Title: A continuum of K\"unneth theorems for persistent homology

Abstract: Given a parametrized filtration of topological spaces, the goal of persistent homology is understanding how homology groups of the spaces vary with the choice of parameters. This information is encoded in an algebraic object called a persistence module. If the parameters are one-dimensional such as $\mathbb{R}$, this is called one-parameter persistence, while parameters with more than one dimension such as $\mathbb{R}^n$ yield multi-parameter persistence. Persistence modules have been shown to have a particularly rich algebraic structure.

We develop new aspects of the homological algebra theory for persistence modules, in both the one-parameter and multi-parameter settings. For each $p\in (0,\infty]$, we define a novel tensor product of persistence modules, $\otimes_{\ell^p}$. We prove that each $\otimes_{\ell^p}$ is left adjoint to an internal hom functor $\scHom^{\ell^p}$ also depending on $p\in (0,\infty]$. Furthermore, we compute their derived functors, $Tor^{\ell^p}$ and $Ext_{\ell^p}$ explicitly for interval modules. We prove that every $\otimes_{\ell^p}$ yields a \text{K\"unneth} short exact sequence of chain complexes of persistence modules arising from filtered cell complexes. We also prove that every $\scHom^{\ell^p}$ yields a Universal Coefficients Theorem. We show the resulting short exact sequence computes the Borel-Moore homology of an open complex induced from the original filtration on a cell complex. Finally, we show that for every $p\in [1,\infty]$ the associated K\"unneth short exact sequence can be used to significantly speed up and approximate persistent homology computations in a product metric space $(X\times Y,d^p)$ with the distance $d^p((x,y),(x',y'))=||d_X(x,x'),d_Y(y,y')||_p$.