# Equivariant homotopic distance

## TL;DR

This paper introduces the equivariant homotopic distance, a new invariant that generalizes several existing concepts in equivariant topology, providing a flexible framework for analyzing pairs of G-maps.

## Contribution

It defines the equivariant homotopic distance and explores its fundamental properties, connecting it with known invariants like equivariant LS-category and topological complexity.

## Key findings

- Establishes basic properties including homotopy invariance and triangle inequality.
- Provides cohomological and dimension-connectivity bounds.
- Analyzes applications to Hopf G-spaces and equivariant fibrations.

## 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.