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Homotopy Types of Small Semigroups

Dennis Sweeney

TL;DR

This work investigates how the classifying spaces $BS$ of finite semigroups and monoids realize a wide range of homotopy types and develops practical tools to study them. It combines detailed semigroup structure -- notably the minimal ideal $K(S)$ and Rees matrix semigroups -- with topological methods to relate $BS$ to the group completion $GS$, providing constructive proofs and near-linear-time algorithms to compute $GS$ and $H_*(BS)$. The authors identify conditions (notably $K$-thinness) under which $BS$ is homotopy equivalent to $BGS$, and they demonstrate a rich landscape of examples with exotic homotopy types, including suspensions, wedges of spheres, Moore spaces, and even infinite-generation cohomology rings. A key result is that the set of homotopy types of classifying spaces of finite monoids is closed under suspension, and the paper presents extensive computations for semigroups up to size 8, revealing both patterns and counterexamples to conjectures in the area. Overall, the work significantly broadens the understanding of how finite semigroups encode topological information and provides robust computational frameworks for exploring these classifying spaces.

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.

Homotopy Types of Small Semigroups

TL;DR

This work investigates how the classifying spaces of finite semigroups and monoids realize a wide range of homotopy types and develops practical tools to study them. It combines detailed semigroup structure -- notably the minimal ideal and Rees matrix semigroups -- with topological methods to relate to the group completion , providing constructive proofs and near-linear-time algorithms to compute and . The authors identify conditions (notably -thinness) under which is homotopy equivalent to , and they demonstrate a rich landscape of examples with exotic homotopy types, including suspensions, wedges of spheres, Moore spaces, and even infinite-generation cohomology rings. A key result is that the set of homotopy types of classifying spaces of finite monoids is closed under suspension, and the paper presents extensive computations for semigroups up to size 8, revealing both patterns and counterexamples to conjectures in the area. Overall, the work significantly broadens the understanding of how finite semigroups encode topological information and provides robust computational frameworks for exploring these classifying spaces.

Abstract

We algorithmically compute integral Eilenberg-MacLane homology of all semigroups of order at most 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 has a left-simple or right-simple minimal ideal , then the classifying space is homotopy equivalent to the classifying space of the group completion. We also describe an algorithm for computing the group completion of a finite semigroup using asymptotically fewer than semigroup operations. Finally, we show that the set of homotopy types of classifying spaces of finite monoids is closed under suspension.

Paper Structure

This paper contains 14 sections, 26 theorems, 47 equations, 5 figures, 7 algorithms.

Table of Contents

  1. Introduction
  2. Classifying Spaces
  3. McDuff's Result
  4. First Properties
  5. The Structure of the Minimal Ideal
  6. Group Completions
  7. K-thin Semigroups have the Classifying Space of a Group
  8. Discussion from Homology Computations
  9. Joins and Suspensions
  10. Computing Homology as Tor
  11. Tables of Homology
  12. Semigroups of order at most 6
  13. Semigroups of order 7
  14. Semigroups of order 8

Key Result

Theorem 3.1

Every path-connected space has the same weak homotopy type as the classifying space $BS$ of some (discrete) monoid $S$.

Figures (5)

  • Figure 1: An illustration of the low-dimensional cells in $BS$
  • Figure 2: The numbers of semigroups (up to (anti-)isomorphism) of each order that are not $K$-thin based on an exhaustive search, and the percentage of all semigroups that are $K$-thin, from dividing by the totals in smallsemiorder10 and subtracting from $1$.
  • Figure 3: The multiplication table for a semigroup $S$ where $BS$ is a Moore space of type $M(C_2, 3)$.
  • Figure 4: The multiplication table for a semigroup $S$ where $BS$ is a Moore space of type $M(C_2, 2)$.
  • Figure 5: The multiplication table for a semigroup $S$ with $H_6(BS;\mathbb{Z}) \cong \mathbb{Z}^9 \times C_{1494640}$.

Theorems & Definitions (79)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Theorem 3.1: Theorem 1 in McDuff79
  • Remark 3.2
  • Remark 3.3
  • Definition 3.5
  • Proposition 3.6
  • Proposition 4.1
  • ...and 69 more