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Morse Theory and Floer Homology

Michèle Audin and Mihai Damian
Publication Date: 
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[Reviewed by
Michael Berg
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Marston Morse and Andreas Floer: two major names in the discipline formed by intersecting geometric topology, algebraic topology, and dynamics. Morse was the architect of Morse theory, of course, a subject whose main thrust is the effective analysis of geometric singularities (very broadly understood) by topological and dynamical means, with so-called Morse functions doing the heavy lifting. On the other hand, Floer is characterized in a blurb on the back-cover of the present book as the designer of “an infinite dimensional analogue of Morse homology.” Morse homology is attached to a Morse complex, which is in turn attached to a complex manifold equipped with a Morse function. Floer homology is attached to an infinite dimensional manifold equipped with a so-called action functional that sets a whole new game in motion, the prize for the winner being a proof of the Arnol’d conjecture.

On pp. 154–155 of the present work we encounter a marvelously useful flow-chart (if the insiders among you will pardon the pun) outlining what comes about in Floer’s scheme:

(1) the critical points of [the action] functional will be exactly the desired periodic solutions [or trajectories in the sense of the Arnol’d conjecture — see below]; (2) … with these critical points we … construct a complex …(3)… [t]o define the differential [see (7)] of this complex we will use … the gradient (or its negative) of the functional … (4) … then [we] “count” its trajectories … (5) … show that the trajectories of finite energy … connect two critical points [having defined the notion of a trajectory’s energy]; (6) … [use] Gromov compactness … and (7) “[use] another compactness property to define the differential ∂” … (8) [we check] that these spaces of trajectories are manifolds … (9) … verify that ∂◦∂ = 0 … (10) … show that the homology of the complex … depends neither on the choice of the functional nor on that of the vector field … (11) [and finally verify] that the homology we have … constructed, the Floer homology, coincides with the Morse homology of the manifold…

And now the denouement: “The obvious inequality between the dimensions of the vector spaces involved in the complex will give the expected result in a manner that is completely analogous to the Morse inequalities.”

Said expected result is of course the substance of the present case of Arnol’d’s conjecture, to the effect that the sum of the dimensions of the Z2-Morse homology vector spaces on a symplectic manifold W (i.e. we take homology with Z2 coefficients) is a lower bound for the number of 1-periodic trajectories of any Hamiltonian vector field on W. The authors of the book under review point out (p. 127) that “the strategy of the proof, due to Floer, consists in presenting the trajectories in question as critical points — no longer of a function on a manifold but of an ‘action functional’ [Hadamard’s nomenclature for a function on an infinite dimensional manifold] on the space of loops on W (a periodic trajectory is in fact a loop on W)” — see the above flow-chart.

Thus, Morse Theory and Floer Homology is a relatively high-level introduction to, and in fact a full account of, the extremely elegant and properly celebrated solution to the Arnol’d problem by the prodigious and tragic Andreas Floer (a victim in 1991, by suicide, to devastating depression). Floer’s approach built on work by Mikhail Gromov and Ed Witten going back to the 1980s, with the solution of the Arnol’d conjecture coming in 1987. Despite the obvious depth and austerity of this material the authors take pains to make their book self-contained — to wit, they include three appendices under the title, “What you need to know to read this book.” And this turns out to be “a bit of differential geometry,” “a bit of algebraic topology,” and “a bit of analysis.” Fair enough: this is not a misrepresentation of the prerequisites for this book, even as I would certainly add that the serious reader had better be genuinely mathematically mature (with the usual quasi-ineffable meaning: we all know what it means, but it’s hard to define, like the elephant). So the strong graduate student might crack the book, but he’d better be serious.

On the other hand, Audin and Damian do note (p. viii) that their book “result[ed] from a course given to graduate students … for whom we first needed to ‘recall’ what a manifold is,” so even a relative rookie might possibly take his chances. In any case, there are a decent number of exercises to be had and, with the first part (of two) properly devoted to Morse theory (and preceded by an appropriate paean to Milnor’s unsurpassed Morse Theory) and the second part to the Arnol’d conjecture and Morse theory, the book is exceptionally well written. Indeed, this is a very good book on a beautiful and important subject and will richly repay those who take the time to work through it.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.