Differential Topology

Victor Guillemin and Alan Pollack offer an elementary, intuitive introduction to smooth manifolds that has become a standard text in the field. Largely self-contained, it asks only for undergraduate analysis and linear algebra. Transversality serves as the unifying idea, letting the authors reach the main results without heavy machinery or ad hoc techniques, among them Stokes theorem, the Lefschetz fixed-point theorem and the Poincare-Hopf index theorem. Its many exercises range from routine practice to step-by-step guided proofs of fundamental results, the Jordan-Brouwer separation theorem among them, and one exercise section builds de Rham cohomology and proves its homotopy invariance. It can serve advanced undergraduates or a first graduate course.

It is one of the books on the shelves Eric Weinstein posted on Twitter, titles that mattered enough to him to bring home.