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 $\mathsf{iTC}(X)=\mathsf{dTC}(X)=\mathsf{TC}(X)$.

arXiv comment29 pages. Changes made based on the referee reports. To appear in Homology Homotopy Appl