Obstruction theory and the complexity of counting group homomorphisms
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866913048511381504 |
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| author | Samperton, Eric Weiß, Armin |
| author_facet | Samperton, Eric Weiß, Armin |
| contents | Fix a finite group $G$. We study the computational complexity of counting problems of the following flavor: given a group $Γ$, count the number of homomorphisms $Γ\to G$. Our first result establishes that this problem is $\#\mathsf{P}$-hard whenever $G$ is a non-abelian group and $Γ$ is provided via a finite presentation. We give several improvements showing that this hardness conclusion continues to hold for restricted $Γ$ satisfying various promises. Our second result shows that if $G$ is class 2 nilpotent and $Γ= π_1(M^3)$ for some input 3-manifold triangulation $M^3$ with $|H^2(M,Z(G)|$ bounded above, then there is a polynomial time algorithm to compute the number of homomorphisms from $Γ$ to $G$. This algorithm is explained in part by the fact that 3-manifolds are close enough to being Eilenberg-MacLane spaces for us to solve the necessary group cohomological obstruction problems efficiently using the given triangulation. A similar polynomial time algorithm for counting maps to finite, class 2 nilpotent $G$ exists when $Γ$ is itself a finite group encoded via a multiplication table, provided that $|H^2(Γ,Z(G))|$ is similarly bounded from above. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_02885 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Obstruction theory and the complexity of counting group homomorphisms Samperton, Eric Weiß, Armin Group Theory Computational Complexity Geometric Topology Fix a finite group $G$. We study the computational complexity of counting problems of the following flavor: given a group $Γ$, count the number of homomorphisms $Γ\to G$. Our first result establishes that this problem is $\#\mathsf{P}$-hard whenever $G$ is a non-abelian group and $Γ$ is provided via a finite presentation. We give several improvements showing that this hardness conclusion continues to hold for restricted $Γ$ satisfying various promises. Our second result shows that if $G$ is class 2 nilpotent and $Γ= π_1(M^3)$ for some input 3-manifold triangulation $M^3$ with $|H^2(M,Z(G)|$ bounded above, then there is a polynomial time algorithm to compute the number of homomorphisms from $Γ$ to $G$. This algorithm is explained in part by the fact that 3-manifolds are close enough to being Eilenberg-MacLane spaces for us to solve the necessary group cohomological obstruction problems efficiently using the given triangulation. A similar polynomial time algorithm for counting maps to finite, class 2 nilpotent $G$ exists when $Γ$ is itself a finite group encoded via a multiplication table, provided that $|H^2(Γ,Z(G))|$ is similarly bounded from above. |
| title | Obstruction theory and the complexity of counting group homomorphisms |
| topic | Group Theory Computational Complexity Geometric Topology |
| url | https://arxiv.org/abs/2602.02885 |