Intertwining category and complexity

Abstract

We develop the theory of the intertwining distributional versions of the LS-category and the sequential topological complexities of a space $X$, denoted by $\mathsf{icat}(X)$ and $\mathsf{iTC}_m(X)$, respectively. We prove that they satisfy most of the nice properties as their respective distributional counterparts $\mathsf{dcat}(X)$ and $\mathsf{dTC}_m(X)$, and their classical counterparts $\mathsf{cat}(X)$ and $\mathsf{TC}_m(X)$, such as homotopy invariance and special behavior on topological groups. We show that the notions of $\mathsf{iTC}_m$ and $\mathsf{dTC}_m$ are different for each $m \ge 2$ by proving that $\mathsf{iTC}_m(\mathcal{H})=1$ for all $m \ge 2$ for Higman's group $\mathcal{H}$. Using cohomological lower bounds, we also provide various examples of locally finite CW complexes $X$ for which $\mathsf{icat}(X) > 1$, $\mathsf{iTC}_m(X) > 1$, $\mathsf{icat}(X) = \mathsf{dcat}(X) = \mathsf{cat}(X)$, and $\mathsf{iTC}(X) = \mathsf{dTC}(X) = \mathsf{TC}(X)$.