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.

Lets say you have a uniform distribution over all finite simplicial complexes on $n$ vertices. Given $p > 0$ what is the probability the complex you get has $H_K(X,Z)$ or $H^K(X,Z)$ having $p$ torsion?

• $H^1(X)$ with integer coefficients has no torsion. – Ryan Budney Mar 4 '11 at 15:44
• Look at the cohomology-homology universal coefficient theorem. – Ryan Budney Mar 4 '11 at 15:45
• What does probability mean here? Are you putting some distribution on finite simplicial complexes? – Matthew Kahle Mar 4 '11 at 16:08
• Matthew, it sounds like he wants some precise measure of the distribution of torsion in the (co)homology of a random collection of simplicial complexes. Presumably 2-torsion is the "most common" but perhaps rhl wants to quantify that statement somehow. – Ryan Budney Mar 4 '11 at 16:26
• @rhl: Your reformulation in the above comment looks much better than the original question. I suggest that you edit your question (click on "edit"), and also remove the vague sentence "being a space that has a non-orientable manifold with some other stuff glued/attached to it somehow?". – André Henriques Mar 5 '11 at 14:38