The Morse-Bott inequalities, orientations, and the Thom isomorphism in Morse homology

TL;DR

This paper examines the Morse-Bott inequalities, identifies errors in their usual orientation assumptions, provides a counterexample, and clarifies the correct theoretical framework involving orientations and the Thom isomorphism in Morse homology.

Contribution

The paper corrects misconceptions about orientation assumptions in Morse-Bott inequalities and clarifies their relation to the Thom isomorphism in Morse homology.

Findings

Incorrect orientation assumptions can invalidate Morse-Bott inequalities

Counterexample demonstrates the necessity of proper orientation considerations

Clarification of the role of the Thom isomorphism in Morse homology

Abstract

The Morse-Bott inequalities relate the topology of a closed manifold to the topology of the critical point set of a Morse-Bott function defined on it. The Morse-Bott inequalities are sometimes stated under incorrect orientation assumptions. We show that these assumptions are insufficient with an explicit counterexample and clarify the origin of the mistake.