Symmetric cohomology of groups and Poincaré duality (Q2097244)

The authors investigate properties of the `exterior' (co)homology of finite groups introduced by \textit{A. V. Zarelua} [MAIK Nauka/In\-ter\-pe\-ri\-o\-di\-ca. 190--218 (1999; Zbl 1013.20047)]. By results of the first author [J. Algebra 509, 397--418 (2018; Zbl 1427.20070)], this is a direct summand of the `symmetric' group (co)homology introduced by \textit{M. D. Staic} [Arch. Math. 93, No. 3, 205--211 (2009; Zbl 1209.20046)]. Zarelua's theory is constructed using the Koszul complex \(\mathrm{Kos}^+_* (G)\); this is a finite length, (non-projective) resolution in \(G\)-modules of \(\mathbb{Z}\), such that \(\mathrm{Kos}^+_t (G)\) is the exterior power \(\Lambda^{t+1} (\mathbb{Z} [G])\) for \(t \in \mathbb{N}\). For a \(G\)-module \(M\), the exterior cohomology \(H^*_\lambda (G, M)\) is the homology of \(\hom_G (\mathrm{Kos}^+_*(G), M)\) and the exterior homology \(H^\lambda_* (G,M)\) is defined likewise. There are comparison maps \(H_*(G,M) \rightarrow H^\lambda_* (G,M)\) and \( H^* _\lambda (G,M) \rightarrow H^* (G,M)\) with the usual group (co)homology and, by construction, the exterior (co)homology vanishes in (co)homological degree greater than the order of the group \(G\). There is a related construction, using the cochain complex \(C^* (G)\) given by the exterior algebra with differential defined by multiplication by \(\mathcal{N}:= \sum_{g \in G} g\). The associated homology \(H^\kappa_* (G,M)\) is defined as the homology of the complex \(\hom_G (C^* (G) , M)\). The authors show that, up to a `twist' and appropriate regrading, \(C^* (G)\) is isomorphic to the dual of the augmented Koszul complex \(\mathrm{Kos}_*(G)\). This gives the isomorphism: \[ H_\lambda ^{n-k-1} (G,M) \cong H^\kappa_k (G, M^{\mathrm{tw}}), \] for \(0 \leq k \leq n-2\), where \(n\) is the order of \(G\) and \(M^{\mathrm{tw}}\) denotes the twist of \(M\). (If \(G\) has odd order, then \(M^{\mathrm{tw}}=M\).) The authors exhibit a natural transformation, with domain Tate cohomology, \[ \kappa_k : \hat{H}^{-k} (G, M) \rightarrow H^\kappa_k (G,M) \] for \(k \geq 0\) and show, for example, that this is an isomorphism for \(k=0\). Moreover, for \(k \geq 1\), they show that \(\kappa_{k+1}\) factors across the exterior homology \( H_k^\lambda (G,M) \) as \[ \hat{H}^{-k-1} (G, M) \cong H_k (G,M) \rightarrow H_k^\lambda (G,M) \rightarrow H^\kappa _{k+1} (G,M). \] The main result of the paper is a Poincaré duality statement for exterior (co)homology. The authors first show that, for \(p\) an odd prime such that \(G\) has no elements of order \(q\) for each prime \(q<p\) (in particular \(G\) has odd order), then \(\kappa_k\) is an isomorphism for \(0 \leq k <p-1\). Then, using the above results together with the fact that the comparison \(H_* (G, M) \rightarrow H_*^\lambda (G,M)\) is an isomorphism in an appropriate range (by results of the first author, [loc. cit.]), they deduce the isomorphism \[ H_\lambda ^{n-i-2} (G, M) \cong H^\lambda_ i (G, M) \] for \(1 \leq i < p-2\). This is a weakened version of a property that Zarelua erroneously claimed to hold in greater generality.