Physics for a Mathematician 02-II - Personal Bias vs History
II. Personal Bias vs. History
This is kind of a digression, and is more of a confession of my ignorance of history throughout graduate school than anything else. Confessions are perfect for blog posts, right?
As a symplectic and contact geometer, my eyes are somewhat biased by the fact that Floer theory, which can be seen as an infinite-dimensional generalization of the Morse-Smale-Witten complex, appears everywhere around me. For a while, this meant that my understanding of the history was particularly biased. Often in seminar or conference talks, people simply refer to the finite-dimensional case we have just discussed as the Morse complex instead of the Morse-Smale-Witten complex, but this is historically inaccurate, and the latter name is certainly more appropriate. Somewhat embarrassingly, in the first two years of my graduate school career, I was under the impression that the Morse inequalities came out of studying this chain complex. But it wasn't until I picked up Milnor's Morse Theory (1963) that I learned that viewpoint to be patently false. (In hindsight, it should have been obvious that the Morse inequalities were much more low-tech.) I proceeded to read Milnor's book over two or three days, and was amazed at how much could be proved from Morse theory without all of the fancy language of the Morse-Smale-Witten complex (e.g. Bott periodicity!).
As a symplectic and contact geometer, my eyes are somewhat biased by the fact that Floer theory, which can be seen as an infinite-dimensional generalization of the Morse-Smale-Witten complex, appears everywhere around me. For a while, this meant that my understanding of the history was particularly biased. Often in seminar or conference talks, people simply refer to the finite-dimensional case we have just discussed as the Morse complex instead of the Morse-Smale-Witten complex, but this is historically inaccurate, and the latter name is certainly more appropriate. Somewhat embarrassingly, in the first two years of my graduate school career, I was under the impression that the Morse inequalities came out of studying this chain complex. But it wasn't until I picked up Milnor's Morse Theory (1963) that I learned that viewpoint to be patently false. (In hindsight, it should have been obvious that the Morse inequalities were much more low-tech.) I proceeded to read Milnor's book over two or three days, and was amazed at how much could be proved from Morse theory without all of the fancy language of the Morse-Smale-Witten complex (e.g. Bott periodicity!).
In my third or fourth year of grad school, I bought a copy of Milnor's Lectures on the h-cobordism theorem (1965). My thesis topic involved contact handle attachments, so I wanted to understand the smooth theory of handles extremely well before delving into the contact version. This was sufficient for my purposes, but I was surprised that the chain complex doesn't appear in full generality, though it does appear in the special case that the Morse function is self-indexing, i.e. with $f(p) = \mathrm{ind}(p)$ for $p$ a critical point, stated in slightly more algebraic terminology. It seems from assorted references that this chain complex was known by Smale and Thom. Certainly, this isn't too surprising, especially in the case of Smale, whose proof of the h-cobordism theorem inspired Milnor's book.
Nonetheless, as far as I'm aware, the Morse-Smale-Witten complex, in its general form, interpreted in terms of gradient trajectories, was never written down until Witten's paper (1982). The miracle, then, is that the complex arises as a result of reasoning from physics, a completely circuitous route from the viewpoint of what was already "known" to Morse theory experts.
In Witten's paper, when discussing instanton corrections, the reference he puts down is Milnor's "Lectures on h-cobordism." This greatly surprised me and since I haven't read those lectures, I can't say how they might be related. Do you have any ideas on their relation?
ReplyDeleteAdmittedly, Witten's reference to Milnor seems to be misleading. It should more closely be associated with the answer one obtains from the instanton computations, not the idea of WKB approximations and instantons themselves.
DeleteI have clarified a little bit what I wrote, but here's an explanation of the relation. In Section 7 of "Lectures on the h-cobordism theorem," Milnor essentially describes the Morse-Smale-Witten complex in the special case that the Morse function $f$ is self-indexing, meaning that $f(p) = \mathrm{ind}(p)$ for $p$ a critical point. He phrases this a bit more algebraically, in terms of intersection numbers of attaching and coattaching spheres, but this is just a rephrasing of the "number of gradient trajectories"; in fact it is more general in that it doesn't require the Morse-Smale condition. The differential is defined as a connecting map of a long exact sequence (Corollary 7.3) and it is shown to square to zero (Theorem 7.4). One can always modify a Morse function to make it self-indexing, in fact without any creation or cancellation of critical points, so this is sufficient for most purposes (Theorem 4.8). Going from this case to the not-necessarily-self-indexing case is completely achievable from Milnor's exposition, but it's not trivial, and, perhaps just by my own ignorance, I've never heard anybody state this general case in terms of algebraic intersection numbers of attaching and co-attaching spheres (without requiring Morse-Smale) instead of counts of gradient trajectories.
Hope that helps!
I should add: Theorem 7.4 also proves invariance in the case of a self-indexing Morse function by showing that the Morse homology matches with the singular homology.
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