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I am interested in triple cup product operations on the cohomology ring $H^*(Y;\Bbb Z/p^r)$ of 3-manifolds. Trying to extract the algebra, I am led to the following question.

Let's fix a prime power $p^r$, and consider finite abelian groups $A$ with $p^r A = 0$, equipped with a homomorphism $\phi: \Lambda^3 A \to \Bbb Z/p^r$. Consider the monoid of such groups under direct sum. Does this have a nice presentation? Since it probably doesn't, instead consider the Grothendieck group on this monoid. Is that any better?

In particular, is it ever possible that the triple cup product 'simplifies' under direct sum? I don't know what invariants I can extract from $\phi$, much less which ones clearly cannot die from direct sum.

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  • $\begingroup$ I have seen this but I am not adept enough to carry out the same ideas over a ring of positive characteristic, and I guess I'm hoping the problem of moduli will be more well-behaved since everything in sight is finite. $\endgroup$
    – mme
    Apr 9, 2018 at 21:03
  • $\begingroup$ I also probed this based on yours: math.stackexchange.com/questions/3277549 but it seems that you do not have a Math.SE...? $\endgroup$
    – wonderich
    Jun 29, 2019 at 17:05

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