The basic objects of differential topology are manifolds, introduced by Riemann (as "multiply-extended quantities'') to generalize surfaces to many dimensions. The appeal of manifolds is the richness of available structures that follow from the definition. A manifold M is a topological space that is nice (Hausdorff and second countable) and such that for each point x in M there is a neighborhood Ux that is homeomorphic to an open subset of a fixed Euclidean space Rm. Such neighborhoods may overlap and this gives rise to coordinate transformations. Restricting the class of coordinate transformations determines some of the particular properties of the manifold, for example, if the transformations are complex analytic (a complex manifold), or have Jacobians of positive determinant (an orientable manifold). Furthermore, the neighborhoods may be taken as parameter spaces for other geometric data, glued together with the coordinate transformations into fibre bundles.
In a locally Euclidean space, we can do calculus, and so manifolds admit differentiable functions, vector fields, a tangent space, all intrinsically defined via the coordinate transformations. The main point of differential topology is to sort out the consequences of all this structure, and the interrelations between its various aspects. The problem with a book about manifolds is that the basic definitions are so many, and you need them all to study their interactions with one another. An example of note is the Poincaré-Hopf Index Theorem, which states that the index of a vector field with isolated singular points on a compact, oriented manifold without boundary is equal to the Euler characteristic of the manifold — an analytically determined integer is equal to a topological invariant.
The book of Barden and Thomas is based on courses taught at the University of Cambridge. The direction is topological, leaving the geometric (Riemannian metrics) for another course. With this decisive turn, the authors can strike deep into relevant structure with purpose. After the basic structure is set out, they treat fibre bundles with the Submersion Theorem of Ehresman as goal, illustrating how manifolds with geometry might be classified by the local geometries. The next few chapters treat topological equipment — the exterior derivative and de Rham cohomology, which feature the relations between the calculus and the topology on a manifold. The next chapter gives the reader some real substance for all the structure. Degrees, indices, the Gauss map, and Morse theory employ the available structure in a dramatic way to give back geometric data. A rich source of examples is considered next with a thumbnail introduction to Lie groups. There is a final chapter that provides the background in analysis and algebra to support the course.
This reviewer had several pleasant conversations with Charles Thomas about this book, and about manifolds in general, especially at the time when the solution to the Poincaré conjecture was announced. I am saddened by the death of Professor Thomas. I am glad that we have a record of his notes and the course or Professor Barden; they are rich with insight into how to think about manifolds, introducing students to a subject that is central to mathematics and to our framework for understanding physics. Short and insightful, this is a good way to get started.
John McCleary is Professor of Mathematics at Vassar College.