It has been more than two decades since Raoul Bott and I published Differential Forms in Algebraic Topology. While this book has enjoyed a certain success, it does assume some familiarity with manifolds and so is not so readily accessible to the average first-year graduate student in mathematics. It has been my goal for quite some time to bridge this gap by writing an elementary introduction to manifolds assuming only one semester of abstract algebra and a year of real analysis. Moreover, given the tremendous interaction in the last twenty years between geometry and topology on the one hand and physics on the other, my intended audience includes not only budding mathematicians and advanced undergraduates, but also physicists who want a solid foundation in geometry and topology.
With so many excellent books on manifolds on the market, any author who un- dertakes to write another owes to the public, if not to himself, a good rationale. First and foremost is my desire to write a readable but rigorous introduction that gets the reader quickly up to speed, to the point where for example he or she can compute de Rham cohomology of simple spaces.
A second consideration stems from the self-imposed absence of point-set topology in the prerequisites. Most books laboring under the same constraint define a manifold as a subset of a Euclidean space. This has the disadvantage of making quotient manifolds, of which a projective space is a prime example, difficult to understand. My solution is to make the first four chapters of the book independent of point-set topology and to place the necessary point-set topology in an appendix. While reading the first four chapters, the student should at the same time study Appendix A to acquire the point-set topology that will be assumed starting in Chapter 5.
The book is meant to be read and studied by a novice. It is not meant to be encyclopedic. Therefore, I discuss only the irreducible minimum of manifold theory which I think every mathematician should know. I hope that the modesty of the scope allows the central ideas to emerge more clearly. In several years of teaching, I have generally been able to cover the entire book in one semester.