Homotopy Types of Small Semigroups
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866929703473905664 |
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| author | Sweeney, Dennis |
| author_facet | Sweeney, Dennis |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_05276 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Homotopy Types of Small Semigroups Sweeney, Dennis Algebraic Topology 20M50 (Primary) 55U10, 55-08, 55-11 (Secondary) 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. |
| title | Homotopy Types of Small Semigroups |
| topic | Algebraic Topology 20M50 (Primary) 55U10, 55-08, 55-11 (Secondary) |
| url | https://arxiv.org/abs/2502.05276 |