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‘Real’ Gerbes and Dirac Cones of Topological Insulators

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Abstract

A time-reversal invariant topological insulator occupying a Euclidean half-space determines a ‘Quaternionic’ self-adjoint Fredholm family. We show that the discrete spectrum data for such a family is geometrically encoded in a non-trivial ‘Real’ gerbe. The gerbe invariant, rather than a naïve counting of Dirac points, precisely captures how edge states completely fill up the bulk spectral gap in a topologically protected manner.

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Notes

  1. The self-adjointness of the lift seems to have been left implicit in [6].

  2. Let \(D^{1,1}\) be the unit disc in the plane with involution \((x,y)\mapsto (x,-y)\), whose boundary \(S^{1,1}\) is \({\tilde{S}}^1\). In the KR-theory exact sequence for the pair \((X\times D^{1,1}, X\times S^{1,1})\), substitute the Thom isomorphism \(KR^{-n}(X\times D^{1,1}, X\times S^{1,1})\cong KR^{-n}(X)\) to obtain the Gysin sequence.

  3. This means a complex line bundle with an antiunitary lift of \(\iota \) squaring to \(+1\).

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Acknowledgements

KG is supprted by Japan JSPS KAKENHI Grant Numbers 20K03606 and JP17H06461. GCT is supported by Australian Research Council Discovery Projects Grant DP200100729, and thanks J. Kellendonk for helpful correspondence.

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Correspondence to Guo Chuan Thiang.

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Appendices

Equivariant Cohomology

1.1 Čech formulation

Let G be a compact Lie group and X a space with a left continuous action of G. By definition, the Borel equivariant cohomology \(H^n_G(X)\) is defined to be the singular integral cohomology of the Borel construction

$$\begin{aligned} H^n_G(X) = H^n_G(X; {\mathbb {Z}}) = H^n(EG \times _G X; {\mathbb {Z}}). \end{aligned}$$

Let us assume that

X is a G-CW complex.

This assumption will be good enough for a Čech cohomology description of the Borel equivariant cohomology, as outlined below:

  1. 1.

    In general, if \(X_\bullet \) is a simplicial space and A is a finitely generated abelian group, then the singular cohomology \(H^*(\Vert X_\bullet \Vert ; A)\) of the geometric realisation \(\Vert X_\bullet \Vert \) is isomorphic to the singular cohomology \(H^*(X_\bullet ; A)\) of the simplicial space [16]. The Borel construction \(EG \times _G X\) is homotopy equivalent to the geometric realisation \(\Vert G^\bullet \times X \Vert \) of the simplicial space \(G^\bullet \times X\). In more detail, the latter comprises the sequence of spaces \(\{ G^p \times X \}_{p \ge 0}\) together with the face maps \(\partial _i : G^{p+1} \times X \rightarrow G^{p} \times X\), (\(i = 0, \ldots , p+1\)) given by

    $$\begin{aligned} \partial _i(g_1, \ldots , g_{p+1}, x) = \left\{ \begin{array}{ll} (g_2, \ldots , g_{p+1}, x), &{} (i = 0) \\ (g_1, \ldots , g_{i-1}, g_ig_{i+1}, g_{i+1}, \ldots , g_{p+1}, x), &{} (i = 1, \ldots , p) \\ (g_1, \ldots , g_p, g_{p+1}x), &{} (i = p+1) \end{array} \right. \end{aligned}$$

    and the degeneracy maps \(s_i : G^p \times X \rightarrow G^{p+1} \times X\), (\(i = 0, \ldots , p\)) given by

    $$\begin{aligned} s_i(g_1, \ldots , g_p, x) = (g_1, \ldots , g_i, 1, g_{i+1}, \ldots , g_p, x). \end{aligned}$$

