AN INTRODUCTION to GROMOV's H-PRINCIPLE
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An Introduction to Differential Topology and Surgery Theory
An introduction to differential topology and surgery theory Anthony Conway December 8, 2018 Abstract This course decomposes in two parts. The first introduces some basic differential topology (mani- folds, tangent spaces, immersions, vector bundles) and discusses why it is possible to turn a sphere inside out. The second part is an introduction to surgery theory. Contents 1 Differential topology2 1.1 Smooth manifolds and their tangent spaces......................2 1.1.1 Smooth manifolds................................2 1.1.2 Tangent spaces..................................4 1.1.3 Immersions and embeddings...........................6 1.2 Vector bundles and homotopy groups..........................8 1.2.1 Vector bundles: definitions and examples...................8 1.2.2 Some homotopy theory............................. 11 1.2.3 Vector bundles: the homotopy classification.................. 15 1.3 Turning the sphere inside out.............................. 17 2 Surgery theory 19 2.1 Surgery below the middle dimension.......................... 19 2.1.1 Surgery and its effect on homotopy groups................... 20 2.1.2 Motivating the definition of a normal map................... 21 2.1.3 The surgery step and surgery below the middle dimension.......... 24 2.2 The even-dimensional surgery obstruction....................... 26 2.2.1 The intersection and self intersection numbers of immersed spheres..... 27 2.2.2 Symmetric and quadratic forms......................... 31 2.2.3 Surgery on hyperbolic surgery kernels..................... 33 2.2.4 The even quadratic L-groups.......................... 35 2.2.5 The surgery obstruction in the even-dimensional case............ 37 2.3 An application of surgery theory to knot theory.................... 40 1 Chapter 1 Differential topology The goal of this chapter is to discuss some classical topics and results in differential topology. -
The Golden Age of Immersion Theory in Topology: 1959–1973
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 42, Number 2, Pages 163–180 S 0273-0979(05)01048-7 Article electronically published on January 25, 2005 THE GOLDEN AGE OF IMMERSION THEORY IN TOPOLOGY: 1959–1973. A MATHEMATICAL SURVEY FROM A HISTORICAL PERSPECTIVE DAVID SPRING Abstract. We review the history of modern immersion-theoretic topology during the period 1959–1973, beginning with the work of S. Smale followed by the important contributions from the Leningrad school of topology, including the work of M. Gromov. We discuss the development of the major geometrical ideas in immersion-theoretic topology during this period. Historical remarks are included and technical concepts are introduced informally. 1. Brief overview In this article1 I briefly review selected contributions to immersion-theoretic topology during the early “golden” period, from about 1959 to 1973, during which time the subject received its initial important developments from leading topolo- gists in many countries. Modern immersion-theoretic topology began with the work of Stephen Smale ([51]), ([52]) on the classification of immersions of the sphere Sn into Euclidean space Rq, n ≥ 2, q ≥ n + 1. During approximately the next 15 years the methods introduced by Smale were generalized in various ways to analyse and solve an astonishing variety of geometrical and topological problems. In particular, at Leningrad University during the late 1960s and early 1970s, M. Gromov and Y. Eliashberg developed geometrical methods for solving general partial differential relations in jet spaces where, informally, a partial differential relation of order r, r ≥ 1, is a set of equations (closed relations) or inequalities (open relations) on the partial derivatives of order ≤ r of a function. -
Introduction to the H-Principle
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