Realizing wedges of Moore spaces as classifying spaces of finite semigroups

TL;DR

The paper advances the conjecture that finite simply connected CW complexes can be realized as classifying spaces of finite semigroups by showing that every finite wedge of Moore spaces $M(A_n,n)$ with finitely generated abelian groups $A_n$ is homotopy equivalent to $BM$ for some finite monoid $M$, using a wedge-closure strategy based on suspensions and minimal-ideal rectangular bands. A key technical achievement is the explicit construction of finite monoids realizing Moore spaces $M(C,2)$ for cyclic groups $C$, together with a general method to realize wedge sums of such spaces via a finite monoid $S$ whose minimal ideal remains a rectangular band. The paper also develops an algebraic framework to study global dimension and homology of regular and von Neumann regular monoids, proving finiteness or vanishing results for homology in high degrees and answering several questions from homotopy small about the homology of bands and related monoid algebras. Overall, the results substantially extend the range of topological types realizable as classifying spaces of finite semigroups and provide tools to analyze their (co)homological properties.

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.

Theorems & Definitions (39)

• Lemma 2.1
• Theorem 2.2
• Theorem 3.1
• proof
• Theorem 4.1
• proof
• Theorem 4.2
• proof
• Theorem 5.1: Geigle-Lenzing
• Proposition 5.2