    These facts lead to isomorphisms:

    $$\begin{aligned} H^n_G(X) \cong H^n(\Vert G^\bullet \times X \Vert ; {\mathbb {Z}}) \cong H^n(G^\bullet \times X; {\mathbb {Z}}), \end{aligned}$$

    where \(H^n(G^\bullet \times X; {\mathbb {Z}})\) is the singular integral cohomology of the simplicial space \(G^\bullet \times X\). This is the cohomology associated to the double complex \((C^q(G^p \times X; {\mathbb {Z}}), \delta , \partial )\), where for each p, \((C^q(G^p \times X; {\mathbb {Z}}), \delta )\) is the singular cochain complex of the topological space \(G^p \times X\), while \(\partial : C^q(G^p \times X; {\mathbb {Z}}) \rightarrow C^q(G^{p+1} \times X; {\mathbb {Z}})\) is given by \(\partial = \sum _{i=0}^{p+1} (-1)^i \partial _i^*\), with \(\partial _i : G^{p+1} \times X \rightarrow G^p \times X\) the face maps in the simplicial space \(G^\bullet \times X\).

  2. 2.

    There is an isomorphism

    $$\begin{aligned} H^n(G^\bullet \times X; {\mathbb {Z}}) \cong {\check{H}}^n(G^\bullet \times X; {\mathbb {Z}}). \end{aligned}$$

    The right hand side is the Čech cohomology of the simplicial space \(G^\bullet \times X\) with coefficients in the constant sheaf \({\mathbb {Z}}\), defined as the colimit

    $$\begin{aligned} {\check{H}}^n(G^\bullet \times X; {\mathbb {Z}}) = \varinjlim _{{\mathfrak {U}}^\bullet } {\check{H}}^n({\mathfrak {U}}^\bullet ; {\mathbb {Z}}), \end{aligned}$$

    where \({\check{H}}^n({\mathfrak {U}}^\bullet ; {\mathbb {Z}})\) is the Čech cohomology associated to an open cover \({\mathfrak {U}}^\bullet \) of the simplicial space \(G^\bullet \times X\). Such an open cover consists of a sequence of open covers \({\mathfrak {U}}^0, {\mathfrak {U}}^1, \ldots \), where \({\mathfrak {U}}^p = \{ U^p_i \}_{i \in I^p }\) is an open cover of \(G^p \times X\), the sequence of sets \(\{ I^p \}\) forms a simplicial set, and a compatibility with the face maps is supposed. For example, suppose G is finite, and let \(\{ U_i \}_{i \in I}\) be an invariant open cover of X. Then there is an associated open cover \({\mathfrak {U}}^\bullet \) of \(G^\bullet \times X\), in which \({\mathfrak {U}}^p\) of \(G^p \times X\) consists of open sets

    $$\begin{aligned} U_{(g_1, \ldots , g_p, i)} = \{ (g_1, \ldots , g_p) \} \times U_i \end{aligned}$$

    indexed by \((g_1, \ldots , g_p, i) \in G^p \times I\). (Details can be found in [23].)

    For each \(p = 0, 1, \ldots \), we have the usual Čech complex \(({\check{C}}^q({\mathfrak {U}}^p; {\mathbb {Z}}), \delta )\). From the compatibility, we can define a homomorphism \(\partial : {\check{C}}^q({\mathfrak {U}}^p; {\mathbb {Z}}) \rightarrow {\check{C}}^q({\mathfrak {U}}^{p+1}; {\mathbb {Z}})\). This leads to a double complex \(({\check{C}}^q({\mathfrak {U}}^p; {\mathbb {Z}}), \delta , \partial )\), and its associated cohomology is \({\check{H}}^*({\mathfrak {U}}^\bullet ; {\mathbb {Z}})\).

    Mimicking the idea in [7], we may construct a homomorphism

    $$\begin{aligned} H^n(G^\bullet \times X; {\mathbb {Z}}) \rightarrow {\check{H}}^n(G^\bullet \times X; {\mathbb {Z}}). \end{aligned}$$

    Each of the singular cohomology and the Čech cohomology is defined through a double complex. Hence the above homomorphism becomes an isomorphism when

    $$\begin{aligned} H^n(G^p \times X; {\mathbb {Z}}) \rightarrow {\check{H}}^n(G^p \times X; {\mathbb {Z}}) \end{aligned}$$

    is an isomorphism for all np, as is the case in our setup.

  3. 3.

