Abstract

In algebraic topology it is well-known that, using the Mayer-Vietoris sequence, the homology of a space $X$ can be studied splitting $X$ into subspaces $A$ and $B$ and computing the homology of $A$, $B$, $A\cap B$. A natural question is to which an extent persistent homology benefits of a similar property. In this paper we show that persistent homology has a Mayer-Vietoris sequence that in general is not exact but only of order two. However, we obtain a Mayer-Vietoris formula involving the ranks of the persistent homology groups of $X$, $A$, $B$ and $A\cap B$ plus three extra terms. This implies that topological features of $A$ and $B$ either survive as topological features of $X$ or are hidden in $A\cap B$. As an application of this result, we show that persistence diagrams are able to recognize an occluded shape by showing a common subset of points.