Homotopy Types of Small Semigroups

Abstract

We present a software package for quickly calculating the integral homology of finite semigroups and monoids by a novel approach of exploiting structure in projective resolutions. We describe the usage of this package on semigroups and monoids of small orders. From the results of these calculations, we present a wealth of counterexamples in finite semigroup theory: we give a finite aperiodic semigroup with large torsion in its homology, two finite semigroups with certain Moore spaces as classifying spaces, and a finite semigroup and with nontrivial rational homology in infinitely many dimensions, refuting conjectures of William Nico. We further show that the set of homotopy types of classifying spaces of finite semigroups is closed under suspension.