    Henceforth, suppose G is a finite group. Then, we may show that any open cover \({\mathfrak {U}}^\bullet \) of \(G^\bullet \times X\) admits a refinement associated to an invariant open cover: Since \(G^p \times X\) is paracompact, we may find a refinement \({\mathfrak {V}}^\bullet \) such that the open cover \({\mathfrak {V}}^p = \{ V^p_j \}_{j \in J^p}\) of \(G^p \times X\) is locally finite. Being a CW complex, X is (completely) regular. So, for each \(x \in X\), we can find an open set \(W_x\) such that \(x \in W_x \subset V^0_j\) for all \(j \in J^0\) such that \(x \in V^0_j\). Considering the action of the finite group G, we may take \(W_x\) to be G-invariant, and we would eventually get an invariant open cover \({\mathfrak {W}}\) of X whose associated open cover of \(G^\bullet \times X\) refines \({\mathfrak {V}}^\bullet \).

    As a result, in the case that G is finite, it is enough to consider Čech cohomology groups \({\check{H}}^*({\mathfrak {U}}^\bullet ; {\mathbb {Z}})\) of the open covers \({\mathfrak {U}}^\bullet \) associated to invariant open covers \({\mathfrak {U}}\) of X.

  4. 4.

    Suppose further that \(G = {\mathbb {Z}}_2\). Let \({\mathfrak {U}}= \{ U_i \}_{i \in I}\) be an invariant open cover of X. We have another double complex \(({\check{C}}^{p, q}({\mathfrak {U}}), \delta , \partial )\) comprising Čech q-cochains, with \(\delta \) the usual Čech coboundary, and \(\partial :{\check{C}}^{p,q}({\mathfrak {U}})\rightarrow {\check{C}}^{p+1,q}({\mathfrak {U}})\) given by \(\partial (\omega )=\omega -(-1)^p\iota ^*\omega \). It can be shown that \(({\check{C}}^{p, q}({\mathfrak {U}}), \delta , \partial )\) and \(({\check{C}}^q({\mathfrak {U}}^p; {\mathbb {Z}}), \delta , \partial )\) are quasi-isomorphic: Taking a refinement if necessary, we can assume that \({\mathfrak {U}}^\bullet \) is an open cover of the simplicial space \({\mathbb {Z}}_2^\bullet \times X\) of the form described in item 2 above. Then the complex \(\partial : {\check{C}}^q({\mathfrak {U}}^p; {\mathbb {Z}}) \rightarrow {\check{C}}^q({\mathfrak {U}}^{p+1}; {\mathbb {Z}})\) is identified with the cochain complex of the group \({\mathbb {Z}}_2\) with coefficients in a \({\mathbb {Z}}_2\)-module. It is well-known that the cochain complex of a group is quasi-isomorphic to the normalized cochain complex. In the present case, the normalized cochain complex agrees with \(\partial : {\check{C}}^{p, q}({\mathfrak {U}}) \rightarrow {\check{C}}^{p+q, q}({\mathfrak {U}})\). Thus, by a spectral sequence argument, the total complexes for the two double complexes are quasi-isomorphic.

  5. 5.

    To summarize, if X is a \({\mathbb {Z}}_2\)-CW complex, then

    $$\begin{aligned} H^n_{{\mathbb {Z}}_2}(X) \cong \varinjlim _{{\mathfrak {U}}} H^n({\check{C}}^{*, *}({\mathfrak {U}})), \end{aligned}$$
    (32)

    where \({\mathfrak {U}}\) runs over invariant open covers of X.

