Functoriality in Morse theory on closed manifolds

Details Functoriality in Morse theory on closed manifolds Functoriality in Morse theory on closed manifolds

2008-05-14

arxiv.org/abs/0805.2131v2

Mathematics

Differential Geometry

22 pages; added appendices with proofs that (i) the unstable manifolds of a Morse-Smale gradient flow are the open cells of a

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Functoriality in Morse theory on closed manifolds does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Functoriality in Morse theory on closed manifolds, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Functoriality in Morse theory on closed manifolds will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-56287