Trace map, Cayley transform and LS category of Lie groups

TL;DR

This paper introduces a Cayley transform-based method to compute the LS category of Lie groups and homogeneous spaces, providing explicit open coverings and connecting to height functions for critical point analysis.

Contribution

It presents a simplified approach using the Cayley transform for LS category computation and explicit coverings for specific Lie groups, with insights into critical point structures.

Findings

Explicit categorical open coverings for U(n), U(2n)/Sp(n), U(n)/O(n)

Connection between Cayley transform and height functions in Lie groups

Explicit covering of Sp(2) by categorical open sets

Abstract

The aim of this paper is to use the so-called Cayley transform to compute the LS category of Lie groups and homogeneous spaces by giving explicit categorical open coverings. When applied to U(n), $U(2n)/Sp(n)$ and $U(n)/O(n)$ this method is simpler than those formerly known. We also show that the Cayley transform is related to height functions in Lie groups, allowing to give a local linear model of the set of critical points. As an application we give an explicit covering of $Sp(2)$ by categorical open sets. The obstacles to generalize these results to $Sp(n)$ are discussed.