If X is a G-CW complex with G a compact Lie group, then \(G^p \times X\) is a CW complex, thus locally contractible. As a result, we get the exponential exact sequence of sheaves,

$$\begin{aligned} 0 \longrightarrow {\mathbb {Z}}\longrightarrow {\underline{{\mathbb {R}}}} \overset{\mathrm{exp}\,2\pi i(\cdot )}{\longrightarrow } \underline{{\mathbb {T}}} \longrightarrow 0, \end{aligned}$$
(33)

where \({\mathbb {Z}}\) is the constant sheaf, and \({\underline{A}}\) means the sheaf of germs of A-valued continuous functions. From this, we get the associated exact sequence

$$\begin{aligned} \rightarrow {\check{H}}^n(G^\bullet \times X; {\mathbb {Z}}) \rightarrow {\check{H}}^n(G^\bullet \times X; {\underline{{\mathbb {R}}}}) \rightarrow {\check{H}}^n(G^\bullet \times X; \underline{{\mathbb {T}}}) \rightarrow {\check{H}}^{n+1}(G^\bullet \times X; {\mathbb {Z}}) \rightarrow . \end{aligned}$$

Because \(G^p \times X\) is paracompact for \(p \ge 0\), a spectral sequence argument identifies \({\check{H}}^n(G^\bullet \times X; {\underline{{\mathbb {R}}}})\) with the group cohomology with coefficients in a G-module. Then a standard “averaging argument” leads to \({\check{H}}^n(G^\bullet \times X; {\underline{{\mathbb {R}}}}) = 0\) for \(n \ne 0\), and

$$\begin{aligned} {\check{H}}^n(G^\bullet \times X; \underline{{\mathbb {T}}}) \cong {\check{H}}^{n+1}(G^\bullet \times X; {\mathbb {Z}}) \cong H^{n+1}_G(X) \end{aligned}$$
(34)

for \(n \ge 1\). The details of this argument can be found in [23] (Lemma 4.4). This gives another way to a Čech formulation of the Borel equivariant cohomology.

1.2 Local coefficients

On a connected CW complex X, a choice of a homomorphism \(\ell \in \mathrm {Hom}(\pi _1(X), {\mathbb {Z}}_2) \cong H^1(X; {\mathbb {Z}}_2)\) makes the group \({\mathbb {Z}}\) into a module over the fundamental group \(\pi _1(X)\), and allows us to “twist” the integral cohomology \(H^*(X)\) to produce the cohomology \(H^{\ell + *}(X)\) with local coefficients [29]. As an application of this construction, a choice of a homomorphism \(\phi : G \rightarrow {\mathbb {Z}}_2\) leads to a \(\phi \)-twisted version \(H^{\phi + *}_G(X)\) of the Borel equivariant integral cohomology \(H^*_G(X)\) of a G-CW complex X, since \(\phi \in \mathrm {Hom}(\pi _1(BG)), {\mathbb {Z}}_2) \cong H^1(BG; {\mathbb {Z}}_2)\) leads to an element in \(H^1(EG \times _G X; {\mathbb {Z}}_2)\) by pull-back. In particular, when \(G = {\mathbb {Z}}_2\) and \(\phi = \mathrm {id}\) is the identity map, the \(\phi \)-twisted equivariant cohomology is denoted \(H^*_\pm (X) = H^*_{{\mathbb {Z}}_2}(X; {\mathbb {Z}}(1)) = H^{\phi + *}_{{\mathbb {Z}}_2}(X)\).

The exact sequence of sheaves, Eq. (33), can be made \({\mathbb {Z}}_2\)-equivariant, with the action on \({\mathbb {Z}}\) being \(m\mapsto -m\) (whence the equivariant sheaf is denoted \({\mathbb {Z}}(1)\)). To the equivariant sheaf \({\mathbb {Z}}(1)\) we can associate a sheaf \({\tilde{{\mathbb {Z}}}}\) on the simplicial space \({\mathbb {Z}}_2^\bullet \times X\) as a twisted version of the constant sheaf \({\mathbb {Z}}\), and the same argument leading to Eq. (34) gives

$$\begin{aligned} H^n_\pm (X) \cong {\check{H}}^n({\mathbb {Z}}_2^\bullet \times X; {\tilde{{\mathbb {Z}}}}) \cong {\check{H}}^n({\mathbb {Z}}_2^\bullet \times X; \underline{\tilde{{\mathbb {T}}}}), \end{aligned}$$

where \(\underline{\tilde{{\mathbb {T}}}}\) is likewise a twisted version of the sheaf \(\underline{{\mathbb {T}}}\) on \({\mathbb {Z}}_2^\bullet \times X\). See [22] Appendix A, which is based on [27, 35], for details.

