On the o-minimal LS-category

TL;DR

This paper introduces the o-minimal LS-category for definable sets in o-minimal structures, relating it to classical and semialgebraic categories, and explores its properties for definable groups.

Contribution

It defines the o-minimal LS-category, establishes its relation with classical categories, and studies its behavior in definable groups, linking definable homotopy equivalence to Lie group homotopy.

Findings

The o-minimal LS-category is well-defined for definable sets.

A relation between o-minimal and classical LS-categories is established.

Definably connected compact groups are homotopy equivalent iff their Lie groups are.

Abstract

We introduce the o-minimal LS-category of definable sets in o-minimal expansions of ordered fields and we establish a relation with the semialgebraic and the classical one. We also study the o-minimal LS-category of definable groups. Along the way, we show that two definably connected definably compact definable groups G and H are definable homotopy equivalent if and only if L(G) and L(H) are homotopy equivalent, where L is the functor which associates to each definable group its corresponding Lie group via Pillay's conjecture.