Publication: Topología diferencial : Variedades con borde. Transversalidad. Aproximación
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Addison-Wesley Iberoamericana España
This book provides an introduction to differential topology, mainly to the approximation and transversality theories, and shows how these techniques can be applied to obtain several classical results, such as Brouwer's fixed point theorem, the Jordan-Brouwer theorem asserting that any compact hypersurface without boundary of the affine space disconnects it, the Hopf theorem concerning the classification by homotopy of continuous maps into the sphere, or Brouwer's theorem on the existence of nowhere vanishing vector fields on spheres. The book consists of four chapters and a list of more than fifty exercises that help the reader to become familiar with the concepts presented throughout the text. Each chapter starts with a brief introduction motivating the concepts and results that are developed in it. Chapter 1 introduces basic concepts such as manifolds with boundary, partition of unity, tangent bundle, product of manifolds and the orientation induced on the boundary of an oriented manifold. Results on the existence of diffeotopies and about immersions and submersions are also included. The concept of transversality is introduced in Chapter 2; the motivation is the description of manifolds as inverse images. Before introducing the definition of transversality it is proved that a hypersurface has a global equation if and only if it disconnects the ambient space. Then the transversality of a map and a manifold, of manifolds and of maps is defined and after proving the Brown-Sard theorem concerning the density of regular values of maps, the parametrized theorem of density of the transversality is considered. The final section of the chapter deals with the Whitney embedding theorem. In Chapter 3 the space Cr(X,Y) of Cr -maps between two given manifolds X and Y with boundary is endowed with its strong topology, which ensures that any two sufficiently close maps are homotopic. Also, basic results relating approximation, homotopy and transversality are stated: any continuous map can be approximated by differentiable maps, any continuous map is homotopic to differentiable maps which are transversal to a given manifold, and the fact that continuous homotopy between differentiable maps guarantees differentiable homotopy; proper homotopy is also studied. In Chapter 4 the results of the preceding chapters are combined in order to prove the theorems of Brouwer, Hopf and Jordan-Brouwer mentioned at the beginning. The book is clearly written and self-contained. Mathematical prerequisites reduce to general topology and analysis, so that the book would make an excellent text for an undergraduate course in differential topology.
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