}$. On the other hand, the basis
transformation of ${\cal A}$ to be considered is not a basis transformation
that does not change the structure as $\mathbb{Z}$-module $A$, but a basis
transformation that keeps the integer lattice spanned by ${\cal A}$ invariant.
For example, if we add $r$ times $p_{j}a^{(p_{j})}_{t}$ to
$p_{i}a^{(p_{i})}_{s}$, we have

$\displaystyle
p_{i}a^{(p_{i})}_{s}+rp_{j}a^{(p_{j})}_{t}=p_{i}(a^{(p_{i})}_{s}+r\frac{p_{j}}{p_{i}}a^{(p_{j})}_{t})$
(210)

which is well-defined only if $p_{i}|p_{j}$, i.e., in the case where $i\geq
j$. Therefore, the basis transformations for ${\cal A}$ are given by block
upper triangular unimodular matrices of the form

$\displaystyle U=\left[\begin{array}[]{ccccc}u_{11}&u_{12}&u_{13}&\cdots\\\
O&u_{22}&u_{23}\\\ O&O&u_{33}\\\ \vdots\\\ \end{array}\right].$ (215)

Through these basis transformations, the entension matrix $M$ changes to

$\displaystyle M\mapsto V^{-1}MU.$ (216)

Since the inverse of a lower-triangular block unimodular matrix is also a
lower-triangular block unimodular, the above transformation is nothing but a
“directional” Gauss-Jordan elimination from the upper left to the lower right.
We write the extension matrix $M$ as a block matrix

$\displaystyle M=(M^{(q_{i},p_{j})})_{0\leq i\leq l,1\leq j\leq k},$ (217)
$\displaystyle M^{(q_{i},p_{j})}=(M^{(q_{i},p_{j})}_{st})_{1\leq s\leq
m_{i},1\leq t\leq n_{j}},$ (218)

where $M^{(q_{i},p_{j})}_{st}$ takes values in $\mathbb{Z}_{{\rm
gcd}(p_{j},q_{i})}$. (Here, we set ${\rm
gcd}(p_{j},\infty)=\mathbb{Z}_{p_{j}}$.) The operations allowed in the
“directional” Gauss-Jordan elimination are as follows:

\- Multiply a row vector by $(-1)$.

\- Multiply a column vector by $(-1)$.

\- (Row reduction) Add an integer multiple $rv^{(q_{i})}_{s},r\in\mathbb{Z},$
of a $(i,s)$th row vector $v^{(q_{i})}_{s}=(M^{(q_{i},p_{j})}_{st})_{1\leq
j\leq k,1\leq t\leq n_{j}}$ to other row vector
$v^{(q_{i^{\prime}})}_{s^{\prime}}$ with
$(q_{i},s)\neq(q_{i^{\prime}},s^{\prime})$ and $i\geq i^{\prime}$. Here, when
adding the row vector $rv^{(q_{i})}_{s}$ to the row vector
$v^{(q{i^{\prime}})}_{s^{\prime}}$ with $i>i^{\prime}$, take the modulo ${\rm
gcd}(q_{i^{\prime}},p_{j})$ of the entries $rM^{(q_{i},p_{j})}_{st}$.

\- (Column reduction) Add an integer multiple
$rv^{(p_{j})}_{t},r\in\mathbb{Z},$ of a $(j,t)$th column vector
$v^{(p_{j})}_{t}=(M^{(q_{i},p_{j})}_{st})_{1\leq i\leq l,1\leq s\leq m_{i}}$
to other column vector $v^{(p_{j^{\prime}})}_{t^{\prime}}$ with
$(p_{j},t)\neq(p_{j^{\prime}},t^{\prime})$ and $j\geq j^{\prime}$. Here, when
adding the column vector $rv^{(p_{j})}_{t}$ to the column vector
$v^{(p_{j^{\prime}})}_{t^{\prime}}$ with $j>j^{\prime}$, take the modulo ${\rm
gcd}(q_{i},p_{j^{\prime}})$ of the entries $rM^{(q_{i},p_{j})}_{st}$.

The extension matrix $M$ can be transformed to a standard form to some extent
by using the transformation (216), which performs the Smith decomposition in
order, starting from the top-left block.

$\displaystyle
M=\begin{array}[]{c|c|c}M^{(q_{0},p_{1})}&M^{(q_{0},p_{2})}&\cdots\\\
\hline\cr M^{(q_{1},p_{1})}&M^{(q_{1},p_{2})}\\\ \hline\cr\vdots&&\\\
\end{array}.$ (222)

First, take $M^{(q_{0},p_{1})}$ in Smith normal form.

