$G$-functors, $G$-posets and homotopy decompositions of $G$-spaces

Stefan Jackowski ¹   ; Jolanta Słomińska ²

¹ Institute of Mathematics Warsaw University Banacha 2 02-097 Warszawa, Poland
² Faculty of Mathematics and Information Sciences Warsaw Technical University Pl. Politechniki 1 00-661 Warszawa, Poland

Fundamenta Mathematicae, Tome 169 (2001) no. 3, pp. 249-287

Résumé

We describe a unifying approach to a variety of homotopy decompositions of classifying spaces, mainly of finite groups. For a group $G$ acting on a poset ${\bf W}$ and an isotropy presheaf $d:{\bf W}\rightarrow {\cal S}(G)$ we construct a natural $G$-map $ \mathop {\rm hocolim}\nolimits _{{\cal W}_d}G/d(-)\rightarrow |{\bf W}|$ which is a (non-equivariant) homotopy equivalence, hence $ \mathop {\rm hocolim}\nolimits _{{\cal W}_d}EG\times _GF_d \rightarrow EG\times _G|{\bf W}|$ is a homotopy equivalence. Different choices of $G$-posets and isotropy presheaves on them lead to homotopy decompositions of classifying spaces. We analyze higher limits over the categories associated to isotropy presheaves ${\cal W}_d$; in some important cases they vanish in dimensions greater than the length of ${\bf W}$ and can be explicitly calculated in low dimensions. We prove a cofinality theorem for functors $F:{\cal C}\rightarrow {\cal O}(G)$ into the category of $G$-orbits which guarantees that the associated map $\alpha _F:\mathop {\rm hocolim}\nolimits _{{\cal C}}EG\times _G F(-)\rightarrow BG$ is a mod-$p$-homology decomposition.

Détail

DOI : 10.4064/fm169-3-4

Keywords: describe unifying approach variety homotopy decompositions classifying spaces mainly finite groups group acting poset isotropy presheaf rightarrow cal construct natural g map mathop hocolim nolimits cal rightarrow which non equivariant homotopy equivalence hence mathop hocolim nolimits cal times rightarrow times homotopy equivalence different choices g posets isotropy presheaves lead homotopy decompositions classifying spaces analyze higher limits categories associated isotropy presheaves cal important cases vanish dimensions greater length explicitly calculated low dimensions prove cofinality theorem functors cal rightarrow cal category g orbits which guarantees associated map alpha mathop hocolim nolimits cal times rightarrow mod p homology decomposition