Morse Theory/Morse Homology

The purpose of this talk is to outline the basics of Morse theory and the Morse complex. This will be really important as a toy example of Floer the- ory, which is a version of this in the infinite dimensional setting. Basic ref- erences are [Milnor, Morse Theory] and [Nicolaescu, An Invitation to Morse Theory], while [Milnor, Lectures on the h-cobordism Theorem] is more de- tailed. A good outline for a lecture like this is at http://math.illinois.edu/ jpas- cale/m392c/notes/morse.pdf

Plan Definitions. Morse function, Riemannian metric, gradient flow, Morse-Smale condition, genericity, nondegeneracy, index of critical point, invariance of index

First examples. Closed surfaces, Sn, Rn, height functions

The Morse Lemma. Describe Morse critical points locally. Ascending/descending manifolds for gradient flow. Should be explained with examples above, draw in- dex 0,1,2 critical points in R3 No need to prove it unless you’re feeling ambitious.

Handle decompositions from Morse functions. Inverse images of intervals con- taining critical values are handles. Can use these to chop up manifold into nice pieces(elementary cobordisms)

Examples: Closed surfaces, 3-manifolds. Choose your favorite surface and de- scribe a handle decomposition of it. Do it again for lens spaces, or anything else you like.

If possible, talk about self-indexing Morse functions. Again, no need to prove they exist unless you want to.

Relationship with Spinc-structures and Heegaard Diagrams on 3-manifolds There’s a way to relate this to Heegaard diagrams in a way that we’ll need later on, detailed in [Tweedy pg. 50-54]. Make sure to talk about this.

Notes The references above should be enough, but there are tons of resources about Morse theory that you can search for, or ask people about.

Dependencies This should be one the first few talks, so no real background needed.

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