## Differential TopologyThis book presents some of the basic topological ideas used in studying differentiable manifolds and maps. Mathematical prerequisites have been kept to a minimum; the standard course in analysis and general topology is adequate preparation. An appendix briefly summarizes some of the back ground material. In order to emphasize the geometrical and intuitive aspects of differen tial topology, I have avoided the use of algebraic topology, except in a few isolated places that can easily be skipped. For the same reason I make no use of differential forms or tensors. In my view, advanced algebraic techniques like homology theory are better understood after one has seen several examples of how the raw material of geometry and analysis is distilled down to numerical invariants, such as those developed in this book: the degree of a map, the Euler number of a vector bundle, the genus of a surface, the cobordism class of a manifold, and so forth. With these as motivating examples, the use of homology and homotopy theory in topology should seem quite natural. There are hundreds of exercises, ranging in difficulty from the routine to the unsolved. While these provide examples and further developments of the theory, they are only rarely relied on in the proofs of theorems. |

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### Contents

Introduction | 1 |

Manifolds and Maps | 7 |

Differentiable Maps and the Tangent Bundle | 15 |

Copyright | |

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algebraic algebraic topology analytic apply approximation assume atlas boundary called Chapter charts circle closed compact complete components connected consider construct contains continuous coordinate covering critical points curves defined definition denote dense diffeomorphism differential differential structure dimension disjoint disks easy embedding equivalence Euler example Exercise exists extends fact Figure finite follows genus give given global Hence homeomorphism homotopy identity immersion implies induced integer invariant isomorphism isotopy Lemma linear locally manifold means measure metric morphism Morse function n-manifold natural neat nonorientable normal obtained open set orientation orthogonal pair positive preserves projection Proof prove regular value result reverses sequence submanifold suffices Suppose surface tangent Theorem Theory transverse trivial true tubular neighborhood union unique vector bundle vector field vector space zero