On the Lusternik-Schnirelmann category of Peano continua

TL;DR

This paper introduces a generalized LS-category for spaces using covers by arbitrary subsets, proves its equivalence to the classical category in certain spaces, and computes it for fractal continua.

Contribution

It defines a new generalized LS-category, proves its equivalence to classical LS-category for compact metric ANR spaces, and calculates it for fractal Peano continua.

Findings

cat_g coincides with classical LS-category for compact metric ANR spaces

Provides dimension-theoretic proofs of key theorems

Calculates cat_g for Menger spaces and Pontryagin surfaces

Abstract

We define the LS-category cat_g by means of covers of a space by general subsets, and show that this definition coincides with the classical Lusternik-Schnirelmann category for compact metric ANR spaces. We apply this result to give short dimension theoretic proofs of the Grossman-Whitehead theorem and Dranishnikov's theorem. We compute cat_g for some fractal Peano continua such as Menger spaces and Pontryagin surfaces.