As in Eq. (32) for the untwisted case, we may identify \(H^n_\pm (X)\) with the (colimit of the) cohomology of the double complex \(({\check{C}}^{p, q}({\mathfrak {U}}), \delta , \partial )\) comprising Čech q-cochains, but with \(\partial :{\check{C}}^{p,q}({\mathfrak {U}})\rightarrow {\check{C}}^{p+1,q}({\mathfrak {U}})\) given by \(\partial (\omega )=\omega +(-1)^p\iota ^*\omega \).

1.3 First equivariant cohomology

Let X be a \({\mathbb {Z}}_2\)-CW complex. In the Čech formulation of \(H^n_{{\mathbb {Z}}_2}(X)\), Eq. (32), a 1-cochain

$$\begin{aligned} (\kappa _{ij}, \lambda _i) \in {\check{C}}^{0, 1}({\mathfrak {U}}) \oplus {\check{C}}^{1, 0}({\mathfrak {U}}) = {\check{C}}^1({\mathfrak {U}}; {\mathbb {Z}}) \oplus {\check{C}}^0({\mathfrak {U}}; {\mathbb {Z}}) \end{aligned}$$

consists of locally constant functions \(\kappa _{ij} : U_i \cap U_j \rightarrow {\mathbb {Z}}\) and \(\lambda _i : U_i \rightarrow {\mathbb {Z}}\). The cocycle condition for \((\kappa _{ij}, \lambda _i)\) reads

$$\begin{aligned} (\delta \kappa )_{ijk}&= 0,&(\delta \lambda )_{ij}&= \partial \kappa _{ij},&0&= \partial \lambda _i. \end{aligned}$$

These conditions are equivalent to:

$$\begin{aligned} \kappa _{jk}(x) - \kappa _{ik}(x) + \kappa _{ij}(x)&= 0,&(x \in U_i \cap U_j \cap U_k)\nonumber \\ \lambda _j(x) - \lambda _i(x)&= \kappa _{ij}(x) - \kappa _{ij}(\iota (x)),&(x \in U_i \cap U_j) \nonumber \\ 0&= \lambda _i(x) + \lambda _i(\iota (x)).&(x \in U_i) \end{aligned}$$
(35)

Lemma A.1

Let X be a \({\mathbb {Z}}_2\)-CW complex such that \(H^1_{{\mathbb {Z}}_2}(X) = 0\). Suppose that we have:

  1. 1.

    an invariant open cover \({\mathfrak {U}}= \{ U_\mu \}_{\mu \in A}\) of X,

  2. 2.

    locally constant functions \(r_{\mu \nu } : U_\mu \cap U_\nu \rightarrow {\mathbb {Z}}\) for \(\mu , \nu \in A\) such that

    • \(r_{\mu \xi }(x) = r_{\mu \nu }(x) + r_{\nu \xi }(x)\) for all \(x \in U_{\mu } \cap U_{\nu } \cap U_{\xi }\) and \(\mu , \nu , \xi \in A\);

    • \(r_{\mu \nu }(x) = r_{\mu \nu }(\iota (x))\) for all \(x \in U_{\mu } \cap U_{\nu }\) and \(\mu , \nu \in A\).

Then there is an invariant open cover \({\mathfrak {V}}= \{ V_i \}_{i \in I}\) which refines \({\mathfrak {U}}\) and, we can find locally constant functions \(\tau _i : V_i \rightarrow {\mathbb {T}}\) satisfying

$$\begin{aligned} \exp \pi i r_{ij}(x)&= \tau _j(x) \tau _i(x)^{-1},&(x \in V_i \cap V_j) \\ \overline{\tau _i(x)}&= \tau _i (\iota (x)),&(x \in V_i) \end{aligned}$$

where \(r_{ij} : V_i \cap V_j \rightarrow {\mathbb {Z}}\) are induced from \(r_{\mu \nu }\) through the refinement.

