nLab transgression in group cohomology



Group Theory



Special and general types

Special notions


Extra structure





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

(1)H grp +1(G;A)= H +1(BG;A) τ H n(ΛBG,A) [g]G ad/GH grp n(C g;A) \begin{array}{rcl} H_{grp}^{\bullet + 1}(G;\,A) \;\;=\;\; \\ H^{\bullet + 1}(B G;\, A) & \xrightarrow{ \;\;\;\; \tau \;\;\;\; } & H^n \big( \Lambda B G, \, A \big) \;\simeq \\ && \underset{ \mathclap{ \array{ \phantom{-} \\ [g] \in G^{ad}/G } } }{\oplus} \; H^n_{grp}\big(C_g;\, A \big) \end{array}

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), \begin{array}{l} \tau(c) \big( \gamma, g_{n-1}, \cdots, g_1, g_0 \big) \\ \;=\; \pm \underset{ 0 \leq j \leq n }{\sum} (-1)^j \cdot c\big( g_{n-1}, \cdots,\, g_{n-j},\, Ad_j(\gamma),\, g_{n-j-1},\, \cdots,\, g_0 \big) \,, \end{array}


γg n1Ad 1(γ)g n2g 0Ad n(γ)ΛBG \gamma \xrightarrow{ g_{n-1} } Ad_{1}(\gamma) \xrightarrow{ g_{n-2} } \cdots \xrightarrow{\; g_0 \;} Ad_{n}(\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 nj)(γ) (g n1g nj) 1γ(g n1g nj), \begin{aligned} Ad_{j}(\gamma) & \;\coloneqq\; Ad_{(g_{n-1}\cdots g_{n-j})}(\gamma) \\ & \;\coloneqq\; (g_{n-1} \cdots g_{n-j})^{-1} \cdot \gamma \cdot (g_{n-1} \cdots g_{n-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:


(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


(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 this isomorphically as the nerve of the delooping groupoid, W¯G=N(G*)\overline{W} G \,=\, N (G \rightrightarrows \ast). From this 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{\phantom{--}} \cdots \bullet \xrightarrow{\phantom{--}} \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}} \big( \overline{W}G \big) \;\;\; \in \; Ho(sSet)

(where the first equality reflects that GG is assumed to be discrete).

Group(oid) cohomology

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


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

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

for (the homotopy type of) the Eilenberg-MacLane space with AA in degree nn – here constructed as the underlying simplicial set of the simplicial abelian group which is the image under the Dold-Kan construction of the chain complex that is concentrated on AA in degree nn.


(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 the following hom-set in the classical homotopy category:

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

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:


(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) \,.


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


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¯G,undrlngDK(A[n])) =Ho(sSet)(BG,B 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 ,\, undrlng \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:


(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 basic facts about products of simplicial sets:


(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:


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:


(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.


The groupoid convolution algebra of the inertia groupoid of the delooping groupoid BG\mathbf{B}G is the Drinfeld double (see there for more) of the group algebra of GG.


(minimal simplicial circle)

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].


(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{ \scriptsize{0} } \\ \mathbb{Z} } \;\; \right] \;\simeq\; \mathbb{Z}[1] \,\oplus\, \mathbb{Z} \,.

The following proposition holds on general grounds (duscussed at free loop space of classifying space), but the explicit component-based proof we give now is necessary, further below, in order to understand the transgression-formula for cocycles in the group cohomology of GG to cocycles on the inertia groupoid.


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 \,.


This is a specialization of the general fact that the simplicial nerve repects mapping objects (see this Prop. at nerve). We spell it out for the present case:

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:


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) maps non-degenerate n+1n+1-simpleces of [S,W¯G]×S[S,\overline{W}G] \times S as follows:


This follows by unwinding the component formula for the evaluation map on simplicial mapping complexes (this Prop.):

Recall that we denote by

(9)(γ;g n1,g n2,,g 0)Hom(S×Δ[n],W¯G)(NMap(B,BG)) n \big( \gamma ;\, g_{n-1} ,\, g_{n-2} ,\, \cdots ,\, g_{0} \big) \;\; \in \;\; \mathrm{Hom} \big( S \times \Delta[n] ,\, \overline{W}G \big) \;\simeq\; \Big( N \mathrm{Map} \big( { \mathbf{B}\mathbb{Z} } ,\, { \mathbf{B}G } \big) \Big)_{n}

the nn-cell in the nerve of the inertia groupoid that corresponds to the sequence of natural transformation which start at the functor

γGHom Grp(,G)Hom(B,BG) \gamma \,\in\, G \,\simeq\, \mathrm{Hom}_{\mathrm{Grp}}(\mathbb{Z}, G) \,\simeq\, \mathrm{Hom}\big( \mathbf{B}\mathbb{Z} ,\, \mathbf{B}G \big)

and successively have components g nGg_{n-\bullet} \in G.

