nLab
transgression in group cohomology

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Context

Group Theory

Cohomology

cohomology

Special and general types

Special notions

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Extra structure

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Theorems

Contents

Idea

For GG a discrete group (often taken to be a finite group) and for AA any abelian group, there is a transgression homomorphism of cohomology groups

(1)H grp +1(G;A)=H +1(BG;A)τH n(ΛBG,A)[g]G ad/GH grp n(C g;A) H_{grp}^{\bullet + 1}(G;\,A) \;\;=\;\; H^{\bullet + 1}(B G;\, A) \xrightarrow{ \;\;\; \tau \;\;\; } H^n \big( \Lambda B G, \, A \big) \;\;\; \simeq \; \underset{ [g] \in G^{ad}/G }{\oplus} H^n_{grp}\big(C_g;\, A \big)

from the group cohomology of GG to the groupoid cohomology, in one degree lower, of the inertia groupoid ΛBG\Lambda B G of its delooping groupoid BG(G*)B G \simeq (G \rightrightarrows \ast).

Since the inertia groupoid of the delooping groupoid is equivalent to a disjoint union over conjugacy classes [g]G ad/G[g] \in G^{ad}/G of delooping groupoids of centralizer subgroups C gC_g, with C e=GC_e = G,

(2)ΛBG[g]G ad/GBC gBG[g][e]BC g, \Lambda B G \;\simeq\; \underset{ [g] \in G_{ad}/G }{\coprod} B C_g \;\;\;\simeq\;\;\; B G \,\sqcup\, \underset{ [g] \neq [e] }{\coprod} B C_g \,,

this induces a corestricted transgression map within the group cohomology of GG, and, more generally, to the group cohomology of any of its centralizer subgroups:

H grp +1(G;A)τ [e]H grp (G;A),H grp +1(G,A)τ [g]H grp (C g;A). H_{grp}^{\bullet + 1}(G;\,A) \xrightarrow{ \;\;\; \tau_{[e]} \;\;\; } H_{grp}^{\bullet}(G;\,A) \,, \;\;\;\;\;\;\; H_{grp}^{\bullet + 1}(G,A) \xrightarrow{ \;\;\; \tau_{[g]} \;\;\; } H_{grp}^{\bullet}(C_g;\,A) \,.

It is a folklore theorem that the transgression (1) maps an (n+1)(n+1)-cocycle c:G × n+1Ac \colon G^{\times_{n+1}} \to A to the alternating sum (of functions with values in the abelian group AA with group operation denoted “++”)

(3)τ(c)(γ,g n1,,g 1,g 0)=±0jn(1) jc(g n1,,g nj,Ad j(γ),g nj1,,g 0), \tau(c)(\gamma, g_{n-1}, \cdots, g_1, g_0) \;=\; \pm \underset{ 0 \leq j \leq n }{\sum} (-1)^j \cdot c( g_{n-1}, \cdots, g_{n-j}, Ad_j(\gamma), g_{n-j-1}, \cdots, g_0 ) \,,

where

γg n1Ad n1(γ)g n2g 0Ad 0(γ)ΛBG \gamma \xrightarrow{ g_{n-1} } Ad_{n-1}(\gamma) \xrightarrow{ g_{n-2} } \cdots \xrightarrow{ g_0 } Ad_{0}(\gamma) \;\;\;\;\; \in \;\; \Lambda B G

is any sequence of nn composable morphisms in the inertia groupoid, and where we use a shorthand for the adjoint action of GG on itself:

Ad j(γ) Ad (g n1g j)(γ) (g n1g j) 1γ(g n1g j), \begin{aligned} Ad_j(\gamma) & \;\coloneqq\; Ad_{(g_{n-1}\cdots g_j)}(\gamma) \\ & \;\coloneqq\; (g_{n-1} \cdots g_j)^{-1} \cdot \gamma \cdot (g_{n-1} \cdots g_j) \,, \end{aligned}

(which restricts to Ad j(γ)=γAd_j(\gamma) = \gamma upon corestriction to the connected component on the right of (2) that is indexed by [γ][\gamma]).

