Equivariant splendid Rickard equivalences

Rickard Equivalences and Block Theory

De nition We say that a subgroup H of G controls the fusion of p subgroups of G if the following two conditions are ful lled C H contains a Sylow p subgroup Sp of G C whenever P is a subgroup of Sp and g is an element of G such that gPg Sp there exist z in the centralizer CG P of P in G and h in H such that g hz Basic example We denote by Op G the largest normal subgroup of G with order prime t...

Block Theory via Stable and Rickard Equivalences

Equivariant weak n-equivalences

The notion of n-type was introduced by J.H.C. Whitehead ([22, 23]) where its clear geometric meaning was presented. Following J.L. Hernandez and T. Porter ([12, 13]) we use the term weak n-equivalence for a map f : X → Y of path-connected spaces which induces isomorphisms πk(f) : πk(X)→ πk(Y ) on homotopy groups for k ≤ n. Certainly, weak n-equivalence of a map determines its n-connectedness bu...

Cohomology of groups, abelian Sylow subgroups and splendid equivalences

Let G be a finite group and let R be a complete discrete valuation domain of characteristic 0 with residue field k of characteristic p and let S be R or k. The cohomology rings H∗(K,S) for subgroups K of G together with restriction to subgroups of G, transfer from subgroups of G and conjugation by elements of G gives H∗(−, S) the structure of a Mackey functor. Moreover, the group HSplenS(K) of ...

Spaces of Equivariant Self-equivalences of Spheres

Let F(ST) denote the identity component of the space of homotopy self-equivalences of STM and let F = inj limw FiS" ). This paper studies the homotopy properties of certain equivariant analogs of the infinite loop space F.