🤖 AI Summary
This study investigates the computational complexity of counting homomorphisms from an input group Γ to a finite group G. The problem is shown to be #P-hard when G is non-abelian and Γ is given by a finite presentation. In contrast, a polynomial-time algorithm exists when G is a 2-step nilpotent group and Γ is either the fundamental group of a 3-manifold or a finite group. By integrating techniques from computational group theory, group cohomology, obstruction theory, and triangulations of 3-manifolds, this work establishes—for the first time—a connection between the approximability of Eilenberg–MacLane spaces and the existence of efficient algorithms, thereby highlighting the profound influence of topological structure on computational complexity.
📝 Abstract
Fix a finite group $G$. We study the computational complexity of counting problems of the following flavor: given a group $\Gamma$, count the number of homomorphisms $\Gamma \to G$. Our first result establishes that this problem is $\#\mathsf{P}$-hard whenever $G$ is a non-abelian group and $\Gamma$ is provided via a finite presentation. We give several improvements showing that this hardness conclusion continues to hold for restricted $\Gamma$ satisfying various promises. Our second result, in contrast, shows that if $G$ is class 2 nilpotent and $\Gamma = \pi_1(M^3)$ for some input 3-manifold triangulation $M^3$, then there is a polynomial time algorithm. The difference in complexity is explained by the fact that 3-manifolds are close enough to being Eilenberg-MacLane spaces for us to be able 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 $\Gamma$ is itself a finite group encoded via a multiplication table.