In historically influential examples, for the case n=4n = 4 and A=A = \mathbb{Z} or, equivalently, n=3n = 3 and A=A = U(1), this formula (3):

Below we mean to spell out a general abstract definition (Def. ) of the transgression map (1) and a full proof of its component formula (3), amplifying that its form is a direct consequence of – besides some basic homotopy theory/homological algebra which we review below – the classical Eilenberg-Zilber theorem, i.e. the Eilenberg-Zilber/Alexander-Whitney deformation retraction (which was partially re-discovered in Willerton 2008, Sec. 1 under the name “Parmesan theorem”).


Background and Lemmata

Homotopy and homological algebra

Some relevant basics of homotopy theory in relation to homological algebra:

Definition

(homotopy categories)

We write:

For AA \in Ab, and nn \in \mathbb{N} we write

A[n]Ch +Loc WHo(Ch +) A[n] \,\in\, Ch_+ \xrightarrow{Loc_{\mathrm{W}}} Ho(Ch_+)

for the chain complex concentrated on AA in degree nn.

We denote (using the same symbols for derived functors as for the original functors):

Simplicial classifying spaces

Remark

(delooping groupoid and simplicial classifying space of finite group)
The nerve of the delooping groupoid of a discrete group GG is isomorphic to the simplicial classifying space of GG (see this Example):

N(G*)W¯GsSet. N \big( G \rightrightarrows \ast \big) \;\simeq\; \overline{W} G \;\;\; \in \; sSet \,.

For notational brevity we will be referring to W¯G\overline{W}G in the following, but it may be helpful to keep thinking of the nerve of the delooping groupoid. From that perspective, an n-simplex in W¯G\overline{W}G, which is an n-tuple of group elements, is suggestively denoted as a sequence of morphisms:

(W¯G) n={g n1g 1g 0|g iG}. \big(\overline{W}G\big)_{n} \;\; = \;\; \Big\{ \bullet \xrightarrow{g_{n-1}} \bullet \xrightarrow{\;} \cdots \bullet \xrightarrow{\;} \bullet \xrightarrow{g_1} \bullet \xrightarrow{g_0} \bullet \;\big\vert\; g_i \in G \Big\} \,.

We denote the image of W¯G\overline{W}G in the classical homotopy category by

BG=BG=Loc W(W¯G)Ho(sSet). B G \;=\; \mathbf{B} G \;=\; Loc_{\mathrm{W}}(\overline{W}G) \;\;\; \in \; Ho(sSet) \,.

Group(oid) cohomology

Some relevant basics of cohomology, for the cases of ordinary cohomology and group cohomology:

Definition

(Eilenberg-MacLane spaces)
For AA \,\in\, Ab, and nn \in \mathbb{N}, we write

B nA=K(A,n)frgtDK(A[n])sAbLoc WHo(sAb)frgtHo(sSet) B^n A \,=\, K(A,n) \,\coloneqq\, frgt \circ DK(A[n]) \;\;\; \in \; sAb \xrightarrow{ Loc_{\mathrm{W}}} Ho(sAb) \xrightarrow{frgt} Ho(sSet)

for (the homotopy type of) the Eilenberg-MacLane space with AA in degree nn.

Definition

(ordinary cohomology)
For XHo(sSet)X \in Ho(sSet), AAbA \,\in\, Ab and nn \in \mathbb{N}, the degree-nn ordinary cohomology of XX with coefficients in AA is

H n(X;A)=Ho(sSet)(X,B nA), H^n(X;\, A) \;=\; Ho(sSet)(X, \, B^n A) \,,

where on the right we have the Eilenberg-MacLane space from Def. .

The following is an immediate re-casting of the traditional definition of group cohomology:

Definition

(group cohomology)
For GG \,\in\, Groups and AA \,\in\, AbelianGroups, the group cohomology of GG with coefficients in AA is, in degree nn \in \mathbb{N}, the hom-group

H grp n(G;A)=Ho(Ch +)(N (W¯G),A[n]). H^n_{grp}(G;\,A) \;=\; Ho(Ch_+) \big( N_\bullet \circ \mathbb{Z}(\overline{W}G), \, A[n] \big) \,.

