Truncated Quillen coplexes of p-groups

2014/01/29 by Francesco Fumagalli, Fumagalli, Francesco, John Shareshian +1
Mathematics · #05E45 #20 E15 #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #math.GR #msc:05E45 #msc:20 #msc:E15

paper · pdf · doi:10.48550/arxiv.1401.7513

arxiv created 2014/01/29 · openalex publication_date 2014/01/29 · arxiv updated 2014/01/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p be an odd prime and let P be a p-group. We examine the order complex of the poset of elementary abelian subgroups of P having order at least p². S. Bouc and J. Thévenaz showed that this complex has the homotopy type of a wedge of spheres. We show that, for each nonnegative integer l, the number of spheres of dimension l in this wedge is controlled by the number of extraspecial subgroups X of P having order p^2l+3 and satisfying Omega₁(C_P(X))=Z(X). We go on to provide a negative answer to a question raised by Bouc and Thévenaz concerning restrictions on the homology groups of the given complex.

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