Differential TopologyDifferential Topology provides an elementary and intuitive introduction to the study of smooth manifolds. In the years since its first publication, Guillemin and Pollack's book has become a standard text on the subject. It is a jewel of mathematical exposition, judiciously picking exactly the right mixture of detail and generality to display the richness within. The text is mostly self-contained, requiring only undergraduate analysis and linear algebra. By relying on a unifying idea?transversality?the authors are able to avoid the use of big machinery or ad hoc techniques to establish the main results. In this way, they present intelligent treatments of important theorems, such as the Lefschetz fixed-point theorem, the Poincar‚?Hopf index theorem, and Stokes theorem. The book has a wealth of exercises of various types. Some are routine explorations of the main material. In others, the students are guided step-by-step through proofs of fundamental results, such as the Jordan-Brouwer separation theorem. An exercise section in Chapter 4 leads the student through a construction of de Rham cohomology and a proof of its homotopy invariance. The book is suitable for either an introductory graduate course or an advanced undergraduate course. |
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a₁ algebra arbitrary ball Chapter Check closed codim compact manifold coordinate corollary critical points curve defined definition deg f denote derivative diffeomorphism differential dimension dimensional dx₁ Euclidean space Euler characteristic Exercise exists f₁ Figure finite function f g₁ global HINT homotopic identity immersion implies integral intersection number intersection theory Inverse Function Theorem isomorphism k-dimensional manifold Lefschetz fixed point Lefschetz number Lemma Let f linear map local diffeomorphism manifold with boundary map f map g matrix measure zero Morse function neighborhood open set open subset ordered basis oriented manifolds p-form parametrization point of f positively oriented preimage Proof Prove regular value Sard's theorem Section Show smooth function smooth map submanifold submersion subspace Suppose that f T₂(X tangent space tion topology transversal U₁ v₁ vector field vector space w₁ x₁