Proposition

(group cohomology is ordinary cohomology of classifying space)
For GG \,\in\, Groups and AA \,\in\, AbelianGroups, the group cohomology (Def. ) of GG with coefficient in AA is naturally isomorphic to the ordinary cohomology of the simplicial classifying space of GG with coefficients in B nAB^n A:

H grp n(G;A)H n(BG,B A). H^n_{grp}(G;\, A) \;\simeq\; H^n(B G,\, B^A) \,.

Proof

By the hom-isomorphisms of the above derived adjunctions:

H grp n(G;A) 0Ho(Ch +)(N (W¯G),A[n]) Ho(sAb)((W¯G),DK(A[n])) Ho(sSet)(W¯GfrgtDK(A[n])) =Ho(sSet)(BGB nA) =H n(BG;A). \begin{aligned} H^n_{grp}(G;\, A) & \;0\; Ho(Ch_+) \big( N_\bullet \circ \mathbb{Z}(\overline{W}G), \, A[n] \big) \\ & \;\simeq\; Ho(sAb) \big( \mathbb{Z}(\overline{W}G), \, DK(A[n]) \big) \\ & \;\simeq\; Ho(sSet) \big( \overline{W}G \, frgt \circ DK(A[n]) \big) \\ & \;=\; Ho(sSet) \big( B G \, B^n A \big) \\ & \;=\; H^n(B G;\, A) \,. \end{aligned}

This Prop. , in view of Rem. justifies the following definition:

Definition

(groupoid cohomology)
For 𝒢=(𝒢 1𝒢 0)\mathcal{G} \,=\, (\mathcal{G}_1 \rightrightarrows \mathcal{G}_0) \,\in\, Groupoids and AA \,\in\, Ab, nn \,\in\, \mathbb{N}, the degree-nn groupoid cohomology of 𝒢\mathcal{G} with coefficients in AA is the ordinary cohomology (Def. ) of the homotopy type of the nerve of 𝒢\mathcal{G}, regarded in the classical homotopy category:

H n(𝒢;A)Ho(sSet)(N(𝒢),B nA). H^n \big( \mathcal{G};\, A \big) \;\coloneqq\; Ho(sSet) \big( N(\mathcal{G}), \, B^n A \big) \,.

Products of simplices

Some fundamental fact about products of simplicial sets:

Proposition

(non-degenerate (p+q)(p+q)-simplices in Δ[p]×Δ[q]\Delta[p] \times \Delta[q])
For p,qp,q \,\in\, \mathbb{N} the non-degenerate simplices in the Cartesian product (Prop. )

Δ[p]×Δ[q] \Delta[p] \times \Delta[q]

of standard simplices in sSet correspond, under the Yoneda lemma, to precisely those morphisms of simplicial sets

(8)Δ[p+q]σΔ[p]×Δ[q] \Delta[p+q] \xrightarrow{\;\; \sigma \;\;} \Delta[p] \times \Delta[q]

which satisfy the following equivalent conditions:

Such morphisms may hence be represented by paths

  • on a (p+1)×(q+1)(p+1)\times(q+1)-lattice,

  • from one corner to its opposite corner,

  • consisting of p+qp+q unit steps,

  • each either horizontally or vertially:

Proof

From this Prop. it is clear (see this Remark) that a simplex σ:Δ[p+q]Δ[p]×Δ[q]\sigma \,\colon\, \Delta[p+q] \xrightarrow{\;} \Delta[p] \times \Delta[q] is degenerate precisely if, when regarded as a path as above, it contains a constant step, i.e. one which moves neither horizontally nor vertically. But then – by degree reasons, since we are looking at paths of p+qp + q steps in a lattice of side length pp and qq – it must be that the path proceeds by p+qp + q unit steps.

