Homotopy Types of Small Semigroups

TL;DR

This paper computes the homology of small semigroups, explores their classifying spaces, refutes some conjectures, and introduces algorithms for homotopy and group completion computations, revealing new topological properties of finite semigroups.

Contribution

It provides the first systematic computation of semigroup homology for orders up to 8, offers new proofs of topological properties, and introduces efficient algorithms for group completion.

Findings

Computed homology for all semigroups of order ≤8.

Identified semigroups with notable classifying spaces.

Refuted conjectures of Nico regarding semigroup topology.

Abstract

We algorithmically compute integral Eilenberg-MacLane homology of all semigroups of order at most $8$ and present some particular semigroups with notable classifying spaces, refuting conjectures of Nico. Along the way, we give an alternative topological proof of the fact that if a finite semigroup $S$ has a left-simple or right-simple minimal ideal $K(S)$, then the classifying space $BS$ is homotopy equivalent to the classifying space $B(GS)$ of the group completion. We also describe an algorithm for computing the group completion $GS$ of a finite semigroup $S$ using asymptotically fewer than $|S|^2$ semigroup operations. Finally, we show that the set of homotopy types of classifying spaces of finite monoids is closed under suspension.