In the 1980s, gauge theory methods in low-dimensional topology gave rise to a powerful offspring: Floer homology. Floer took up a suggestion of Witten to consider Morse theoretical arguments in supersymmetric quantum field theory, and in doing so, created a tool which has since yielded penetrating insights and diverse applications. This came on the heels of Donaldson’s use of the Yang-Mills equations to produce a 4-manifold invariant. Floer’s instanton homology provided a fine topological invariant for 3-manifolds, which fractionated Donaldson’s invariants in much the same way ordinary homology may be said to fractionate the Euler characteristic. Their foundations lie in a thorough accounting of the analytic details of the moduli space of solutions to the Yang-Mills equations.
The Morse-theoretic approach is indeed versatile, and has led to several distinct versions of Floer homology, beginning with another, symplectic version due to Floer. This version is especially developed in the modern Lagrangian Intersection Floer theory. In low-dimensional topology, Oszvath and Szabo’s Heegard Floer homology has recently provided a more computable, combinatorial invariant. And closest to home, just as Donaldson’s 4-manifold invariants were revolutionized in the 1990’s by the Seiberg-Witten equations, so may the Floer homology of 3-manifolds be redefined using these equations, instead of the Yang-Mills equations. This is the modern approach most in line with the focus on the fundamental analysis, and is the subject of the book under review.
The goal of Kronheimer and Mrowka is to provide a complete foundation, with detailed proofs, for the Seiberg-Witten, or “monopole” Floer homology, something which has incidentally never been fully achieved for the instanton Floer theory. The Floer groups are here defined for any compact, connected, oriented 3-manifold, which is notable in a theory known for requiring special conditions to achieve applicability.
It may strike the uninitiated topologist that in this Floer theory, everything familiar is different. Morse theory is circle-valued. Homology is graded by a geometrically defined set not isomorphic to the integers. It is a testament to the authors’ adroit motivation of the arguments that these technicalities meld seamlessly into the landscape. The claim on the overleaf of “assuming only a basic grounding in differential geometry and analysis” is certainly a brazen bit of advertising (presumably by an editor), and is a minor discredit to what the authors in fact achieve. As an 800-page book on an intricate and difficult subject, it is admirably focused and coherent, with the final effect that the book seems small in comparison with what it contains.
The principal benefit conferred by considering the Seiberg-Witten equations, compactness, is explained at the start, in Chapter II. Also developed early is the complicating and requisite perturbation method. The first half of the book overall is devoted to detailed study of the moduli space qua Hilbert manifold, and defining the monopole Floer homology as Morse theory of the Chern-Simons-Dirac functional. After arriving at the definition on page 443, a long chapter follows dedicated to cobordism/topological invariance, and then perturbations are developed further, exploring the implications of the circle-valued Morse theory.
An important question for any book is: when do the examples and applications begin? The answer here, “in due time,” reflects one of the great strengths of the book: a long and thorough introduction. Thus by the time we are quoted the Floer homology groups of something other than the 3-sphere on page 78, we have seen a quite complete sketch of the topological and gauge theoretical background, and the structure of the arguments to come has been rendered clear. For the reader lacking context, it may help to read the notes and references section at the end of each chapter first.
This last point is important. While, for instance, familiarity with Morse theory is not assumed, this book will be markedly clearer to readers already having experience with gauge theory, Hilbert manifolds, and slice theorems. As an exposition of the analytic story behind Floer homology, it is gentle, precise, and evidently complete. It should be of interest to any mathematician faced with an infinite-dimensional moduli space of some sort. It does not pretend to teach the prerequisites, and for graduate students, this is a committing field of study. The book also does not place emphasis on remaining challenges or open problems, particularly within this gauge theoretic perspective. But for the reader who knows that this is where they want to go, this book will serve as an adept guide.
Noah Kieserman teaches at Colby College in Waterville, ME.