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《高等代数研究》课程教学资源（书籍文献）流形 Universitext，Editorial Board，S. Axler、K.A. Ribet，An Introduction to Manifolds

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Contents xiii 19 The Exterior Derivative . 189 19.1 Exterior Derivative on a Coordinate Chart . . . . . . . . . . . . . . . . . . . . . . 190 19.2 Local Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 19.3 Extension of a Local Form to a Global Form . . . . . . . . . . . . . . . . . . . . 191 19.4 Existence of an Exterior Differentiation . . . . . . . . . . . . . . . . . . . . . . . . 192 19.5 Uniqueness of Exterior Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . 192 19.6 The Restriction of a k-Form to a Submanifold . . . . . . . . . . . . . . . . . . . 193 19.7 A Nowhere-Vanishing 1-Form on the Circle . . . . . . . . . . . . . . . . . . . . 193 19.8 Exterior Differentiation Under a Pullback . . . . . . . . . . . . . . . . . . . . . . 195 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 Part VI Integration 20 Orientations. 201 20.1 Orientations on a Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 20.2 Orientations and n-Covectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 20.3 Orientations on a Manifold. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 20.4 Orientations and Atlases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 21 Manifolds with Boundary . 211 21.1 Invariance of Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 21.2 Manifolds with Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 21.3 The Boundary of a Manifold with Boundary . . . . . . . . . . . . . . . . . . . . 215 21.4 Tangent Vectors, Differential Forms, and Orientations . . . . . . . . . . . . 215 21.5 Boundary Orientation for Manifolds of Dimension Greater than One . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 21.6 Boundary Orientation for One-Dimensional Manifolds . . . . . . . . . . . 218 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 22 Integration on a Manifold . 221 22.1 The Riemann Integral of a Function on Rn . . . . . . . . . . . . . . . . . . . . . 221 22.2 Integrability Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 22.3 The Integral of an n-Form on Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 22.4 The Integral of a Differential Form on a Manifold . . . . . . . . . . . . . . . 225 22.5 Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 22.6 Line Integrals and Green’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Part VII De Rham Theory 23 De Rham Cohomology . 235 23.1 De Rham Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 23.2 Examples of de Rham Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 23.3 Diffeomorphism Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 23.4 The Ring Structure on de Rham Cohomology . . . . . . . . . . . . . . . . . . . 240 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

xiv Contents 24 The Long Exact Sequence in Cohomology . 243 24.1 Exact Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 24.2 Cohomology of Cochain Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . 245 24.3 The Connecting Homomorphism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 24.4 The Long Exact Sequence in Cohomology . . . . . . . . . . . . . . . . . . . . . 247 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 25 The Mayer–Vietoris Sequence . 249 25.1 The Mayer–Vietoris Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 25.2 The Cohomology of the Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 25.3 The Euler Characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 26 Homotopy Invariance . 257 26.1 Smooth Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 26.2 Homotopy Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 26.3 Deformation Retractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 26.4 The Homotopy Axiom for de Rham Cohomology . . . . . . . . . . . . . . . . 261 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 27 Computation of de Rham Cohomology . 263 27.1 Cohomology Vector Space of a Torus . . . . . . . . . . . . . . . . . . . . . . . . . . 263 27.2 The Cohomology Ring of a Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 27.3 The Cohomology of a Surface of Genus g . . . . . . . . . . . . . . . . . . . . . . 267 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 28 Proof of Homotopy Invariance . 273 28.1 Reduction to Two Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 28.2 Cochain Homotopies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 28.3 Differential Forms on M × R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 28.4 A Cochain Homotopy Between i ∗ 0 and i∗ 1 . . . . . . . . . . . . . . . . . . . . . . . 276 28.5 Verification of Cochain Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 Part VIII Appendices A Point-Set Topology . 281 A.1 Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 A.2 Subspace Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 A.3 Bases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 A.4 Second Countability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 A.5 Separation Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 A.6 The Product Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 A.7 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 A.8 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

A Brief Introduction Undergraduate calculus progresses from differentiation and integration of functions on the real line to functic s on the plane and in 3-space.Then on ters vector valued functior s and learns about inte extends and integral ca to R".This b extensio the calculus of curves and surfaces to higher dimensions The higher-dimensional analogues of smooth curves and surfaces are called man- ifolds.The constructions and theorems of vector calculus become simpler in the more general setting of manifolds:gradient.curl.and divergence are all special cases of the exterior derivative,and the fundamental theorem for line integrals,Green's theorem, Stokes'theorem,and the divergence theorem are different manifestations of a single general Stokes'theorem for manifolds. Higher-dimensional manifoldsrise even ifoneis interested ony in the three dime sional s ace which we inhabit.For ple,if we ca rotation followed by n an affine m then the nR3isasix dimensiona manifold.Moreover,this six mens sionl manifold is not Ro We consider two manifolds to be topologically the same if there is a homeo morphism between them,that is,a bijection that is continuous in both directions.A topological invariant of a manifold is a property such as compactness that remains unchanged under a homeomorphism.Another example is the number of connected components of a manifold.Interestingly,we can use differential and integral calculus alled the de Rhar f the manifold O n is as follo recast calculus on in a way uitable for alizat on to manifolds e do this by giving meani ng to the symbol and dz,so that they assume a life of their own,as differential forms,instead of being mere notations as in undergraduate calculus. While it is not logically necessary to develop differential forms on R"before the theory of manifolds-after all,the theory of differential forms on a manifold in Part V subsumes that onR,from a pedagogical point of view it is advantageous to treatR" separately first,since it isonthat the essential simplicity of differential forms and xterior differ ntiation mes most apparent

0 A Brief Introduction Undergraduate calculus progresses from differentiation and integration of functions on the real line to functions on the plane and in 3-space. Then one encounters vectorvalued functions and learns about integrals on curves and surfaces. Real analysis extends differential and integral calculus from R3 to Rn. This book is about the extension of the calculus of curves and surfaces to higher dimensions. The higher-dimensional analogues of smooth curves and surfaces are called manifolds. The constructions and theorems of vector calculus become simpler in the more general setting of manifolds; gradient, curl, and divergence are all special cases of the exterior derivative, and the fundamental theorem for line integrals, Green’s theorem, Stokes’ theorem, and the divergence theorem are different manifestations of a single general Stokes’ theorem for manifolds. Higher-dimensional manifolds arise even if one is interested only in the threedimensional space which we inhabit. For example, if we call a rotation followed by a translation an affine motion, then the set of all affine motions in R3 is a six-dimensional manifold. Moreover, this six-dimensional manifold is not R6. We consider two manifolds to be topologically the same if there is a homeomorphism between them, that is, a bijection that is continuous in both directions. A topological invariant of a manifold is a property such as compactness that remains unchanged under a homeomorphism. Another example is the number of connected components of a manifold. Interestingly, we can use differential and integral calculus on manifolds to study the topology of manifolds. We obtain a more refined invariant called the de Rham cohomology of the manifold. Our plan is as follows. First, we recast calculus on Rn in a way suitable for generalization to manifolds. We do this by giving meaning to the symbols dx, dy, and dz, so that they assume a life of their own, as differential forms, instead of being mere notations as in undergraduate calculus. While it is not logically necessary to develop differential forms on Rn before the theory of manifolds—after all, the theory of differential forms on a manifold in Part V subsumes that on Rn, from a pedagogical point of view it is advantageous to treat Rn separately first, since it is on Rn that the essential simplicity of differential forms and exterior differentiation becomes most apparent

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