Proof

The locally constant functions \(r_{\mu \nu }\) define a 1-cocycle

$$\begin{aligned} (\kappa _{\mu \nu }, \lambda _\mu ) = (r_{\mu \nu }, 0) \in {\check{C}}^{0, 1}({\mathfrak {U}}) \oplus {\check{C}}^{1, 0}({\mathfrak {U}}). \end{aligned}$$

By the assumption \(0 = H^1_{{\mathbb {Z}}_2}(X) \cong \varinjlim H^1({\check{C}}^{*, *}({\mathfrak {U}}))\), there exists an invariant open cover \({\mathfrak {V}}= \{ V_i \}_{i \in I}\) which refines \({\mathfrak {U}}\) and \(H^1({\check{C}}^{*, *}({\mathfrak {V}})) = 0\). Let \((r_{ij}, 0) \in {\check{C}}^{0, 1}({\mathfrak {V}}) \oplus {\check{C}}^{1, 0}({\mathfrak {V}})\) be the induced 1-cocycle. Because of the vanishing \(H^1({\check{C}}^{*, *}({\mathfrak {V}})) = 0\), the 1-cocyle is a coboundary, so that there exists

$$\begin{aligned} (m_i) \in {\check{C}}^{0, 0}({\mathfrak {V}}) = {\check{C}}^0({\mathfrak {V}}; {\mathbb {Z}}) \end{aligned}$$

such that

$$\begin{aligned} (\delta m)_{ij}&= r_{ij},&\partial m_i&= 0. \end{aligned}$$

Thus, we have locally constant functions \(m_i : V_i \rightarrow {\mathbb {Z}}\) for \(i \in I\) such that

$$\begin{aligned} m_j(x) - m_i(x)&= r_{ij}(x),&(x \in V_i \cap V_j) \\ m_i(x) - m_i(\iota (x))&= 0.&(x \in V_i) \end{aligned}$$

Now, \(\tau _i : V_i \rightarrow {\mathbb {T}}\) is given by \(\tau _i(x) = \exp \pi i m_i(x)\). \(\square \)

Remark A.2

If we assume that X is connected, then the choice of \((m_i)\) can be constrained: Let \((m'_i)\) be another choice which cobounds the 1-cocycle \((r_{ij}, 0)\). The difference cocycle \(d_i = m'_i - m_i\) gives rise to global constant \(d \in {\mathbb {Z}}\). Hence the difference \(\tau '(x) / \tau (x) = \exp \pi i d = \pm 1\) is a global sign.

1.3.1 Other formulations of first equivariant cohomology

Let \({\mathbb {T}}_{\mathrm{triv}}\) be the unit circle in the complex plane equipped with the trivial involution, and \({\mathbb {T}}_{\mathrm{conj}}\) be the one equipped with the complex conjugation involution.

Lemma A.3

Let X be a \({\mathbb {Z}}_2\)-CW complex. Then there are isomorphisms of groups

$$\begin{aligned} H^1_{{\mathbb {Z}}_2}(X)&\cong [X, {\mathbb {T}}_{\mathrm {triv}}]_{{\mathbb {Z}}_2},\\ H^1_{\pm }(X)&\cong [X, {\mathbb {T}}_{\mathrm {conj}}]_{{\mathbb {Z}}_2}. \end{aligned}$$

Proof

A proof of \(H^1_{\pm }(X) \cong [X, {\mathbb {T}}_{\mathrm {conj}}]_{{\mathbb {Z}}_2}\) is given in Proposition A.2 of [22]. The proof of the first statement is similar, and detailed below. Write \(G={\mathbb {Z}}_2\) for brevity. We have \(H^1_G(X) = H^1_G(X; {\mathbb {Z}})\cong {\check{H}}^1(G^\bullet \times X; {\mathbb {Z}})\), and the long exact sequence

$$\begin{aligned} \rightarrow {\check{H}}^n(G^\bullet \times X; {\mathbb {Z}}) \rightarrow {\check{H}}^n(G^\bullet \times X; {\underline{{\mathbb {R}}}}) \rightarrow {\check{H}}^n(G^\bullet \times X; \underline{{\mathbb {T}}}) \overset{\delta }{\rightarrow } {\check{H}}^{n+1}(G^\bullet \times X; {\mathbb {Z}}) \rightarrow \end{aligned}$$

induced by the short exact sequence of sheaves, Eq. (33).

