Morse theory, which Marston Morse himself referred to as “the calculus of variations in the large,” typically deals with the study of critical points on manifolds. The main idea is, very roughly, that as one travels along a manifold by means of a so-called Morse function, a certain associated homology tags the critical points in an elegant and revealing way. Furthermore, this situation can be exploited to yield the famous Morse inequalities, which have profound topological and geometrical significance. For instance, if we define the k-th Morse number of a Morse function on a given manifold to be the number of critical points of Morse index k for that function, then the manifold’s Euler-Poincaré characteristic equals the alternating sum of these Morse numbers (cf. p. 48 of the book under review). Here, the Morse index of a critical point is just the number of negative eigenvalues of the Hessian of the given Morse function at that point. This is by anyone’s standard a very beautiful result.
This fundamental homological nature of things Morse, so to speak, can also be gleaned from a quick consideration of what Raoul Bott called “the ‘prime’ example all of us use when we explain Morse theory to the uninitiated” (cf. p. 105 of his “Morse Theory Indomitable,” Publ. Math. IHES, no. 68(1988), pp. 99–114), namely, the doughnut standing up on a table. In this setting the Morse function par excellence is just the height-off-the-table function, and one associates to each height (or level) the sublevel set consisting of all the points of the doughnut at or below the that height. As we move (vertically) through a critical point, e.g. the doughnut hole’s bottom point or top point, the sublevel set (as a submanifold) changes topologically, e.g. by gluing a handle onto the earlier chunk.
This is obviously a job for homology, and this critically important perspective is brought out appropriately early in Nicolaescu’s An Invitation to Morse Theory, in the first section of his second chapter, titled, “Surgery, handle attachment, and cobordism.” Of course, Milnor, in his 1973 standard, Morse Theory (Ann. Math. Studies no. 51, Princeton University Press), starts off his book (which is to say, his lectures) already on p. 1 with this homological perspective, via a detailed treatment of “the prime example,” while in Nicolaescu’s book we have to wait until p. 49 to find our first doughnut. However, Nicolaescu makes excellent use of the preceding forty-eight pages by presenting a thorough discussion of what it takes to be a Morse function in general (chapter 1) and by covering (in the first part of chapter 2) a fairly large amount of the prerequisite topology and geometry. This is proper, of course, for an introductory text.
Then the remainder of chapter 2 of An Invitation to Morse Theory swiftly deals with connections with dynamics, Bott functions, Floer homology, and min-max theory. Accordingly, in less than ninety pages, Nicolaescu is ready for applications, among which I wish to single out that in § 3.3, “Symplectic manifolds and Hamilton flows.” This inclusion of a consideration of flows from a Morse theoretic perspective is apt indeed, seeing that it provides something of a point of departure for studying some hugely important hyper-modern themes characterized by applications of hard analysis to geometry: Perelman’s conquest of the Poincaré conjecture comes to mind right away, of course. And this feature is rather unusual in an introductory text.
Nicolaescu adds two more chapters to An Invitation to Morse Theory, the first dealing with complex Morse theory, the second being a collection of thirty eight problems followed by solutions for many of them. It is worth noting the chapter on complex Morse theory includes coverage of the Hard Lefschetz Theorem and the Picard-Lefschetz Formula, conveying the considerable depth Nicolaescu achieves in what, at only around 230 pages, is really a rather short book.
This brevity fits in with Nicolaescu’s warning, issued at the close of his Preface, that “[p]enetrating the inherently eclectic subject of Morse theory requires quite a varied background. The… book is addressed to a reader familiar with the basics of algebraic topology (fundamental group, singular (co)homology, Poincaré duality …) and the basics of differential geometry (vector fields and their flows, Lie and exterior derivative, integration on manifolds, basics of Lie groups and Riemannian geometry…).”
Taking this proviso into account, then, that the reader had better have a pretty strong topology and geometry background already, An Invitation to Morse Theory is a fine book from which to start learning Morse theory. I cannot resist, however, recommending that the reader also keep a copy of Milnor’s Morse Theory close at hand, for a complementary perspective and for a look at other topics.
Michael Berg is Professor of Mathematics at Loyola Marymount University in California.