$\displaystyle\to\begin{array}[]{c|c|c}\begin{array}[]{c|c}\Lambda^{(q_{0},p_{1})}&O\\\
\hline\cr O&O\\\ \end{array}&M^{(q_{0},p_{2})}&\cdots\\\ \hline\cr
M^{(q_{1},p_{1})}&M^{(q_{1},p_{2})}\\\ \hline\cr\vdots&&\\\ \end{array},$
(228)

where $\Lambda^{(q_{0},p_{1})}$ is a diagonal matrix. In the $(q_{0},p_{1})$
block, the components of the $(q_{0},p_{1})$ block remain unchanged at 0 even
after performing basis transformations on the rows and columns that have
component 0. So, the lower half of $M^{(q_{0},p_{2})}$ can be taken in Smith
normal standard form.

$\displaystyle\to\begin{tabular}[]{c|c|c|c|c}$\Lambda^{(q_{0},p_{1})}$&$O$&\lx@intercol\hfil$*$\hfil\lx@intercol\vrule\lx@intercol&\hbox{\multirowsetup$M^{(q_{0},p_{3})}$}\\\
\cline{1-4}\cr\hbox{\multirowsetup$O$}&\hbox{\multirowsetup$O$}&$\Lambda^{(q_{0},p_{2})}$&$O$&\\\
\cline{3-4}\cr&&$O$&$O$&\\\
\cline{1-5}\cr\lx@intercol\hfil$M^{(q_{1},p_{1})}$\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil$M^{(q_{1}p_{2})}$\hfil\lx@intercol\vrule\lx@intercol&$M^{(q_{1},p_{3})}$\\\
\end{tabular},$ (233)

where $*$ is some integer matrix. Similarly, $M^{(q_{1},p_{1})}$ can also be
taken in Smith normal form on the right side.

$\displaystyle\to\begin{tabular}[]{c|cc|cc|c}$\Lambda^{(q_{0},p_{1})}$&\lx@intercol\hfil$O$\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil$*$\hfil\lx@intercol\vrule\lx@intercol&\hbox{\multirowsetup$M^{(q_{0},p_{3})}$}\\\
\cline{1-5}\cr\hbox{\multirowsetup$O$}&\lx@intercol\hfil\hbox{\multirowsetup$O$}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil$\Lambda^{(q_{0},p_{2})}$\hfil\lx@intercol\vrule\lx@intercol&$O$&\\\
\cline{4-5}\cr&&&\lx@intercol\hfil$O$\hfil\lx@intercol\vrule\lx@intercol&$O$&\\\
\cline{1-6}\cr\hbox{\multirowsetup$*$}&\lx@intercol\hfil$\Lambda^{(q_{1},p_{1})}$\hfil\lx@intercol\vrule\lx@intercol&$O$&\lx@intercol\hfil\hbox{\multirowsetup$M^{(q_{1},p_{2})}$}\hfil\lx@intercol\vrule\lx@intercol&\hbox{\multirowsetup$M^{(q_{1},p_{3})}$}\\\
\cline{2-3}\cr&\lx@intercol\hfil$O$\hfil\lx@intercol\vrule\lx@intercol&$O$&&&\\\
\end{tabular}.$ (239)

In this way, taking the Smith normal form of the extension matrix $M$ in order
from the upper left, we can partially reduce it. The matrix
$\Lambda^{(q_{i},p_{j})}$ is a $\mathbb{Z}_{{\rm gcd}(p_{j},q_{i})}$-valued
diagonal matrix, but its entries can be limited from 1 to the integer no
larger than ${\rm gcd}(p_{j},q_{i})/2$ due to the sign change of the basis.
Furthermore, if an entry of the diagonal matrix $\Lambda^{(q_{i},p_{j})}$ is
$\lambda$, then all entries in the blocks to the right and below it (denoted
by $*$ in the above matrix) can be replaced by their remainders when divided
by $\lambda$. In particular, if $\lambda=1$, the entry can be replaced by 0.

The standard form of the extension matrix $M$ in the sense described above can
be enumerated by the following procedure. First, determine the non-zero
entries of the Smith normal form $\Lambda^{(q_{i},p_{j})}$. This procedure
simply constructs a set of pairs between the direct summands of $A$ and $B$:
$A=\bigoplus_{i=1}^{k}\mathbb{Z}_{p_{i}}^{\oplus
n_{i}},B=\bigoplus_{j=0}^{l}\mathbb{Z}_{q_{j}}^{\oplus m_{j}}$. To enumerate
all such pairs where there are $d$ pairs and
${\tilde{p}_{1},\dots,\tilde{p}_{d}}$ has no more than $n_{i}$ occurrences of
$p_{i}$ and ${\tilde{q}_{1},\dots,\tilde{q}_{d}}$ has no more than $m_{j}$
occurrences of $q_{j}$, we need to consider all possible sets of $d$ pairs
$\left\\{(\tilde{p}_{r},\tilde{q}_{r})\in\\{p_{1},\dots,p_{k}\\}\times\\{q_{0},\dots,q_{l}\\}\right\\}_{r=1}^{d}$.
Moreover, the upper bound for the number of pairs $d$ is given by ${\rm
min}(\sum_{i=1}^{k}n_{k},\sum_{j=0}^{l}m_{j})$. Therefore, the set of possible
sets of pairs is given by the following expression:

