On the Topological Complexity of Maps

TL;DR

This paper introduces a new homotopy-invariant measure called TC(f) for the topological complexity of maps, extending classical invariants and applying it to group homomorphisms, enriching the understanding of topological and algebraic structures.

Contribution

It defines and develops TC(f), a homotopy invariant for maps, and explores its properties and relationships with existing invariants like TC and cat.

Findings

TC(f) interacts with TC(X) and TC(Y) similarly to how cat(f) interacts with cat(X) and cat(Y)

TC(f) satisfies analogous inequalities as TC(X) and cat(X)

Application to group homomorphisms f:H→G

Abstract

We define and develop a homotopy invariant notion for the topological complexity of a map $f:X \to Y$, denoted TC($f$), that interacts with TC($X$) and TC($Y$) in the same way cat($f$) interacts with cat($X$) and cat($Y$). Furthermore, TC($f$) and cat($f$) satisfy the same inequalities as TC($X$) and cat($X$). We compare it to other invariants defined in the papers [15,16,17,18,20]. We apply TC($f$) to studying group homomorphisms $f:H\to G$.