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Differential Manifolds
Antoni A. Kosinski
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eBook - ePub
Differential Manifolds
Antoni A. Kosinski
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The concepts of differential topology form the center of many mathematical disciplines such as differential geometry and Lie group theory. Differential Manifolds presents to advanced undergraduates and graduate students the systematic study of the topological structure of smooth manifolds. Author Antoni A. Kosinski, Professor Emeritus of Mathematics at Rutgers University, offers an accessible approach to both the h-cobordism theorem and the classification of differential structures on spheres.
`How useful it is,` noted the Bulletin of the American Mathematical Society, `to have a single, short, well-written book on differential topology.` This volume begins with a detailed, self-contained review of the foundations of differential topology that requires only a minimal knowledge of elementary algebraic topology. Subsequent chapters explain the technique of joining manifolds along submanifolds, the handle presentation theorem, and the proof of the h-cobordism theorem based on these constructions. There follows a chapter on the Pontriagin Construction—the principal link between differential topology and homotopy theory. The final chapter introduces the method of surgery and applies it to the classification of smooth structures of spheres. The text is supplemented by numerous interesting historical notes and contains a new appendix, `The Work of Grigory Perelman,` by John W. Morgan, which discusses the most recent developments in differential topology.
`How useful it is,` noted the Bulletin of the American Mathematical Society, `to have a single, short, well-written book on differential topology.` This volume begins with a detailed, self-contained review of the foundations of differential topology that requires only a minimal knowledge of elementary algebraic topology. Subsequent chapters explain the technique of joining manifolds along submanifolds, the handle presentation theorem, and the proof of the h-cobordism theorem based on these constructions. There follows a chapter on the Pontriagin Construction—the principal link between differential topology and homotopy theory. The final chapter introduces the method of surgery and applies it to the classification of smooth structures of spheres. The text is supplemented by numerous interesting historical notes and contains a new appendix, `The Work of Grigory Perelman,` by John W. Morgan, which discusses the most recent developments in differential topology.
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Informations
Sujet
MathematicsSous-sujet
Differential GeometryVI
Operations on Manifolds
In this chapter we describe various operations on manifolds: connected sum, attachment of handles, and surgery. All of these are special cases of a general construction, joining of two manifolds along a submanifold, presented in Sections 4 and 5. However, since all important features are already present in the special cases of connected sum and connected sum along the boundary, we discuss these two cases first in Sections 1 and 3, respectively.
The general construction is specialized to attaching of handles in Section 6. We are particularly interested in the question when the attachment of two handles of consecutive dimensions results in no change to the manifold, that is when the second handle âdestroysâ the first. The main result in this direction, SmaleĂšs Cancellation Lemma, is proved in Section 7. The proof is based on an elementary but far-reaching theorem concerning attachment of disc bundles along a cross section in the boundary.
In Section 8 we look at handle attachment from a different point of view, more convenient for homology calculations. Section 9 introduces the operation of surgery, and in Section 10 we calculate some related homological results. In Section 11 we define handlebodies and investigate their structure. Some important examples are constructed in Section 12 using the plumbing construction. The results of the last two sections will not be used until Chapter VIII.
1 Connected Sum
Connected sum is the operation of âjoining two manifolds by a tube.â
Given two connected m-dimensional manifolds M1, M2, let hi,: Rm â Mi, i = 1, 2, be two imbeddings. If both manifolds are oriented, then we assume that h1 preserves the orientation and h2 reverses it.
Let α : (0, â) â (0, â) be an arbitrary orientation reversing diffeomorphism. We define αm: Rm â 0 â Rm â 0 by
The connected sum M1 # M2(h1, h2, α) is the space obtained from the (disjoint) union of M1 â h1(0) and M2 â h2(0) by identifying h1(Ï
) with h2(αM(Ï
)) ...