# Is the cohomology ring of a finite group computable?

1. Is there an algorithm which halts on all inputs that takes as input a finite group ($p$-group if you like) and outputs a finite presentation of the cohomology ring (with trivial coefficients $\mathbb{F}_p$) in terms of generators and relations.

For concreteness, let's say the input is given by a generating set of matrices, or permutations, or by giving its multiplication table - from the computability point of view, these are all equivalent.

I've read much of David J. Green's book "Grobner bases and the computation of group cohomology," in which he presents an algorithm that produces "partial presentation" of the cohomology ring degree-by-degree. There is a sufficient criterion due to J. F. Carlson which says when you're done - that is, when this partial presentation is actually a correct presentation of the cohomology ring - but the book seems to indicate that Carlson's criterion either is not necessary or at least not known to be necessary (as of its writing, 2003).

Now, the algorithm used in Green's book will eventually get a complete presentation of the cohomology ring, but the issue is whether the algorithm can tell when it's reached a high enough degree to be done. Following this strategy, a related question is:

1. Consider the function $b_p:FinGrp \to \mathbb{N}$ defined by $b_p(G)$ is the least $n \in \mathbb{N}$ such that the cohomology ring of $G$ with coefficients in $\mathbb{F}_p$ is completely determined by the partial presentation one gets by going up to degree $n$. Is $b_p$ bounded by a computable function?

As I understand it this follows from Benson's Regularity Conjecture, proved by Symonds fairly recently. It says that $b_p = 2(|G|-1)$ will do.

• Great! Here's the link: ams.org/journals/jams/2010-23-04/S0894-0347-10-00666-1/… May 9, 2018 at 6:46
• At least for a module presentation over $H^*(BU(n);\Bbb F_p)$ or $H^*(BSO(n);\Bbb F_p)$ when $G$ has a faithful unitary or orthogonal representation, there is a better bound: in the first case, only generators up to degree $n^2$ and relators up to degree $n^2+1$, or in the latter case replace $n^2$ by $n(n-1)/2$. I learned this from Burt Totaro's book "Group Cohomology and Algebraic Cycles". I guess there might be related bounds for ring relations - perhaps $2n^2+1$ resp $n(n-1)+1$ suffices - but I don't remember this off the top of my head.
– mme
Dec 6, 2018 at 11:01
• Finding a bound for a module presentation over $H^*(BU(n);\mathbb{F}_p)$ has been known a lot longer than Symonds' result giving a bound for a ring presentation. Symonds' result was a major advance.
– IJL
Jan 21, 2019 at 16:53

This seems to be addressed in the paper of Gonzalez-Diaz and Real (not paywalled).

González-Díaz, R.; Real, P., Computation of cohomology operations on finite simplicial complexes, Homology Homotopy Appl. 5, No.2, 83-93 (2003). ZBL1028.55011.

• I looked at the paper and don't see how it answers the question. The paper deals with finite simplicial complexes, whose cohomology in particular has a trivial upper bound (the dimension of the largest simplex). Whereas in some sense the question of existence of a computable upper bound is the whole question here (or at least a significant part of it). Did I miss something in the paper? May 9, 2018 at 6:08
• Group cohomology is about an infinite-dimensional cell complex, right? (the classifying space of the group) May 9, 2018 at 10:51