Download e-book for iPad: Algebraic Topology: An Introduction by William S. Massey

By William S. Massey

ISBN-10: 0387902716

ISBN-13: 9780387902715

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William S. Massey Professor Massey, born in Illinois in 1920, obtained his bachelor's measure from the collage of Chicago after which served for 4 years within the U.S. military in the course of international conflict II. After the warfare he acquired his Ph.D. from Princeton college and spent extra years there as a post-doctoral study assistant. He then taught for ten years at the college of Brown college, and moved to his current place at Yale in 1960. he's the writer of various study articles on algebraic topology and similar issues. This e-book constructed from lecture notes of classes taught to Yale undergraduate and graduate scholars over a interval of a number of years.

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1 Let S be a compact surface. 1 by prov- ing that S is homeomorphic to a polygon with the edges identified in pairs as indicated by one of the symbols listed at the end of Section 5. First step. From the discussion in the preceding section, we may assume that S is triangulated. Denote the number of triangles by n. We assert that we can number the triangles T1, T2, . -_1, 2 g 2' g 11.. 1 / l9 angles T1; for T2 choose any triangle that has an edge in common with T1, for T3 choose any triangle that has an edge in common with T1 or T2, etc.

A noncompact manifold with boundary may or may not have a countable basis of open sets. In any 36 / CHAPTER ONE Two-Dimensional Manifolds case, it is always locally compact. We should note that the boundary of a connected manifold may be disconnected; also, the boundary of a noncompact manifold may be compact. The concepts of orientability and nonorientability apply to manifolds with boundary exactly as in the case of manifolds. For example, a Mobius strip is a nonorientable manifold with boundary, whereas the cylinder {(x,y,z)6R3=x2+y2=1,0§2§1l is an orientable manifold with boundary.

The sphere SP1 is the boundary, and the open disc U" is the interior. 2 Another example is the “half-space,” {x E R" : x1 g 0}. 3 The Mobius strip, as it is usually defined, is a 2-dimensional manifold with boundary. 4 Other examples of 2-dimensional manifolds with boundary may be obtained by removing a collection of small, open discs from a 2—dimensional manifold. It is quite plausible and can be proved rigorously that the set of boundary points and the set of interior points are mutually disjoint.

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Algebraic Topology: An Introduction by William S. Massey


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