Bott-Thom isomorphism, Hopf bundles and Morse theory

TL;DR

This paper develops a Morse-theoretic deformation approach to analyze mapping spaces of spheres into orthogonal groups, leading to new stable decompositions of vector bundles and classical Bott-Thom isomorphism results.

Contribution

It introduces a Morse-theoretic deformation framework for mapping spaces, establishing their homotopy equivalence to Clifford representation spaces and deriving new stable bundle decompositions.

Findings

Mapping spaces are weakly homotopy equivalent in a stable range

Stable decomposition of vector bundles over sphere bundles

Implications for classical Bott-Thom isomorphism theorems

Abstract

Based on Morse theory for the energy functional on path spaces we develop a deformation theory for mapping spaces of spheres into orthogonal groups. This is used to show that these mapping spaces are weakly homotopy equivalent, in a stable range, to mapping spaces associated to orthogonal Clifford representations. Given an oriented Euclidean bundle $V \to X$ of rank divisible by four over a finite complex $X$ we derive a stable decomposition result for vector bundles over the sphere bundle $\mathord{\mathbb S}( \mathbb{R} \oplus V)$ in terms of vector bundles and Clifford module bundles over $X$. After passing to topological K-theory these results imply classical Bott-Thom isomorphism theorems.