The Witten complex for singular spaces of dimension two with cone-like singularities

✍ Scribed by Ursula Ludwig

• Publisher: John Wiley and Sons
• Year: 2011
• Tongue: English
• Weight: 240 KB
• Volume: 284
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200810167

✦ Synopsis

Abstract

The Witten deformation is a method proposed by Witten which, given a function \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$f:M \longrightarrow \mathbb {R}$\end{document} on a smooth compact manifold M, allows to prove the Morse inequalities. Witten’s proof of the Morse inequalities is analytical and can thus be applied to situations where the Thom‐Smale method is not accessible. In these notes we generalise the Witten deformation to certain singular Riemannian manifolds X which are metric models for singular algebraic curves, and functions on X which we call admissible Morse functions. They are particular examples of stratified Morse functions in the sense of the theory developed by Goresky and MacPherson. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim