(Multi-)Persistent Homology and Topological Robotics

Abstract

After developing the relevant background and proving some general results in the early chapters, the main novel content of this thesis is the computation of the $i$-th homology groups of the second configuration spaces of metric graphs $\mathsf{Star_k}$ and $\hat{\mathcal{H}}{m,n}$, with two restraint parameters. These configuration spaces are filtered by the poset $(\mathbb{R},\leq)^{\op}\times(\mathbb{R},\leq)$. We study the persistence modules $PH_i((\mathsf{Star_k})^2{-,-};\mathbb{F})$ and $PH_i((\hat{\mathcal{H}}{m,n}){-,-}^2;\mathbb{F})$ where $i=0,1$, since higher homology vanishes for these spaces. Next, we construct a new representation over the poset given by the hyperplane arrangement of the configuration spaces of the finite graph. There is no loss of information when we restrict to the poset of chambers because the functor $PH_i(-)$ factors through the poset of chambers. Using this machinery and the homology groups we calculated, we find the direct sum decomposition of the $2$-parameter persistence modules $PH_i((\mathsf{Star_k})^2_{-,-};\mathbb{F})$ and $PH_i((\hat{\mathcal{H}}{m,n}){-,-}^2;\mathbb{F})$, where each summand is indecomposable. In particular, we show that $PH_0((\hat{\mathcal{H}}{m,n}){-,-}^2;\mathbb{F})$ and $PH_1((\hat{\mathcal{H}}{m,n}){-,-}^2;\mathbb{F})$ can be written as a direct sum of polytope modules.