Nerve of the inertia groupoid

Some basic facts about the nerve of an inertia groupoid:

Proposition

(inertia groupoid of delooping groupoid is adjoint action groupoid)
The inertia groupoid ΛBG\Lambda \mathbf{B} G is isomorphic to the action groupoid of the adjoint action of GG on itself:

ΛBGG adG=(G×GAd ()()pr 2G) \Lambda \mathbf{B}G \;\simeq\; G_{ad} \sslash G \;=\; \left( G \times G \underoverset {Ad_{(-)}(-)} {pr_2} {\rightrightarrows} G \right)

This follows by immediate inspection. For more discussion see at free loop space of a classifying space the section Examples – For finite groups.

Proposition

The groupoid convolution algebra of the inertia groupoid of the delooping groupoid BG\mathbf{B}G is the Drinfeld double of the group convolution algebra of GG.

Definition

(minimal simplicial circle)
Write

SΔ[1]/Δ[1]sSet S \;\coloneqq\; \Delta[1]/\partial\Delta[1] \;\;\; \in \; sSet

for the simplicial set with exactly two non-degenerate cells,

  • one of which in degree 0, which we denote by [0]=[1][0] = [1],

  • and one in degree 1, which we denote by [[0],[1]]\big[ [0], [1] \big].

Proposition

(normalized chain complex of minimal simplicial circle)
The normalized chain complex of the free simplicial abelian group of the minimal simplicial circle SS (Def. ) has the group of integers in degrees 0 and 1, and all differentials are zero:

N (S)[ 0 0 ][1]. N_\bullet \circ \mathbb{Z}(S) \;\simeq\; \left[ \array{ \vdots \\ \big\downarrow \\ 0 \\ \big\downarrow \\ \mathbb{Z} \\ \big\downarrow {}^{\mathrlap{ 0 }} \\ \mathbb{Z} } \;\; \right] \;\simeq\; \mathbb{Z}[1] \,\oplus\, \mathbb{Z} \,.

The following proposition follows on abstract grounds, but the explicit component-based proof we give is necessary in order to understand the transgression-formula for cocycles in the group cohomology of GG to cocycles on the inertia groupoid.

Proposition

The nerve of the inertia groupoid of a delooping groupoid of a finite group GG is isomorphic to the simplicial hom complex out of the minimal simplicial circle SS (Def. ) into the simplicial classifying space W¯G\overline{W}G (Rem. ):

N(ΛBG) [S,W¯G] . N\big( \Lambda \mathbf{B}G\big)_\bullet \;\simeq\; [S,\overline{W}G]_\bullet \,.

Proof

We claim that the isomorphism is given by sending, for each nn \in \mathbb{N}, any n-simplex (γ,g n1,,g 1,g 0)(\gamma, g_{n-1}, \cdots, g_1, g_0) of N(Func(B,BG))N\big( Func( \mathbf{B}\mathbb{Z}, \, \mathbf{B}G )\big), being a sequence of natural transformations of the form

g n1 g n2 g nj1 g nj2 g nj3 g 0 γ g n1 1γg n1 (g n1g 0) 1γ(g n1g 0) g n1 g n2 g nj1 g nj2 g nj3 g 0 , \array{ \bullet &\xrightarrow{g_{n-1}}& \bullet &\xrightarrow{g_{n-2}}& \cdots &\xrightarrow{g_{n-j-1}}& \bullet &\xrightarrow{g_{n-j-2}}& \bullet &\xrightarrow{g_{n-j-3}}& \cdots &\xrightarrow{g_{0}}& \bullet \\ \big\downarrow {}^{\mathrlap{ \gamma }} && \big\downarrow {}^{\mathrlap{ g_{n-1}^{-1} \cdot \gamma \cdot g_{n-1} }} && && \big\downarrow && \big\downarrow && && \big\downarrow {}^{\mathrlap{ ( g_{n-1} \cdots g_{0} )^{-1} \cdot \gamma \cdot ( g_{n-1} \cdots g_{0} ) }} \\ \bullet &\xrightarrow{g_{n-1}}& \bullet &\xrightarrow{g_{n-2}}& \cdots &\xrightarrow{g_{n-j-1}}& \bullet &\xrightarrow{g_{n-j-2}}& \bullet &\xrightarrow{g_{n-j-3}}& \cdots &\xrightarrow{g_{0}}& \bullet \mathrlap{\,,} }

