On Morse-Smale-Witten type cohomologies for orbifolds
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.
scientific article; zbMATH DE number 7401512
scientific article