A criterion on trivial homotopy

TL;DR

This paper introduces a new framework that generalizes existing results on trivial homotopy, extending prior work by Jackowski and McClure on vanishing cohomology in classifying spaces.

Contribution

It develops a broad framework that extends previous results on trivial homotopy and vanishing cohomology, applicable to functors with Mackey complements over categories with direct products.

Findings

Generalizes Jackowski and McClure's results

Provides conditions for trivial homotopy in broader contexts

Enhances understanding of cohomology vanishing phenomena

Abstract

In "Homotopy decomposition of classifying spaces via elementary Abelian subgroups", Stephan Jackowski and James McClure show, for functors admitting a Mackey complement over categories holding a direct product, a general result on vanishing cohomology. We develop a framework leading to a general result on trivial homotopy which partially generalizes Jackowski and McClure's result in two different directions.