Equivariant Moore spaces and the Dade group

2016/06/11 by Ergün Yalçın, Yalcin, Ergun · 2 citations
Mathematics · #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Primary: 57S17 #Secondary: 20C20

paper · pdf · doi:10.48550/arxiv.1606.03607

openalex publication_date 2016/06/11 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

Let G be a finite p-group and k be a field of characteristic p. A topological space X is called an n-Moore space if its reduced homology is nonzero only in dimension n. We call a G-CW-complex X an \underlinen-Moore G-space over k if for every subgroup H of G, the fixed point set X^H is an \underlinen(H)-Moore space with coefficients in k, where \underlinen(H) is a function of H. We show that if X is a finite \underlinen-Moore G-space, then the reduced homology module of X is an endo-permutation kG-module generated by relative syzygies. A kG-module M is an endo-permutation module if \rm End_k (M) =M ⊗ _k M^* is a permutation kG-module. We consider the Grothendieck group of finite Moore G-spaces M(G), with addition given by the join operation, and relate this group to the Dade group generated by relative syzygies.

Cited by

• The Euler characteristic of an endotrivial complex
• Backwards uniqueness for Mean curvature flow with asymptotically conical singularities

Related

• A Filtration on Equivariant Borel-Moore Homology
• The structure of Ext(A,Z) and GCH: possible co-Moore spaces
• A classifying space for Phillips' equivariant K-theory
• Multicurves and equivariant cobordism
• T-equivariant cohomology of cell complexes and the case of infinite Grassmannians