# Monotonicity of the Schwarz genus

## TL;DR

This paper investigates the Schwarz genus of fibrations, establishing conditions under which it remains invariant under certain morphisms, and explores implications for topological complexity of spaces and their skeleta.

## Contribution

It proves that Schwarz genus is preserved under fiberwise maps inducing high enough fiber equivalences, linking it to topological complexity of spaces and their skeleta.

## Key findings

- Schwarz genus remains unchanged under specific fiberwise maps.
- Topological complexity of skeleta increases with dimension in high complexity spaces.
- Results connect fibrations' morphisms with topological invariants of spaces.

## Abstract

Schwarz genus $\mathsf{g}(\xi)$ of a fibration $\xi\colon E\to B$ is defined as the minimal integer $n$, such that there exists a cover of $B$ by $n$ open sets that admit partial section to $\xi$. Many important concepts, including Lusternik-Schnirelmann category, Farber's topological complexity and Smale-Vassiliev's complexity of algorithms can be naturally expressed as Schwarz genera of suitably chosen fibrations. In this paper we study Schwarz genus in relation with certain type of morphisms between fibrations. Our main result is the following: if there exist a fibrewise map $f\colon E\to E'$ between fibrations $\xi\colon E\to B$ and $\xi'\colon E'\to B$ which induces an $n$-equivalence between respective fibres for a sufficiently big $n$, then $\mathsf{g}(\xi)=\mathsf{g}(\xi')$. From this we derive several interesting results relating the topological complexity of a space with the topological complexities of its skeleta and subspaces (and similarly for the category). For example, we show that if a CW-complexes has high topological complexity (with respect to its dimension and connectivity), then the topological complexity of its skeleta is an increasing function of the dimension.