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The famous Kunneth formula expresses the homology of a product manifold as the tensor product of the two algebras.

Now suppose we know that a manifold $X$ has a decomposition $H_*(X) \simeq A \otimes B$ as algebras. When is $X$ actually a product of manifolds?

What I thought. Firstly, one should find manifolds $M, N$ such that $H_*(M) \simeq A, H_*(N) \simeq B$. Secondly, there should be a topological map from $M \times N \to X $ or viceversa that induces such isomorphism. At last, one should turn this homological (weak?) equivalence into a diffeomorphism, which could be a consequence of the smooth structure. I am almost sure here surgery theory comes into play, so that the answer could be much easier for higher dimensions. In my mind, each of these steps would produce some obstructions.

Motivation. Suppose you a random data set distributed among a product manifold $M\times N$. You want to understand that this geometry hides a product decomposition. The modern toolkit we have is persistent homology, and I was thinking if there is a way to catch the existence of a decomposition via persistent homology. The very special case of a product of $S_1$'s or in general spheres is probably easier because of the 'thin' homology they have.

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  • $\begingroup$ Are you working in some specific characteristic? The Künneth formula is a tensor product over a field (and then, only for finite-type spaces). Over a PID you get $\mathrm{Tor}$ terms. Over an arbitrary ring... $\endgroup$
    – Najib Idrissi
    Sep 23, 2021 at 13:07
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    $\begingroup$ Usually one considers cohomology to be an algebra. What is the algebra structure of $H_*(M)$ if $M$ is just a differential (or topological) manifold? $\endgroup$
    – Georges Elencwajg
    Sep 23, 2021 at 13:11
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    $\begingroup$ Suppose you work over the rationals, us ecohomology, and assume everything is simply-connected. The the Sullivan-Barge Theorem may tell you when each of hte factors $A$ and $B$ is the cohomology of a manifold: mathoverflow.net/a/115921/8103. This method might only give you a rational homotopy equivalence, but it's a start. $\endgroup$
    – Mark Grant
    Sep 23, 2021 at 14:22
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    $\begingroup$ For your motivation, I think one problem is that a product requires two projections, so you need to analyse the data set to find two directions (sort of generalised principal axes) that have good properties. You want to then project the data set in those two directions (Big problems as those axes may not be straight!) I think your approach may need some refining before the first question links well with the motivation. $\endgroup$
    – Tim Porter
    Sep 23, 2021 at 16:10
  • $\begingroup$ @Najib: since mine is mère curiosity, I am ok with studying rational and finite field homology. $\endgroup$
    – Andrea Marino
    Sep 23, 2021 at 16:24

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