The suspended free loop space of a symmetric space

TL;DR

This paper uses Morse theory to analyze the homotopy type of the free loop space on certain symmetric spaces, revealing a decomposition into Thom spaces over the tangent sphere bundle.

Contribution

It provides an explicit homotopy equivalence for the suspension spectrum of the free loop space on specific symmetric spaces, connecting it to Thom spaces over tangent sphere bundles.

Findings

Homotopy equivalence between suspension spectra of free loop spaces and Thom spaces.

Explicit description of vector bundles over tangent sphere bundles.

Application of Morse theory to symmetric space loop spaces.

Abstract

Let M be one of the projective spaces CP^n, HP^n for n>1 or the Cayley projective plane OP^2, and let LM denote the free loop space on M. Using Morse theory methods, we prove that the suspension spectrum of (LM)_+ is homotopy equivalent to the suspension spectrum of M_+ wedge a family of Thom spaces of explicit vector bundles over the tangent sphere bundle of M.