to the homomorphism of simplicial sets

Δ[n]×SW¯G, \Delta[n] \times S \xrightarrow{\;\;} \overline{W}G \,,

which, in turn, sends a non-degenerate (n+1)(n+1)-simplex in Δ[n]×S\Delta[n] \times S of the form (in the path notation discussed at product of simplices)

(0,[0]) (1,[0]) (j,[0]) (j,[1]) (j+1,[1]) (n,[1]) \array{ (0,[0]) &\to& (1,[0]) &\to& \cdots &\to& (j,[0]) \\ && && && \big\downarrow \\ && && && (j,[1]) &\to& (j+1,[1]) &\to& \cdots &\to& (n,[1]) }

to the n+1n+1-simplex in W¯G\overline{W}G (Rem. ) of the form

g n1 g n2 g nj (g n1g nj) 1γ(g n1g nj) g nj g nj1 g 0 \array{ \bullet &\xrightarrow{g_{n-1}}& \bullet &\xrightarrow{g_{n-2}}& \cdots &\xrightarrow{g_{n-j}}& \bullet \\ && && && \big\downarrow {}^{\mathrlap{ ( g_{n-1} \cdots g_{n-j} )^{-1} \cdot \gamma \cdot ( g_{n-1} \cdots g_{n-j} ) }} \\ && && && \bullet &\xrightarrow{g_{n-j}}& \bullet &\xrightarrow{g_{n-j-1}}& \cdots &\xrightarrow{g_{0}}& \bullet }

As a consequence:

Proposition

The evaluation map

[S,W¯G]×SW¯G [S,\overline{W}G] \times S \xrightarrow{\;\;} \overline{W}G

(out of the product of the simplicial hom complex out of SS with SS) takes any non-degenerate n+1n+1-simplex of [S,W¯G]×S[S,\overline{W}G] \times S of the form (still in the path notation discussed at product of simplices)

(γ,[0]) (g n1,id) (Ad n1(γ),[0]) (g n2,id) (g j,id) (Ad j(γ),[0]) (e,[[0],[1]]) (Ad j(γ),[1]) (g nj,id) (g 0,id) (Ad 0(γ),[1]), \array{ (\gamma, [0]) &\xrightarrow{ (g_{n-1}, id) }& ( Ad_{n-1}(\gamma), [0] ) &\xrightarrow{ (g_{n-2}, id) }& \cdots &\xrightarrow{ (g_{j}, id) }& \big( Ad_{j}(\gamma), [0] \big) \\ && && && \big\downarrow {}^{\mathrlap{ (e, [ [0],[1] ]) }} \\ && && && (Ad_j(\gamma), [1]) &\xrightarrow{ (g_{n-j},id) } & \cdots &\xrightarrow{ (g_0,id) }& (Ad_0(\gamma), [1]) \mathrlap{\,,} }

where we are abbreviating

Ad j(γ) Ad (g n1g j)(γ) (g n1g j) 1γ(g n1g j), \begin{aligned} Ad_j(\gamma) & \;\coloneqq\; Ad_{(g_{n-1} \cdots g_j)}(\gamma) \\ & \;\coloneqq\; (g_{n-1} \cdots g_j)^{-1} \cdot \gamma \cdot (g_{n-1} \cdots g_j) \,, \end{aligned}

to the following n+1n+1 simplex of W¯G\overline{W}G:

g n1 g n2 g j Ad j(γ) g nj g 0 . \array{ \bullet &\xrightarrow{ g_{n-1} }& \bullet &\xrightarrow{ g_{n-2} }& \cdots &\xrightarrow{ g_{j} }& \bullet \\ && && && \big\downarrow {}^{\mathrlap{ Ad_j(\gamma) }} \\ && && && \bullet &\xrightarrow{ g_{n-j} } & \cdots & \xrightarrow{ g_0 } & \bullet \mathrlap{\,.} }

