The genus and the Lyusternik-Schnirelmann category of preimages

TL;DR

This paper introduces an axiomatic generalization of the Lyusternik-Schnirelmann category, extending it to include sectional category and Schwarz genus, and provides lower bounds for preimages under continuous maps.

Contribution

It develops a unified axiomatic framework for various topological invariants related to preimages, expanding the theoretical understanding of these concepts.

Findings

Provides lower bounds on the generalized category for preimages

Unifies several topological invariants under an axiomatic scheme

Extends the applicability of category theory to new classes of problems

Abstract

In this paper some axiomatic generalization (function of open subsets) of the relative Lyusternik-Schnirelmann category is considered, incorporating the sectional category and the Schwarz genus as well. For this function and a given continuous map of the underlying space to a finite-dimensional metric space some lower bounds on the value of this function on the (neighborhood of) preimage of some point are given.