$\displaystyle\Big{\\{}\big{\\{}(\tilde{p}_{r},\tilde{q}_{r})\in\\{p_{1},\dots,p_{k}\\}\times\\{q_{0},\dots,q_{l}\\}\big{\\}}_{r=1}^{d}\Big{|}\sum_{r=1}^{d}\delta_{p_{i},\tilde{p}_{r}}\leq
n_{i}{\rm\ for\ }i=1,\dots,k,$ $\displaystyle\quad{\rm\ and\
}\sum_{r=1}^{d}\delta_{q_{j},\tilde{q}_{r}}\leq m_{j}{\rm\ for\
}j=1,\dots,l,{\rm\ and\ }d=0,\dots,{\rm
min}(\sum_{i=1}^{k}n_{i},\sum_{j=0}^{l}m_{j})\Big{\\}}.$ (240)

Next, for each set of pairs ${(\tilde{p}_{r},\tilde{q}_{r})}_{r=1}^{d}$, we
assign possible integer values to the diagonal entries of the diagonal matrix
$\Lambda^{(\tilde{q}_{r},\tilde{p}_{r})}$. Based on this assignment, we
enumerate all possible assignments of entries for the right and bottom blocks
of the diagonal matrix $\Lambda^{(q_{j},p_{i})}$ to obtain a set of reduced
extension matrices $M$.

As an example, consider the case where $A=\mathbb{Z}_{4}^{\oplus
2}\oplus\mathbb{Z}_{2}^{\oplus 3}$ and $B=\mathbb{Z}^{\oplus
3}\oplus\mathbb{Z}_{4}^{\oplus 2}\oplus\mathbb{Z}_{2}^{\oplus 2}$, and the set
of pairs is chosen to be ${(4,\infty),(2,\infty),(4,4),(2,2)}$. Using this, we
obtain the following initial extension matrix $M_{0}$.

$\displaystyle
M_{0}=\begin{array}[]{c|cc|ccc}&\mathbb{Z}_{4}&\mathbb{Z}_{4}&\mathbb{Z}_{2}&\mathbb{Z}_{2}&\mathbb{Z}_{2}\\\
\hline\cr\mathbb{Z}&\lambda_{1}&&&&\\\ \mathbb{Z}&&&\lambda_{2}&&\\\
\mathbb{Z}&&&&&\\\ \hline\cr\mathbb{Z}_{4}&&\lambda_{3}&&&\\\
\mathbb{Z}_{4}&&&\\\ \hline\cr\mathbb{Z}_{2}&&&&\lambda_{4}&\\\
\mathbb{Z}_{2}&&&&&\\\ \end{array}.$ (249)

Here, $\lambda_{1},\lambda_{3}\in{1,2}$ and $\lambda_{2},\lambda_{4}\in{1}$.
Blanks represent $0$. For each of the four possible combinations of
$(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4})$, namely $(1,1,1,1)$,
$(1,1,2,1)$, $(2,1,1,1)$, and $(2,1,2,1)$, we obtain the extension matrices
$M$ of the following forms.

