A variant of the topological comlexity of a map

TL;DR

This paper introduces a new variant of topological complexity for pairs of maps, generalizing existing concepts and establishing invariance and interpolation properties related to classical topological invariants.

Contribution

It defines a fiberwise topological complexity for two maps, proves its invariance under fiberwise homotopy, and shows it interpolates between classical invariants.

Findings

Defines relative topological complexity for pairs of maps.

Proves invariance under fiberwise homotopy.

Shows interpolation between category and topological complexity.

Abstract

In this paper, we associate to two given continuous maps $f,g: X\rightarrow Z$, on a path connected space $X$, the relative topological complexity $TC^{(f, g, Z)}(X):=TC_X(X\times _ZX)$ of their fiber space $X\times _ZX$. When $g=f$ we obtain a variant of the topological complexity $TC(f)$ of $f: X\longrightarrow Z$ generalizing Farber's topological complexity $TC(X)$ in the sens that $TC(X)=TC(cst_{x_0})$; being $cst_{x_0}$ the constant map on $X$. Moreover, we prove that $TC(f)$ is a fiberwise homotopy equivalence invariant. When $(X,x_0)$ is a pointed space, we prove that $TC^{(f, cst_{x_0}, Z)}(X)$ interpolates $cat(X)$ and $TC(X)$ for any continuous map $f$.