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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?
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    $\begingroup$ Looking for freebies is not in the scope. $\endgroup$
    – Igor Rivin
    Commented Apr 13, 2018 at 0:03
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    $\begingroup$ @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. :) $\endgroup$
    – paul garrett
    Commented Apr 13, 2018 at 1:16
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    $\begingroup$ 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 ;-) $\endgroup$
    – Chris Gerig
    Commented Apr 13, 2018 at 1:50
  • $\begingroup$ Thanks! Seems surprising that no one has written lecture/course notes on this topic! $\endgroup$
    – gradstudent
    Commented Apr 13, 2018 at 2:29
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    $\begingroup$ 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. $\endgroup$
    – Liviu Nicolaescu
    Commented Apr 13, 2018 at 10:19

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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.

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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).

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

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also in the M. W. Hirsch, Differential topology ,Chapter 6 : Morse Theory.

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Nicolaescu, Liviu I.. “An Invitation to Morse Theory.” (2007).

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