$\displaystyle\begin{array}[]{c|cc|ccc}&\mathbb{Z}_{4}&\mathbb{Z}_{4}&\mathbb{Z}_{2}&\mathbb{Z}_{2}&\mathbb{Z}_{2}\\\
\hline\cr\mathbb{Z}&1&&&&\\\ \mathbb{Z}&&&1&&\\\ \mathbb{Z}&&&&&\\\
\hline\cr\mathbb{Z}_{4}&&1&&&\\\ \mathbb{Z}_{4}&&&\\\
\hline\cr\mathbb{Z}_{2}&&&&1&\\\ \mathbb{Z}_{2}&&&&&\\\ \end{array},$ (258)
$\displaystyle\begin{array}[]{c|cc|ccc}&\mathbb{Z}_{4}&\mathbb{Z}_{4}&\mathbb{Z}_{2}&\mathbb{Z}_{2}&\mathbb{Z}_{2}\\\
\hline\cr\mathbb{Z}&1&&&&\\\ \mathbb{Z}&&&1&&\\\ \mathbb{Z}&&&&&\\\
\hline\cr\mathbb{Z}_{4}&&2&*&*&*\\\ \mathbb{Z}_{4}&&&\\\
\hline\cr\mathbb{Z}_{2}&&*&&1&\\\ \mathbb{Z}_{2}&&*&&&\\\ \end{array},$ (267)
$\displaystyle\begin{array}[]{c|cc|ccc}&\mathbb{Z}_{4}&\mathbb{Z}_{4}&\mathbb{Z}_{2}&\mathbb{Z}_{2}&\mathbb{Z}_{2}\\\
\hline\cr\mathbb{Z}&2&&*&*&*\\\ \mathbb{Z}&&&1&&\\\ \mathbb{Z}&&&&&\\\
\hline\cr\mathbb{Z}_{4}&*&1&&&\\\ \mathbb{Z}_{4}&*&&\\\
\hline\cr\mathbb{Z}_{2}&*&&&1&\\\ \mathbb{Z}_{2}&*&&&&\\\ \end{array},$ (276)
$\displaystyle\begin{array}[]{c|cc|ccc}&\mathbb{Z}_{4}&\mathbb{Z}_{4}&\mathbb{Z}_{2}&\mathbb{Z}_{2}&\mathbb{Z}_{2}\\\
\hline\cr\mathbb{Z}&2&&*&*&*\\\ \mathbb{Z}&&&1&&\\\ \mathbb{Z}&&&&&\\\
\hline\cr\mathbb{Z}_{4}&*&2&*&*&*\\\ \mathbb{Z}_{4}&*&&\\\
\hline\cr\mathbb{Z}_{2}&*&*&&1&\\\ \mathbb{Z}_{2}&*&*&&&\\\ \end{array}.$
(285)

The symbol $*\in{0,1}$ is independent, and all combinations are considered.

The set of extension matrices $M$ obtained by the above procedure is still
large, so we further perform the following (I) and (II) reductions:

(I) The row and column reduction. For example, the set of extension matrices
in the form (285) contains the extension matrix in the form

$\displaystyle\begin{array}[]{c|cc|ccc}&\mathbb{Z}_{4}&\mathbb{Z}_{4}&\mathbb{Z}_{2}&\mathbb{Z}_{2}&\mathbb{Z}_{2}\\\
\hline\cr\mathbb{Z}&2&&*&1&\\\ \mathbb{Z}&&&1&&\\\ \mathbb{Z}&&&&&\\\
\hline\cr\mathbb{Z}_{4}&*&2&*&*&1\\\ \mathbb{Z}_{4}&1&&\\\
\hline\cr\mathbb{Z}_{2}&*&*&&1&\\\ \mathbb{Z}_{2}&*&*&&&\\\ \end{array},$
(294)

(where $*$s are arbitrary), but it can be further reduced to

$\displaystyle\to\begin{array}[]{c|cc|ccc}&\mathbb{Z}_{4}&\mathbb{Z}_{4}&\mathbb{Z}_{2}&\mathbb{Z}_{2}&\mathbb{Z}_{2}\\\
\hline\cr\mathbb{Z}&2&&&1&\\\ \mathbb{Z}&&&1&&\\\ \mathbb{Z}&&&&&\\\
\hline\cr\mathbb{Z}_{4}&&2&&&1\\\ \mathbb{Z}_{4}&1&&\\\
\hline\cr\mathbb{Z}_{2}&&*&&&\\\ \mathbb{Z}_{2}&&*&&&\\\ \end{array}.$ (303)

(II) Furthermore, for the direct summands of $\mathbb{Z}_{2}$, we transform
the off-diagonal blocks to Hermite normal form by a basis transformation: we
decompose the extension matrix $M$ into $\mathbb{Z}_{2}$ and
non-$\mathbb{Z}_{2}$ components as

$\displaystyle M=\begin{pmatrix}M^{(*,*)}&M^{(*,2)}\\\
M^{(2,*)}&M^{(2,2)}\end{pmatrix}.$ (304)

We use unimodular transformations $U$ and $V$ to bring $H^{(2,*)}=UM^{(2,*)}$
into the row-style Hermite normal form, and $H^{(*,2)}=M^{(*,2)}V$ into the
column-style Hermite normal form to get

$\displaystyle M$ $\displaystyle\mapsto\begin{pmatrix}1\\\ &U\\\
\end{pmatrix}M\begin{pmatrix}1\\\ &V\\\ \end{pmatrix}$
$\displaystyle=\begin{pmatrix}M^{(*,*)}&H^{(*,2)}\\\ H^{(2,*)}&UM^{(2,2)}V\\\
\end{pmatrix}.$ (305)

Repeat steps (I) and (II) in this order twice to further compress the set of
extension matrices $\\{M\\}$. For a given extension matrix $M$, the extension
corresponding to the matrix $M$ is computed by the Baer sum. Thus, we can
collect all the possible $\mathbb{Z}$-modules obtained by extensions.
Supplemental Material [52] lists the set of $\mathbb{Z}$-modules obtained as
extensions of $A$ by $B$ for a given $A$ and $B$ that appeared in the
calculations of this paper.