Stability of higher-dimensional interval decomposable persistence modules

The algebraic stability theorem for pointwise finite dimensional (p.f.d.) R-persistence modules is a central result in the theory of stability for persistence modules. We present a stability theorem for n-dimensional rectangle decomposable p.f.d. persistence modules up to a constant (2n− 1) that is a generalization of the algebraic stability theorem. We give an example to show that the bound cannot be improved for n = 2. The same technique is then applied to free n-dimensional modules and what we call triangle decomposable modules, where we obtain smaller constants. The result for triangle decomposable modules combined with work by Botnan and Lesnick proves a version of the algebraic stability theorem for zigzag modules and the persistent homology of Reeb graphs. We also prove slightly weaker versions of the results for interval decomposable modules that are not assumed to be p.f.d. This work grew out of my master’s degree at the Department of Mathematical Sciences at NTNU [4].