Equivariant homotopic distance
Navnath Daundkar, J. M. García-Calcines
TL;DR
The paper defines the equivariant homotopic distance $D_G(f,g)$ for $G$-maps and demonstrates that it encompasses equivariant sectional category, equivariant LS-category, and equivariant topological complexity as key instances. By relating $D_G$ to pullbacks of the free path fibration, the authors derive a triangle inequality via categorical methods, along with cohomological and dimension bounds that sharpen orbit-space estimates. They apply the framework to Hopf $G$-spaces, establishing $\mathrm{TC}_G(X)=\mathrm{cat}_G(X)$ under natural hypotheses, and extend the theory to equivariant fibrations to obtain inequalities connecting total space, fibre, and base. Overall, $D_G$ provides a flexible, map-centered unifying approach to equivariant LS-category and TC, with sharp bounds via Borel cohomology and dimension arguments and concrete applications to equivariant fibrations and Hopf $G$-spaces.
Abstract
We introduce and study the notion of \emph{equivariant homotopic distance} $D_G(f,g)$ between $G$-maps $f,g \colon X \to Y$. We show that the equivariant Lusternik-Schnirelmann category and the equivariant topological complexity are particular cases of this notion. This invariant also connects naturally with the equivariant sectional category. What makes $D_G$ distinctive, however, is that it provides a flexible framework centered on pairs of maps, within which one can derive results that are not immediate from the general setting. In particular, we establish its basic properties, including homotopy invariance and a categorical proof of the triangle inequality valid in the equivariant context. We also obtain cohomological and dimension-connectivity bounds, and analyze structural applications to Hopf $G$-spaces and equivariant fibrations.