For \(n=0\), the zeroth Čech cohomology is the group of global sections of the coefficient sheaf. Since \(G^p \times X\) is paracompact and G is finite, \({\check{H}}^1(G^\bullet \times X; {\underline{{\mathbb {R}}}}) = 0\), thus we have

$$\begin{aligned} C(X, {\mathbb {Z}})^G \rightarrow C(X, {\mathbb {R}})^G \rightarrow C(X, {\mathbb {T}})^G \overset{\delta }{\rightarrow } H^1_G(X) \rightarrow 0, \end{aligned}$$

where C(XA) stands for the group of continuous functions \(f : X \rightarrow A\), and \(C(X, A)^G\) is the subgroup consisting of invariant functions: \(f(gx) = f(x)\) for all \(x \in X\) and \(g \in G\). Note that the group of integers \(A = {\mathbb {Z}}\) is given the discrete topology, while \(A = {\mathbb {R}}, {\mathbb {T}}\) are given the standard topologies.

The homomorphism \(\delta : C(X, {\mathbb {T}})^G \rightarrow H^1_G(X)\) descends to give a homomorphism \(\delta : [X, {\mathbb {T}}_{\mathrm {triv}}]_G \rightarrow H^1_G(X)\). To see this, let \(\pi : X \times [0, 1] \rightarrow X\) be the projection, and \(i_t : X \rightarrow X \times [0, 1]\) the map \(i_t(x) = (x, t)\) for each \(t \in [0, 1]\). By the homotopy axiom, \(i_t^* : H^1_G(X \times [0, 1]) \rightarrow H^1_G(X)\) is an isomorphism for any \(t \in [0, 1]\) and its inverse is \(\pi ^* : H^1_G(X) \rightarrow H^1_G(X \times [0, 1])\). From the commutative diagram

it follows that \(\delta (i_0^*{\tilde{f}}) = i_0^* \delta ({\tilde{f}}) = i_1^*\delta ({\tilde{f}}) = \delta (i_1^*{\tilde{f}})\) for all \({\tilde{f}} \in C(X, {\mathbb {T}})^G\). This means that homotopic maps in \(C(X, {\mathbb {T}})^G\) have the same image under \(\delta \), and we get a well-defined \(\delta : [X, {\mathbb {T}}_{\mathrm {triv}}]_G \rightarrow H^1_G(X)\).

Now, let \([f] \in [X, {\mathbb {T}}_{\mathrm {triv}}]_G\) be the element represented by \(f \in C(X, {\mathbb {T}})^G\). Suppose that \(\delta ([f]) = \delta (f) = 0\). By the exact sequence, there is a continuous map \(h : X \rightarrow {\mathbb {R}}\) such that \(f(x) = \exp 2\pi i h(x)\) and \(h(gx) = h(x)\) for all \(x \in X\) and \(g \in G\). If we define \({\tilde{f}} : X \times [0, 1] \rightarrow {\mathbb {T}}\) by \({\tilde{f}}(x, t) = \exp 2 \pi i t h(x)\), then \({\tilde{f}}\) is an equivariant homotopy between f and the constant map 1. Therefore the epimorphism

$$\begin{aligned} \delta : [X, {\mathbb {T}}_{\mathrm {triv}}]_G \rightarrow H^1_G(X) \end{aligned}$$

has trivial kernel, thus it is an isomorphism. \(\square \)

Remark A.4

We can describe the homomorphism \(\delta : [X, {\mathbb {T}}_{\mathrm {triv}}]_{{\mathbb {Z}}_2} \rightarrow H^1_{{\mathbb {Z}}_2}(X)\) explicitly. Let \({\mathfrak {U}}= \{ U_i \}_{i \in I}\) be an invariant open cover. Taking a refinement if necessary, we can assume that \({\mathfrak {U}}\) is such that \(H^1(U_i) = 0\). Given \(\varphi \in C(X, {\mathbb {T}})^{{\mathbb {Z}}_2}\), there are continuous maps \({\tilde{\varphi }}_i : U_i \rightarrow {\mathbb {R}}\) such that \(\varphi (x) = \exp 2\pi \sqrt{-1}{\tilde{\varphi }}_i(x)\) for \(x \in U_i\). Now, by setting