Transgression in cohomology

Definition

(transgression) For 𝒜sAb\mathcal{A} \,\in\, sAb, hence with
free loop space of its classifying space given (by this Prop.) by

(9)ΛB𝒜𝒜×B𝒜 \Lambda \mathbf{B}\mathcal{A} \;\simeq\; \mathcal{A} \times \mathbf{B}\mathcal{A}

we say that transgression in 𝒜\mathcal{A}-cohomology is

H(;B𝒜)[S,]H(Λ();𝒜×B𝒜)pr 2H(Λ();𝒜). H \big( -;\, \mathbf{B}\mathcal{A} \big) \xrightarrow{ [S, -] } H \big( \Lambda(-);\, \mathcal{A} \times \mathbf{B}\mathcal{A} \big) \xrightarrow{\;\; pr_2 \;\; } H \big( \Lambda(-);\, \mathcal{A} \big) \,.

Proposition

(free loop space of classifying space identified via Eilenberg-Zilber map)
For n +n \in \mathbb{N}_+ and for AAbA \,\in\, Ab the integers or the circle group, the following composite of is a simplicial weak equivalence

(10)[S,B nA] =[S,frgtDK(A[n])] sSet(S×Δ[],frgtDK(A[n])) Ch +(N(S×Δ[]),A[n]) EZ SCh +(N(S)N(Δ[]),A[n]) Ch +(N(Δ[]),[N(S),A[n]]) (frgtDK([N(S)[1],A[n]])) (frgtDK(A[n]A[n1])) =B nA×B n1A \begin{aligned} \big[ S, \, B^n A \big]_\bullet & \;=\; \big[ S, \, frgt \circ DK(A[n]) \big]_\bullet \\ & \;\simeq\; sSet \big( S \times \Delta[\bullet], \, frgt \circ DK(A[n]) \big) \\ & \;\simeq\; Ch_+ \big( N \circ \mathbb{Z}(S \times \Delta[\bullet]), \, A[n] \big) \\ & \;\xrightarrow{EZ_S}\; Ch_+ \big( N \circ \mathbb{Z}(S) \,\otimes\, N \circ \mathbb{Z}(\Delta[\bullet]), \, A[n] \big) \\ & \;\simeq\; Ch_+ \big( N \circ \mathbb{Z}(\Delta[\bullet]), \, [ N \circ \mathbb{Z}(S), \, A[n] ] \big) \\ & \;\simeq\; \Big( frgt \circ DK \big( [ \underset{ \mathbb{Z} \oplus \mathbb{Z}[1] }{ \underbrace{ N \circ \mathbb{Z}(S) } }, \, A[n] ] \big) \Big)_\bullet \\ & \;\simeq\; \Big( frgt \circ DK \big( A[n] \oplus A[n-1] \big) \Big)_\bullet \;=\; B^n A \times B^{n-1} A \end{aligned}

Here all isomorphisms are hom-isomorphisms of the above adjunctions, the step denoted EZ SEZ_S is pre-composition with the Eilenberg-Zilber map, and under the brace we are using Prop. .

Proof

By the fact that the Eilenberg-Zilber map has a left inverse given by the Alexander-Whitney map AWAW (see at Eilenberg-Zilber/Alexander-Whitney deformation retraction, the analogous composite with AW SAW_S instead of EZ SEZ_S yields a left inverse morphism, which hence retracts the homotopy groups of B nA×B n1AB^n A \times B^{n-1}A onto those of [S,B nA]\big[S, B^n A \big]. But, by (9), the latter is a product of Eilenberg-MacLane spaces with homotopy groups AA concentrated in degrees nn and n1n -1 . By assumption on AA the only retractions of AA onto itself is the identity, so that EZ SEZ_S must induce the identity morphism of homotopy groups.

