LS category on laminations with transverse invariant measure

TL;DR

This paper introduces a new version of the LS category for laminations with transverse invariant measures, which is invariant under leafwise homotopy and relates to critical sets, with potential applications in variational problems.

Contribution

It defines a measured LS category for laminations that extends classical concepts and proves its invariance and semicontinuity properties, connecting topology with measure theory.

Findings

The measured LS category is invariant under leafwise homotopy preserving transverse measures.

The measured category is semicontinuous under variations of the foliated structure.

The measured category relates to the number of critical sets, aiding variational problem analysis.

Abstract

A version of the tangential LS category is introduced for topological laminations with a transverse invariant measure. Here, we use the transverse measure of the contraction of a tangential categorical open set instead of counting this set. This new measured category is invariant by leafwise homotopy equivalences preserving the transverse measures, and the condition of being zero or positive is a transverse invariant. The usual tangential LS category is also bounded by the number of certain critical sets. It is also proved that the measured category is semicontinuous when the foliated structure and the transverse invariant measure varies on a fixed manifold, which is a version of a result of W. Singhof and E. Vogt for the tangential category. Hopefully, this relation between LS category and critical sets will be useful to deal with foliated versions of variational problems, like the existence of closed leafwise geodesics, and even to simplify the proofs of classical results on manifolds.