On Morse-Smale-Witten type cohomologies for orbifolds
type: Item
instance of: scholarly article
title: On Morse-Smale-Witten type cohomologies for orbifolds
zbMATH Open document ID: 1476.57048
author: A-Li Liao
author: Cheng-Yong Du
published in: Topology and its Applications
publication date: 2021-09-27T00:00:00Z
review text: Let \(M\) be an oriented, smooth, closed manifold, and let \(G\) be a connected, compact Lie group. Assume \(G\) acts on \(M\) by a smooth, effective, locally free, orientation preserving action. Then the quotient \(X=M/G\) is a compact, effective, orientable orbifold. Morse theory is an important tool for understanding the topology of manifolds, and it can be generalized to study orbifolds. Let \(\bar{f}\colon X\to {\mathbb{R}}\) be a Morse function that has a \(G\)-invariant lift \(f\colon M\to {\mathbb{R}}\). Assume that the lift \(f\) is a Morse-Bott function. Associated to \(f\) is a Morse-Bott cochain complex. The cohomology of this complex is called the Morse-Bott cohomology of \(f\), and it computes the \(G\)-equivariant cohomology \(H^\ast_G(M;{\mathbb{R}})\). The orbifold Morse cohomology of \(\bar{f}\) is, by definition, the cohomology of the Morse-Smale-Witten cochain complex associated to \(\bar{f}\). This cohomology is isomorphic to the singular cohomology \(H^\ast(M/G; {\mathbb{R}})\). Since the action of \(G\) on \(M\) is locally free, it follows that there is an isomorphism \(H^\ast(M/G;{\mathbb{R}})\cong H^\ast_G(M;{\mathbb{R}})\). Therefore, there is an isomorphism between the Morse-Bott cohomology of \(f\) and the orbifold Morse cohomology of \(\bar{f}\). However, this isomorphism is not explicit. In this paper, the authors construct a new, filtered cochain complex by using basic forms, and homomorphisms from this complex to the Morse-Bott cochain complex of \(f\) and to the orbifold Morse-Smale-Witten cochain complex of \(\bar{f}\), respectively. They show that, on cohomology groups, the induced homomorphisms of these two homomorphisms are both isomorphisms.
reviewed by: Marja Kankaanrinta
Mathematics Subject Classification ID: 57R18
Mathematics Subject Classification ID: 37D15
zbMATH DE Number: 7401512
zbMATH Keywords: Morse-Bott cohomology
zbMATH Keywords: locally free actions
zbMATH Keywords: orbifold Morse cohomology
zbMATH Keywords: basic forms
MaRDI profile type: Publication
full work available at URL: j.topol.2020.107527
OpenAlex ID: W3112773089
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DOI: 10.1016/J.TOPOL.2020.107527
description: scientific article; zbMATH DE number 7401512