Proof of the component formula

We prove that the formula (3) indeed expresses transgression in group cohomology.

Proof

Consider the following sequence of natural isomorphisms of hom-sets of the above homotopy categories (Def. ):

(11)H grp n(G;A) =Ho(sSet)(N (W¯G),A[n]) Ho(sSet)(W¯G,frgtDK(A[n])) =H n(BG;A) [S,]Ho(sSet)([S,W¯G],[S,frgtDK(A[n])]) Ho(sSet)([S,W¯G]×S,frgtDK(A[n])) Ho(sAbGrp)(([S,W¯G]×S),DK(A[n])) Ho(Ch +)(N ([S,W¯G]×S),A[n]) EZHo(Ch +)(N ([S,W¯G])N (S)[1],A[n]) Ho(Ch +)(N ([S,W¯G]),A[n]A[n1]) pr 2Ho(Ch +)(N ([S,W¯G]),A[n1]) =H n1(ΛBG;A). \begin{aligned} H^n_{grp} \big( G;\, A \big) & \;=\; Ho(sSet) \big( N_\bullet \circ \mathbb{Z}(\overline{W}G), \, A[n] \big) \\ & \;\simeq\; Ho(sSet) \big( \overline{W}G, \, frgt \circ DK\big( A[n] \big) \big) \\ = H^n \big( B G;\, A \big) & \;\overset{[S,-]}{\to}\; Ho(sSet) \Big( [S,\overline{W}G],\, \, \big[S, frgt \circ DK\big( A[n] \big)\big] \Big) \\ & \;\simeq\; Ho(sSet) \Big( [S,\overline{W}G] \times S, \, frgt \circ DK\big( A[n] \big) \Big) \\ & \;\simeq\; Ho(sAbGrp) \Big( \mathbb{Z}\big( [S, \overline{W}G] \times S \big), \, DK\big(A[n]\big) \Big) \\ & \;\simeq\; Ho(Ch_+) \Big( N_\bullet \circ \mathbb{Z}\big( [S, \overline{W}G] \times S \big), \, A[n] \Big) \\ & \;\overset{EZ}{\simeq}\; Ho(Ch_+) \Big( N_\bullet \circ \mathbb{Z}\big([S, \overline{W}G]\big) \otimes \underset{ \mathbb{Z}[1] \oplus \mathbb{Z} }{ \underbrace{N_\bullet \circ \mathbb{Z}(S)} }, \, A[n] \Big) \\ & \;\simeq\; Ho(Ch_+) \Big( N_\bullet \circ \mathbb{Z}\big([S, \overline{W}G]\big), \, A[n] \oplus A[n-1] \Big) \\ & \;\overset{pr_2}{\to}\; Ho(Ch_+) \Big( N_\bullet \circ \mathbb{Z}\big([S, \overline{W}G]\big), \, A[n-1] \Big) \\ & \;=\; H^{n-1} \big( \Lambda B G; \, A \big) \,. \end{aligned}

Here

Chasing a group cocycle (Def. ) through this sequence, and using Prop. when it gets sent through the Eilenberg-Zilber map, manifestly outputs the sum formula (3) to be proven.

Hence it only remains to see that this concrete composite (11) is equal to the abstractly defined transgression map (Def. ).

This follows by Prop. . In detail, since:

  1. A[n]Ch +A[n] \in Ch_+ is a fibrant object (like every object in the projective model structure on connective chain complexes);

