Functoriality in Morse theory on closed manifolds
Abstract
We develop functoriality for Morse theory, namely, to a pair of Morse-Smale systems and a generic smooth map between the underlying manifolds we associate a chain map between the corresponding Morse complexes, which descends to the correct map on homology. This association does not in general respect composition. We give sufficient conditions under which composition is preserved. As an application we provide a new proof that the cup product as defined in Morse theory on the chain level agrees with the cup product in singular cohomology. In appendices we present a proof (due to Paul Biran) that the unstable manifolds of a Morse-Smale system are the open cells of a CW structure on the underlying manifold, and also we show that the Morse complex of the triple is canonically isomorphic to the cellular complex of the CW structure. This gives a new proof that the Morse complex is actually a complex and that it computes the homology of the manifold.
Keywords
Cite
@article{arxiv.0805.2131,
title = {Functoriality in Morse theory on closed manifolds},
author = {Avraham Aizenbud and Frol Zapolsky},
journal= {arXiv preprint arXiv:0805.2131},
year = {2009}
}
Comments
22 pages; added appendices with proofs that (i) the unstable manifolds of a Morse-Smale gradient flow are the open cells of a CW structure on the underlying manifold (due to Paul Biran); (ii) the Morse-Thom-Witten complex is canonically isomorphic to the cellular complex of the above CW structure; various minor corrections