As is the case with all *exposés *published in the *Astérisque* series by the SMF (*Société Mathématique de France*), this volume, no. 399, is aimed at presenting serious contemporary mathematical themes to a suitable audience. The present monograph is aimed at specialists in low-dimensional topology as it is informed and influenced by (hyper)modern physics. It deals largely with research results, specifically work by one of the book’s authors, Ciprian Manolescu, concerning a particular kind of Floer homology.

Floer homology was developed by the late Andreas Floer in the 1980s, and has become one of the mainstays of low-dimensional topology. Says Simon Donaldson:

The concept of Floer homology is one of the most striking developments in differential geometry over the past 20 [now going on 40] years. ... The ideas have led to great advances in the areas of low-dimensional topology and symplectic geometry and are intimately related to developments in Quantum Field Theory … the full richness of Floer’s theory is only beginning to be explored (cf. https://en.wikipedia.org/wiki/Andreas_Floer).

There are some telltale phrases in this remark by Donaldson, e.g. symplectic geometry and QFT, and it is perhaps apposite to recall that symplectic structure is part and parcel of canonical quantization or geometric quantization, and indeed if we look upon QFT from a Hamiltonian perspective, this is precisely what drives the train. It is one of the marvels of modern mathematics that these two areas, low-dimensional topology and quantum field theory, should conspire to such a beautiful synergy.

It is also worth emphasizing that Floer homology can be characterized as a very sophisticated flavor of Morse theory. There’s a wonderful, if compact and for-members-only, on-line lecture by Peter Kronheimer in which the speaker describes (a) Floer hoology as the Morse homology of a chain complex generated by the critical points of a (perturbed) Chern-Simons action or functional, in which the boundary map counts instantons. Digging a little deeper (i.e. watching Kronheimer’s talk more carefully), what’s going on is this. One introduces a Riemannian structure to a 3-manifold, and thereby obtains an \(L^2\)-structure on the tangent space to the space of all connections on the manifold. Then, parameterizing the connections, one obtains a formal gradient flow involving the according Chern-Simons functional(s) and gets a differential equation (involving the Hodge star of the curvature of the parameterized connections) whose solutions are the aforementioned instantons. Some care has to be taken here, because the earlier 1-parameter family needs to be regarded as a single connection on an associated bundle in 4-space, and one subsequently exploits the ensuing fact that the given differential equation can be interpreted as dealing with anti-self dual Yang-Mills equations — hence the instantons.

That this is in fact a high-octane variant of Morse theory can be gleaned from the marvelous description given by the late Raoul Bott in his beautiful article “Morse Theory Indomitable,” in which Morse theory is presented as a natural theme in Steven Smale’s work on dynamical systems.

Well, this is just the tip of the iceberg as far as the required background for this book is concerned. It is beautiful mathematics, very much alive, very much in the mainstream, but it is not for beginners — by no means, in fact. Lidman and Manolescu address two specific variants of Floer homologies, both being qualified by the prefix “Seiberg-Witten,” and the main thrust here is that, now, the Chern-Simons functional is in fact the Chern-Simons-Dirac functional, the corresponding \(L^2\)-gradient takes on a more elaborate form, and the critical points of the Chern-Simons-Dirac functional become solutions to the Seiberg-Witten equations (cf. p. 32 of the book). The purpose of this monograph is to show that regarding this “Seiberg-Witten Floer homology,” if we’re dealing with “rational homology spheres, the definitions given by Kronheimer-Mrowka … and by the second author [Manolescu] are equivalent.” As I said, beautiful mathematics, but not for the beginner.

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