Strong fusion control and stable equivalences

On stable homotopy equivalences

A fundamental construction in the study of stable homotopy is the free infinite loop space generated by a space X. This is the colimit QX = lim −→ ΩΣX. The i homotopy group of QX is canonically isomorphic to the i stable homotopy group of X. Thus, one may obtain stable information about X by obtaining topological results about QX. One such result is the Kahn-Priddy theorem [7]. In another direc...

Metrizable Shape and Strong Shape Equivalences

In this paper we construct a functor Φ : proTop → proANR which extends Mardešić correspondence that assigns to every metrizable space its canonical ANR-resolution. Such a functor allows one to define the strong shape category of prospaces and, moreover, to define a class of spaces, called strongly fibered, that play for strong shape equivalences the role that ANRspaces play for ordinary shape e...

Block Theory via Stable and Rickard Equivalences

On iterated almost ν-stable derived equivalences

In a recent paper [5], we introduced a classes of derived equivalences called almost ν-stable derived equivalences. The most important property is that an almost ν-stable derived equivalence always induces a stable equivalence of Morita type, which generalizes a well-known result of Rickard: derived-equivalent self-injective algebras are stably equivalent of Morita type. In this paper, we shall...

Self-equivalences of Stable Module Categories

Let P be an abelian p-group, E a cyclic p′-group acting freely on P and k an algebraically closed field of characteristic p > 0. In this work, we prove that every self-equivalence of the stable module category of k[P oE] comes from a self-equivalence of the derived category of k[P o E]. Work of Puig and Rickard allows us to deduce that if a block B with defect group P and inertial quotient E is...