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‘Real’ Gerbes and Dirac Cones of Topological Insulators

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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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  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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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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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

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

    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,

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

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

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

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:

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

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.


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

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


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

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


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).

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