Abstract:
Given a chain complex of finitely generated free abelian groups and a prime $p$, we consider the problem of determining whether the homology of the complex contains $p$-torsion, i.e., whether there are nonzero homology classes of exponent $p$. We focus on concrete examples and discuss some different approaches to the problem for these examples. For instance, given a finite group $G$ acting on the complex, we may form a smaller chain complex consisting of all sums $\sum_{g\in G} g(c)$ such that $c$ is an element in the original complex. As long as $p$ does not divide the order of $G$, the larger complex contains $p$-torsion whenever the smaller does. Another approach is to cut the complex into an "upper" and a "lower" part and associate the homology of the original complex to the homology of the two parts via a long exact sequence. In favorable instances, a close examination of the maps between the homology groups may provide information about the existence of $p$-torsion.

We warmly encourage input from the audience on other useful methods to detect torsion in homology groups.