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For references of embedded Morse theory, or so-called relative morse theory of a pair, I have found R.W. Sharpe's "Total Absolute Curvature and Embedded Morse numbers" totally not helpful. For further details, Sharpe refers the reader to B. Perron's "Pseudo-Isotopies de plongements en codimension 2", another reference which for me is totally opaque.

When learning morse theory for the first time, there is the standard example of the height function on the torus in 3-space--and this an example which one 'sees' and bears in mind throughout. However I find myself without any such 'basic' example for the relative morse theory. Even the instance of considering how the height function would yield a handlebody decomposition of the complement of the torus in 3-space is something that I don't 'see'.

Can anybody refer me to some basic examples of the embedded morse theory, or to some further literature?

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    $\begingroup$ I found Martin Scharlemann's paper "The four-dimensional Schoenflies conjecture is true for genus two imbeddings," quite useful in understanding embedded Morse theory. It's not a textbook, but rather a short article which uses the technique and explains it well. $\endgroup$
    – Jim Conant
    Commented May 14, 2012 at 16:54

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I found the treatment in Goresky-MacPherson's book "Stratified Morse theory" (available from Goresky's webpage here) very enlightening. The focus is not on low-dimensional topology, but the treatment is very geometric.

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  • $\begingroup$ The 'relative' morse theory (RMT) i'm trying to learn concerns the following set-up: Given a manifold $X$, an embedded submanifold $Y$, a Morse function $f$ on $Y$ (and a consequent handlebody decomposition of $Y$), describe a handlebody decomposition of $X \setminus Y$. Sharpe's paper claims this set-up is part of morse theory 'folk-lore'. Now the RMT of Goresky/Macpherson as outlined in I.9 concerns another situation. I don't know whether or not the G/M RMT contains the RMT i'm trying to learn. Explicitly i'm trying to understand more of G.Mess' "Torelli groups of genus 2 and 3 surfaces". $\endgroup$
    – JHM
    Commented May 14, 2012 at 19:34
  • $\begingroup$ @J. Martel : The G/M approach can also give the stuff that Sharpe does. If your main interest is Mess's paper, then I recommend looking at Hain's paper arxiv.org/abs/math.AG/0203057, which uses G/M-style Morse theory to strengthen Mess's results. Hain's paper is the one that made Mess's proof understandable to me. $\endgroup$
    – Andy Putman
    Commented May 14, 2012 at 20:21
  • $\begingroup$ (I should say that the results you need in G/M might not be called "relative Morse theory" -- I recommend looking at the chapters where they discuss the topology of complements of affine subspaces in Euclidean spaces, which is very analogous to what Mess is doing). $\endgroup$
    – Andy Putman
    Commented May 14, 2012 at 20:27
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The main set of ideas that you want to learn is the following description of an m-dimensional manifold Y sitting in $R^n$, in such a way that the standard height function is a Morse function when restricted to Y. Thus this function, say f, gives a handle decomposition of Y; as you pass a critical point of f|Y, you add an index k handle to Y. Simultaneously, you add a (k + n -m -1)-handle to the complement of Y. This is described in reasonable detail in Section 6.2 of the book of Gompf and Stipsicz, "An Introduction to 4-manifolds and Kirby Calculus".

I don't know the original source for this description; I learned it more or less as folklore. The informal explanation that Kirby used to give of this involved sitting in a bathtub (I think the person in the bathtub was supposed to be Y) and watching the topology of the water change as it passed various critical points. (An alternate version, helpful for thinking about knots, was a wire in a bucket of water being filled up.) A nice example to think about is how to build a handle decomposition (or Heegaard splitting) of a knot complement. For instance, for a 2-bridge knot, you should see a handle decomposition with a 0-handle, two 1-handles, and a single 2-handle.

As a side remark, the term relative Morse theory, as I understood it, has to do with the study of the Morse function on a manifold Y induced by a Morse function on a larger manifold. In this form, it was extensively studied (in the PL case) in the 60's, in order to give results on embeddings. For instance, various theorems of the form "concordance implies isotopy" in high codimension are proved in this way.

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In addition to the classics like Perron, Sharpe , Rourke and Section 6.2 in Gompf and Stipsicz that Danny mentioned, there is a nice extensive modern treatment in Borodzik and Powell "Embedded Morse theory and relative splitting of cobordisms of manifolds" or Borodzik, Nemathi and Ranicki "Codimension 2 embeddings, algebraic surgery and Seifert forms".

Also one really good source for geometric intuition (which deals with the codimension two case) is Michael Freedman's Ph.D. thesis "Codimension-Two Surgery".

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