同调理论与计数群同态的复杂性
Obstruction theory and the complexity of counting group homomorphisms
AI总结：
研究给定群Γ计数映射Γ→G的计算复杂性，证明当G为非阿贝尔群时该问题为#P难，针对特定Γ提供多项式时间算法计算同态数量。
摘要：
固定一个有限群$G$。我们研究以下风味的计数问题的计算复杂性：给定一个群$Γ$，计数从$Γ$到$G$的同态数量。我们的第一个结果表明，当$G$是非阿贝尔群且$Γ$通过有限呈现提供时，该问题是$\#\mathsf{P}$难的。我们给出了一些改进，显示这种硬度结论在满足各种承诺的受限$Γ$中仍然成立。我们的第二个结果表明，如果$G$是类2 nilpotent群且$Γ= π_1(M^3)$对于某个输入3流形三角化$M^3$，且$|H^2(M,Z(G)|$被上界限制，则存在多项式时间算法来计算从$Γ$到$G$的同态数量。该算法部分由3流形接近于Eilenberg-MacLane空间的事实解释，使我们能够高效地使用给定的三角化解决必要的群上同调障碍问题。当$Γ$本身是一个有限群并通过乘法表编码时，存在类似的多项式时间算法来计数到有限、类2 nilpotent群$G$的映射，前提是$|H^2(Γ,Z(G))|$也被上界限制。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
