Differential Topology: First StepsKeeping mathematical prerequisites to a minimum, this undergraduate-level text stimulates students' intuitive understanding of topology while avoiding the more difficult subtleties and technicalities. Its focus is the method of spherical modifications and the study of critical points of functions on manifolds. No previous knowledge of topology is necessary for this text, which offers introductory material regarding open and closed sets and continuous maps in the first chapter. Succeeding chapters discuss the notions of differentiable manifolds and maps and explore one of the central topics of differential topology, the theory of critical points of functions on a differentiable manifold. Additional topics include an investigation of level manifolds corresponding to a given function and the concept of spherical modifications. The text concludes with applications of previously discussed material to the classification problem of surfaces and guidance, along with suggestions for further reading and study. |
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2-sphere B₁ C₁ circle cobounding compact differentiable manifold connected sum construct containing coordinate neighborhoods coordinate system corresponding critical level cross-cap curves defined Definition 3-1 described diffeomorphic differentiable function differentiable manifold differentiable map differential equations differential topology directly embedded disjoint En-r Er+1 Euclidean coordinates Euclidean space example finite number function ƒ given hemisphere holes homeomorphic hyperplanes identified intersection Jacobian determinant last exercise Lemma Let f let ƒ M₁ manifold with boundary map f modification of type n-space noncritical level nondegenerate critical point nonzero Note obtained one-to-one open cell open set orientable 2-manifold orthogonal trajectories pair point of f projective plane proof result sequence of modifications set of points shrinking spherical modification submanifold subset subspace Suppose surface tangent line Theorem topological product topological space torus trace transformation tubular neighborhood U₁ union Xn+1 y₁ zero