Differential Topology"This book is written for mathematics students who have had one year of analysis and one semester of linear algebra. Included in the analysis background should be familiarity with basic topological concepts in Euclidean space: openness, connectedness, compactness, etc. We borrow two theorems from analysis which some readers may not have studied: the inverse function theorem, which is used throughout the text; and the change of variable formula for multiple integration, which is needed only for Chapter 4"--Page xiii. |
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a₁ arbitrary ball Check closed codim codimension compact manifold corollary critical point curve D₁ defined definition deg f denote derivative diffeo diffeomorphism dimension dimensional dx₁ Euclidean space Euler characteristic Exercise exists f₁ Figure finite g₁ 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 nondegenerate 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 topological transversal Tx(X U₁ v₁ vector field vector space w₁ x₁ дх