Symplectic geometry is a branch of both differential geometry and differential topology. It is very different from the Riemannian geometry with which we are more familiar as it is an even-dimensional geometry that measures the signed area of complex curves rather than length or distance as in Riemannian geometry. Symplectic geometry studies concepts that are initially expressed in the smooth category such as in terms of differential forms, but do not involve derivatives; therefore one may talk about symplectic topology. The field is significantly inspired by the interaction with mathematical physics, low-dimension topology, dynamical systems, quantization, equivariant cohomology and more.

Symplectic geometry has been around as long as classical mechanics has, but the subject did not really become what it is today until the mid-eighties when Gromov introduced the method of using pseudo-holomorphic curves in the study of global symplectic geometry. “Virtual fundamental chain and cycle” determined by the moduli spaces of pseudo-holomorphic curves is a technique to resolve some of the restrictive assumptions of these moduli spaces. This technique is also believed to be needed for the development of symplectic geometry without restrictive topological or geometric assumptions on the symplectic manifold.

The book *Virtual Fundamental Cycles in Symplectic Topology* can be considered extended lecture notes on pseudo-holomorphic curves and it helps deepen understanding of the different approaches to the basics of the virtual fundamental chain and cycle theory in symplectic geometry. The book consists of three main chapters: Notes on Kuranishi Atlases by McDuff, Gromov-Whitten Theory via Kuranishi Structures by Tehrani and Fukaya and Kuranishi Spaces as a 2-category by Joyce. All three chapters create a fundamental cycle and chain associated with the moduli space of pseudoholomorphic maps of curves to a symplectic manifold in different ways. Here the symplectic manifold has compatible almost complex structure.

The first chapter entitled “Notes on Kuranishi Atlases” covers the concept of the virtual fundamental cycle for the moduli space of pseudoholomorpic maps. In this chapter, mostly definitions and remarks are given and they are supported with well-explained examples. On the other hand, the proofs of the theorems and propositions presented in this chapter are not detailed. How to construct a virtual fundamental class for Gromov-Written moduli spaces of closed curves by using finite-dimensional reductions is also discussed in this chapter.

The second chapter entitled “Gromov-Witten Theory via Kuranishi Structures” contains detailed information about the construction of a virtual fundamental class on a space with a Kuranishi structure. In this chapter, the details of the abstract theory of Kuranishi structure are given. Also the construction of a Kuranishi structure on the moduli space of pseudo holomorphic topologies is discussed. The moduli spaces of pseudo holomorphic maps are subjects in symplectic geometry and string theory, particularly in the study of Gromov-Witten invariants. The author supports his discussion with clear remarks and examples.

The third chapter is a survey of Kuranishi spaces and symplectic geometry. In this chapter, the author introduces an alternate definition of Kuranishi space and, with this definition, discuses different uses of Kuranishi atlases.

In my opinion, this book is an advanced level book and will be of benefit to the experts in the area as well as graduate students. The content will be seen as difficult by new learners. Adding an index to the book might be helpful to the reader. However, at the same time the book is interesting in three ways. First, it covers relatively new material in the area and expands symplectic topology in a unique way, while assisting with a better understanding of the basics of the theory of virtual fundamental chain and cycle. Second, it contains many well-explained relevant examples. Finally, all three chapters complement each other yet can be studied individually.

Filiz Dogru, Ph.D is Professor of Mathematics Department at Grand Valley State University, Allendale MI. E-mail:

dogruf@gvsu.edu