Morse inequalities for manifolds with boundary

TL;DR

This paper provides an analytic proof of Morse inequalities tailored for manifolds with boundary, extending classical results to a broader class of geometric objects.

Contribution

It offers a complete, detailed proof of Morse inequalities for manifolds with boundary using Witten's analytic approach, which was previously known but not fully documented.

Findings

Established Morse inequalities for manifolds with boundary.

Demonstrated the effectiveness of Witten's analytic method in this context.

Provided a rigorous proof accessible to experts and newcomers alike.

Abstract

The aim of this paper is to provide a proof for a version of Morse inequality for manifolds with boundary. Our main results are certainly known to the experts on Morse theory, nevertheless it seems necessary to write down a complete proof for it. Our proof is analytic and is based on J. Roe's account of Witten's approach to Morse Theory.