Throughout we are writing “HomHom” for hom-sets and “MapMap” for hom-objects, i.e. for internal homs and we keep tacitly going back and forth through the bijections in (9).

If we abbreviate (this follows conventions familiar in discussion of transgression in group cohomology)

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

then this corresponds to a sequence of composable morphisms in the inertia groupoid of this form:

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

The simplicial map (9) maps the non-degenerate (n+1)(n+1)-cells in the product of simplicial sets (see the discussion there) of SS (?) with the simplicial n n -simplex as follows:

This is not exactly what we need unwinding the evaluation map, but it is close: the image of (9) under the kkth degeneracy map evidently gives the following mapping:

This is the type of mapping that appears in the component formula for the evaluation map on function complexes (from that Prop.), and so the claim follows.

Transgression in cohomology


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

(10)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) \,.


(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

(11) [S,B nA] =[S,undrlngDK(A[n])] sSet(S×Δ[],undrlngDK(A[n])) Ch +(N(S×Δ[]),A[n]) EZ SCh +(N(S)N(Δ[]),A[n]) Ch +(N(Δ[]),[N(S),A[n]]) (undrlngDK([N(S)[1],A[n]])) (undrlngDK(A[n]A[n1])) =B nA×B n1A \begin{aligned} & \big[ S, \, B^n A \big]_\bullet \\ & \;=\; \big[ S, \, undrlng \circ DK(A[n]) \big]_\bullet \\ & \;\simeq\; sSet \big( S \times \Delta[\bullet], \, undrlng \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( undrlng \circ DK \big( [ \underset{ \mathbb{Z} \oplus \mathbb{Z}[1] }{ \underbrace{ N \circ \mathbb{Z}(S) } }, \, A[n] ] \big) \Big)_\bullet \\ & \;\simeq\; \Big( undrlng \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. .


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 (10), 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.


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

(12)H grp n(G;A) =Ho(sSet)(N (W¯G),A[n]) Ho(sSet)(W¯G,undrlngDK(A[n])) =H n(BG;A) [S,]Ho(sSet)([S,W¯G],[S,undrlngDK(A[n])]) Ho(sSet)([S,W¯G]×S,undrlngDK(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, \, undrlng \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, undrlng \circ DK\big( A[n] \big)\big] \Big) \\ & \;\simeq\; Ho(sSet) \Big( [S,\overline{W}G] \times S, \, undrlng \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}


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 (12) 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 (12) 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 (12) is covered by

sSet([S,W¯G],[S,undrlngDK(A[n])]) sSet(limΔ[k]el([S,W¯G])Δ[k],[S,undrlngDK(A[n])]) limΔ[k]el([S,W¯G])sSet(Δ[k],[S,undrlngDK(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],undrlngDK(A[n]A[n1])) sSet(limΔ[k]el([S,W¯G])Δ[k],undrlngDK(A[n]A[n1])) sSet([S,W¯G],undrlngDK(A[n]A[n1])). \begin{aligned} & sSet \Big( \big[S, \overline{W}G \big], \, \big[S, undrlng \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, undrlng \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, undrlng \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], \, undrlng \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], \, undrlng \circ DK \big( A[n] \oplus A[n-1] \big) \Big) \\ & \;\simeq\; sSet \Big( [S,\overline{W}G], \, undrlng \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 (11) from Prop. .


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”):

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:

Generalization to Real /2\mathbb{Z}/2-equivariant cohomology (as appropriate for twists of KR-theory):

The above discussion and proof that transgression is essentially the internal hom out of the circle into the cocycle-as-a-functor is taken from:

following earlier observations in:

Last revised on November 4, 2023 at 08:45:18. See the history of this page for a list of all contributions to it.