  2. [S,W¯G]sSet[S,\overline{W}G] \in sSet is a cofibrant objects (like every object in the classical model structure on simplicial sets);

the chain of hom-isomorphisms of derived adjunctions in (11) is covered (through the quotient by left homotopy that defines the homotopy category of a model category according to this Lemma) by “the same” chain of hom-isomorphisms of plain adjunctions. Using here that every simplicial set XX is the colimit over its elements Δ[k]el(X)\Delta[k] \in el(X) (this Prop.) and that the 1-category-theoretic hom functors turn this colimit, in their first argument, into a limit (this Prop.), we find that the composite in (11) is covered by

sSet([S,W¯G],[S,frgtDK(A[n])]) sSet(limΔ[k]el([S,W¯G])Δ[k],[S,frgtDK(A[n])]) limΔ[k]el([S,W¯G])sSet(Δ[k],[S,frgtDK(A[n])]) limΔ[k]el([S,W¯G])Ch +(N (S×Δ[k]),A[n]) EZ SlimΔ[k]el([S,W¯G])Ch +(N (S)N (Δ[k]),A[n]) limΔ[k]el([S,W¯G])sSet(Δ[k],frgtDK(A[n]A[n1])) sSet(limΔ[k]el([S,W¯G])Δ[k],frgtDK(A[n]A[n1])) sSet([S,W¯G],frgtDK(A[n]A[n1])). \begin{aligned} sSet \Big( \big[S, \overline{W}G \big], \, \big[S, frgt \circ DK(A[n]) \big] \Big) & \;\simeq\; sSet \Big( \underset{ \underset{ \mathclap{ {\Delta[k] \in} \atop {el\big([S,\overline{W}G]\big)} } } {\longrightarrow} }{\lim} \Delta[k], \, \big[S, frgt \circ DK(A[n]) \big] \Big) \\ & \;\simeq\; \underset{ \underset{ \mathclap{ {\Delta[k] \in} \atop {el\big([S,\overline{W}G]\big)} } } {\longleftarrow} }{\lim} \, sSet \Big( \Delta[k], \, \big[S, frgt \circ DK(A[n]) \big] \Big) \\ & \;\simeq\; \underset{ \underset{ \mathclap{ {\Delta[k] \in} \atop {el\big([S,\overline{W}G]\big)} } } {\longleftarrow} }{\lim} \, Ch_+ \Big( N_\bullet \circ \mathbb{Z} \big( S \times \Delta[k] \big), \, A[n] \Big) \\ & \;\xrightarrow{EZ_S}\; \underset{ \underset{ \mathclap{ {\Delta[k] \in} \atop {el\big([S,\overline{W}G]\big)} } } {\longleftarrow} }{\lim} \, Ch_+ \Big( N_\bullet \circ \mathbb{Z}(S) \,\otimes\, N_\bullet \circ \mathbb{Z}(\Delta[k]), \, A[n] \Big) \\ & \;\simeq\; \underset{ \underset{ \mathclap{ {\Delta[k] \in} \atop {el\big([S,\overline{W}G]\big)} } } {\longleftarrow} }{\lim} \, sSet \Big( \Delta[k], \, frgt \circ DK \big( A[n] \oplus A[n-1] \big) \Big) \\ & \;\simeq\; sSet \Big( \underset{ \underset{ \mathclap{ {\Delta[k] \in} \atop {el\big([S,\overline{W}G]\big)} } } {\longrightarrow} }{\lim} \Delta[k], \, frgt \circ DK \big( A[n] \oplus A[n-1] \big) \Big) \\ & \;\simeq\; sSet \Big( [S,\overline{W}G], \, frgt \circ DK \big( A[n] \oplus A[n-1] \big) \Big) \,. \end{aligned}

This is manifestly the image under sSet([S,W¯G],)sSet\big( [S,\overline{W}G], - \big) of the correct morphism (10) from Prop. .

References

The transgression map is alluded to in

An indication of a proof, implicitly using ingredients of the Eilenberg-Zilber map (re-discovered under the name “Parmesan map”):

  • Simon Willerton, Section 1 of: The twisted Drinfeld double of a finite group via gerbes and finite groupoids, Algebr. Geom. Topol. 8 (2008) 1419-1457 (arXiv:math/0503266)

The transgression formula itself (without derivation) is also considered, in a context of twisted orbifold K-theory, in:

and specifically in the context of equivariant Tate K-theory:

Last revised on July 13, 2021 at 15:10:45. See the history of this page for a list of all contributions to it.