Monotonicity of the Schwarz genus

Abstract:

The 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 sections to $\xi$. Many important concepts, including the 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 types of morphisms between fibrations. Our main result is the following: if there exists 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-complex 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.