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I'm interested in understanding the probability that given a prime $p$, $p$ divides the order of the torsional part of $H^k(X,Z)$, where $X$ is a finite simplicial complex.

So, I'm wondering given an arbitrary

Lets say you have a uniform distribution over all finite simplicial complex, complexes on $X$, is there a correspondence between torsion in n$vertices. Given$H^1(X,Z)$and being a space that p > 0$ what is the probability the complex you get has a non-orientable manifold with some other stuff glued/attached to it somehow$H_K(X,Z)$ or $H^K(X,Z)$ having $p$ torsion?

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# connection between non-orientable manifolds and torsion in 1D (co) homology

I'm interested in understanding the probability that given a prime $p$, $p$ divides the order of the torsional part of $H^k(X,Z)$, where $X$ is a finite simplicial complex.

So, I'm wondering given an arbitrary finite simplicial complex, $X$, is there a correspondence between torsion in $H^1(X,Z)$ and being a space that has a non-orientable manifold with some other stuff glued/attached to it somehow?