# What are good Morse Theory lecture notes and books?

Searching on the net I couldnt find any recent lecture/course notes on Morse Theory. I found an old set of notes (http://www.math.toronto.edu/mgualt/Morse%20Theory/mfp.pdf) by Mike Hutchings and these incomplete notes by Ralph Cohen (http://math.stanford.edu/~ralph/morsecourse/biglectures.pdf)

[..I really want a reference which has a detailed description of the gradient flow line" perspective as in the chater 4,5,6 of Ralph's notes. Just that Ralph's notes are very hard to follow given that all the diagrams are missing!..]

I found this book that has been made legally freely available by the author, https://www3.nd.edu/~lnicolae/Morse2nd.pdf and I have read quite a bit of the old book by Milnor.

• Are there other other good references (particularly lecture/course notes) that I am missing?
• Looking for freebies is not in the scope. – Igor Rivin Apr 13 '18 at 0:03
• I am happy to know of modern book references too :) – gradstudent Apr 13 '18 at 0:29
• @IgorRivin, although I agree that "looking for freebies" doesn't sound good, perhaps "looking for freely-available sources" sounds ok? In my own life, I've exerted a good bit of effort to generate "freely available" (and reasonably high quality) documents available... and I have a tendency to think that all academic mathematicians should do so... The possibility of vulgar-sounding references doesn't deter me. :) – paul garrett Apr 13 '18 at 1:16
• Morse homology by Schwarz. Also another one by Audin-Damian (might have misspelled). These are the typical references now (and then a book by Banyaga). But Hutchings’ notes are amazing ;-) – Chris Gerig Apr 13 '18 at 1:50
• Look at these notes www3.nd.edu/~lnicolae/Morse2nd.pdf Floer homology is discussed in Sec 2.5. Chapter 4 is devoted to a rather detailed investigation of the gradient flow dynamics. In particular it is hown that the Morse-Smale condition is equivalent to the fact that the stratification by unstable manifolds is a Whitney stratification. This leads to a more sophisticated view of the Floer homology in Sec 4.5. – Liviu Nicolaescu Apr 13 '18 at 10:19

If you are looking for the classical approach to Morse theory, I feel nothing beats Milnor's book on the subject:

Milnor, J. Morse theory. Annals of Mathematics Studies, No. 51 Princeton University Press, Princeton, N.J. 1963

For the Morse homological approach, i.e. counting flowlines, I really like Weber's paper on the subject:

Weber, Joa The Morse-Witten complex via dynamical systems. Expo. Math. 24 (2006), no. 2.

Another standard reference is the book of Banyaga and Hurtubise.

Banyaga, Augustin; Hurtubise, David Lectures on Morse homology. Kluwer Texts in the Mathematical Sciences, 29. Kluwer Academic Publishers Group, Dordrecht, 2004.

A book that is tough to read, but is a gateway to Floer theory is Schwarz' book.

Schwarz, Matthias Morse homology. Progress in Mathematics, 111. Birkhäuser Verlag, Basel, 1993.

I heard good things about the book of Audin and Damian, but I have not read it.

Audin, Michèle; Damian, Mihai Morse theory and Floer homology. Translated from the 2010 French original by Reinie Erné. Universitext. Springer, London; EDP Sciences, Les Ulis, 2014.

• If you would put the full titles of the books your answer would be more useful. – Piotr Hajlasz Apr 13 '18 at 17:17
• @PiotrHajlasz: I have done so (I did not have access to mathscinet at the time of writing) – Thomas Rot Apr 17 '18 at 12:03

These lecture notes were actually mainly devoted to the Morse Complex in the infinite dimensional setting; but they were thought to be suitable for finite dimensional manifolds as well (btw, you don't need to pay for them).

Another classic text is Bott's Lectures on Morse theory, old and new.