Intertwining category and complexity

TL;DR

This paper introduces and studies the intertwining distributional versions of the LS-category and topological complexity, establishing their properties, differences, and providing examples with specific values.

Contribution

It develops the theory of intertwining distributional invariants and compares them to classical and distributional counterparts, highlighting their properties and differences.

Findings

Intertwining invariants satisfy homotopy invariance.

They behave well on topological groups.

Examples show differences between invariants for certain spaces.

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