Realizing wedges of Moore spaces as classifying spaces of finite semigroups

TL;DR

This paper proves that finite wedges of certain Moore spaces can be realized as classifying spaces of finite semigroups, expanding understanding of the homotopy types achievable by such classifying spaces.

Contribution

It demonstrates that all finite wedges of simply connected Moore spaces of finitely generated abelian groups are homotopy equivalent to classifying spaces of finite semigroups, confirming a conjecture.

Findings

Finite wedges of Moore spaces are classifying spaces of finite semigroups.

Homology groups do not prevent certain CW complexes from being classifying spaces.

Supports Fiedorowicz's conjecture on homotopy types of classifying spaces.

Abstract

Fiedorowicz suggested that it was likely that every finite simply connected CW complex is homotopy equivalent to the classifying space of a finite semigroup. We prove that every finite wedge of simply connected Moore spaces of finitely generated abelian groups is homotopy equivalent to the classifying space of a finite semigroup. Consequently, homology groups alone cannot preclude a finite simply connected CW complex from being homotopy equivalent to the classifying space of a finite semigroup.