$$\begin{aligned} \kappa _{ij}(x)&= {\tilde{\varphi }}_j(x) - {\tilde{\varphi }}_i(x),&\lambda _i(x)&= {\tilde{\varphi }}_i(x) - {\tilde{\varphi }}_i(\iota (x)), \end{aligned}$$

we get locally constant functions \(\kappa _{ij} : U_i \cap U_j \rightarrow {\mathbb {Z}}\) and \(\lambda _i : U_i \rightarrow {\mathbb {Z}}\) satisfying the 1-cocycle conditions, Eq. (35), thus giving a Čech 1-cohomology class in \(H^1_{{\mathbb {Z}}_2}(X)\). A similar construction applies to \([X, {\mathbb {T}}_{\mathrm {conj}}]_{{\mathbb {Z}}_2} \rightarrow H^1_\pm (X)\).

Equivariant Cohomology of the Fixed Point Set

Lemma B.1

Let X be a space with involution \(\iota : X \rightarrow X\). If the involution is trivial, \(X^\iota = X\), then there is a natural isomorphism of groups

$$\begin{aligned} H^n_\pm (X)&\cong \bigoplus _{k \ge 0} H^{n - 2k - 1}(X; {\mathbb {Z}}_2) \\&= H^{n-1}(X; {\mathbb {Z}}_2) \oplus H^{n-3}(X; {\mathbb {Z}}_2) \oplus H^{n-5}(X; {\mathbb {Z}}_2) \oplus \cdots . \end{aligned}$$

Proof

The \({\mathbb {Z}}_2\)-equivariant cohomology \(H^n_\pm (X)\) with local coefficients can be identified with the nth cohomology of the total complex associated to the double complex \((C^{p, q}, \delta , \partial )\), where \(C^{p, q} = C^q(X; {\mathbb {Z}})\) is the (Čech) cochain complex of X computing the integral cohomology of X, \(\delta : C^{p, q} \rightarrow C^{p, q+1}\) its coboundary operator, and \(\partial : C^{p, q} \rightarrow C^{p+1, q}\) the map defined by \(\partial c = c + (-1)^p \iota ^*c\).

Under the assumption \(X^\iota = X\), we have \(\partial c = c + (-1)^p c\). This leads us to decompose the total complex \(C^*\) of \(C^{p, q}\) as

$$\begin{aligned} C^* = \bigoplus _{k \ge 0} \mathrm {Cone}( 2 : C^*(X; {\mathbb {Z}}) \rightarrow C^*(X; {\mathbb {Z}}))[-2k-1], \end{aligned}$$

where \(\mathrm {Cone}( f : D^* \rightarrow E^* )\) stands for the cone complex associated to a cochain map \(f : D^* \rightarrow E^*\), and [m] means the degree shift of a cochain complex by m. The cohomology of \(\mathrm {Cone}(2 : C^*(X; {\mathbb {Z}}) \rightarrow C^*(X; {\mathbb {Z}}))\) is naturally isomorphic to the mod 2 cohomology \(H^*(X; {\mathbb {Z}}_2)\), and the proof is completed. \(\square \)

Remark B.2

A similar proof also shows that

$$\begin{aligned} H^n_{{\mathbb {Z}}_2}(X)&\cong H^n(X; {\mathbb {Z}}) \oplus \bigoplus _{k \ge 1} H^{n - 2k}(X; {\mathbb {Z}}_2) \\&= H^n(X; {\mathbb {Z}}) \oplus H^{n-2}(X; {\mathbb {Z}}_2) \oplus H^{n-4}(X; {\mathbb {Z}}_2) \oplus \cdots . \end{aligned}$$

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Gomi, K., Thiang, G.C. ‘Real’ Gerbes and Dirac Cones of Topological Insulators. Commun. Math. Phys. 388, 1507–1555 (2021). https://doi.org/10.1007/s00220-021-04238-0

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