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Bredon homology of partition complexes. (English) Zbl 1360.55005

In this article, the authors study the Bredon homology and cohomology of the unreduced suspension of the nerve of the poset of proper nontrivial partitions of the set \(\{1,\dots,n\}\) viewed as a \(\Sigma_n\)-space, where \(\Sigma_n\) is the symmetric group on the set \(\{1,\ldots,n\}\). The main result of this work states that, for any prime number \(p\), the Bredon homology and cohomology groups of the aforementioned unreduced suspension with coefficients in a Mackey functor for \(\Sigma_n\) which takes values in \(\mathbb{Z}_{(p)}\)-modules (and which satisfies suitable conditions) are trivial if \(n\) is not a power of \(p\) and “computable in terms of constructions with the Steinberg module” if \(n\) is a power of \(p\). This result sharpens some results of [G. Z. Arone and W. G. Dwyer, Proc. Lond. Math. Soc., III. Ser. 82, No. 1, 229–256 (2001; Zbl 1028.55008)] and is related to [J. Grodal, Ann. Math. (2) 155, No.2, 405–457 (2002; Zbl 1004.55008)].
The authors also give several applications of the main theorem of this article such as new proofs or different approaches to results of [G. Arone and M. Mahowald, Invent. Math. 135, No. 3, 743–788 (1999; Zbl 0997.55016); M. Behrens, Mem. Am. Math. Soc. 1026, iii-xi, 90 p. (2012; Zbl 1330.55012); N. J. Kuhn, Math. Proc. Camb. Philos. Soc. 92, 467–483 (1982; Zbl 0515.55005); N. J. Kuhn and S. B. Priddy, ibid. 98, 459–480 (1985; Zbl 0584.55007); N. J. Kuhn, J. Topol. 8, No. 1, 118–146 (2015; Zbl 1325.55006); C. Rezk, “Rings of power operations for Morava \(E\)-theories are Koszul”, Preprint, arXiv:1204.4831].

MSC:

55N91 Equivariant homology and cohomology in algebraic topology
55R40 Homology of classifying spaces and characteristic classes in algebraic topology
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