The combinatorics of the bar resolution in group cohomology
This paper is devoted to explore the interaction between Combinatorics, Representation Theory and Topology. The work starts by introducing a double complex structure on the bar resolution used in computing the cohomology of a finite group and describing the two spectral sequences which arise from filtering it by rows and columns. This double complex structure turns out to be very closely related to the topology of the \(CW\)-complex \(K_{n}\) whose face poset is isomorphic to the subword ordering on words without repetition from an alphabet of size \(n\). This complex is shellable, and from its shellability it follows that the rank of its top homology is the number of permutations in the symmetric group \(S_{n}\) without fixed points, called derangements. The set of derangements in \(S_{n}\) is denoted by \(D_{n}\). Let us denote by \(L_{S}\) the fundamental quasisymmetric function on the subset \( S\subset [ n-1]=\{ 1,2,\dots ,n-1\} \) and define \( K_{n}( x) =\sum_{\pi \in D_{n}}L_{\text{Des}( \pi ) }\) where Des\((\pi) \) means the descent set of the permutation \(\pi \). The first significant result of this paper says that \(K_{n}( x) = \text{ch}( \varepsilon _{n}\otimes \chi _{n}) \), the Frobenius characteristic map of the homology representation of \(S_{n}\). Now, given a group \(G\), let \(\Delta \) be the simplicial complex whose vertices are the elements of \(G\), and in which every finite subset of \(G\) is a simplex. We may construct the oriented and ordered chain complexes denoted by \(C^{\text{ori}}( \Delta ) \) and \(C^{\text{ord}}( \Delta ) \), respectively, and one shows that there is a chain homotopy equivalence between them, so both of them can be used to compute the group homology \(H_{\ast }( G,M) \) with coefficients in any module \(M\). The associated double complex is defined as follows: we write \(C_{n,0}= \mathbb{Z}V_{n}\) where \(V_{n}\) is the diagonal of the \(( n+1) \) -fold product of copies of \(G\), \(C_{0,n}\) has a \(\mathbb{Z}\)-basis the set of \( ( n+1) \)-tuples of distinct elements of \(G\) and the boundary map is the ordinary one defined in a double chain complex. This double complex, constructed with \(C^{\text{ord}}( \Delta ) \) is extended to give a commutative diagram in which the left hand edge is the chain complex \(C^{ \text{ori}}( \Delta ) \). The second result tells us that this diagram is a proper resolution for \(C^{\text{ori}}( \Delta ) \) by complexes , and as a consequence of this it is derived that for every \(s\) there is an exact sequence \[ 0\leftarrow H_{s}( \Delta ) \leftarrow H_{0,s}^{\text{vert} }( \Delta ) \leftarrow H_{1,s}^{\text{vert}}( \Delta ) \leftarrow \ldots \] The authors make an interesting connection with the combinatorics of the \(CW\)-complex \(K_{n}\), for the first column of the \(E^{0}\) page of the spectral sequence is naturally isomorphic to the cellular chain complex for \(K_{n}\) where \(n=\left| G\right| \) and the remaining columns split into direct sumands naturally isomorphic to cellular chain complexes for links of faces in various \(K_{n}\). The oriented chain complex shows to have a very strong influence on the behavior of this spectral sequence, in fact, it is proved in section 4 that for any infinite group \(G\) the spectral sequence obtained by filtering the double complex \(C^{\text{ord}}( \Delta ) \otimes _{\mathbb{Z}G}M\) by columns has an \(E^{1}\) page which is nonzero only in the left edge; the result for finite groups says that if \(G\) acts freely on a finite set \(\Omega \), the spectral sequence obtained by filtering the double complex \(C^{\text{ord}}( \Delta ) \otimes _{ \mathbb{Z}G}M\) by columns stops at the \(E^{2}\) page and has an \(E^{1}\) page such that \[ 0\leftarrow \mathbb{Z}\leftarrow P_{0}\leftarrow P_{1}\leftarrow \ldots \] is a projective resolution of \(\mathbb{Z}\). This spectral sequence converges to the cohomology of the group and provides a method for computing the group cohomology of a given group in terms of the cohomology of some distinguished subgroups; this is shown in sections 6 and 7 by an examination of the Ext class of this complex.
scientific article; zbMATH DE number 2093461
scientific article