By William S. Massey
William S. Massey Professor Massey, born in Illinois in 1920, bought his bachelor's measure from the college of Chicago after which served for 4 years within the U.S. military in the course of global struggle II. After the battle he bought 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 learn articles on algebraic topology and similar issues. This ebook built from lecture notes of classes taught to Yale undergraduate and graduate scholars over a interval of numerous years.
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Additional info for Algebraic Topology: An Introduction
C) We shall use the Euler characteristic to distinguish between compact surfaces. We shall achieve this purpose with complete rigor in a later chapter by the use of the fundamental group. 1 Let SI and 82 be compact surfaces. The Euler characteristics of SI and 82 and their connected sum, SI # 82, are related by the formula X031 #32) = X031) + X032) — 2PROOF: The proof is very simple; assume SI and 82 are triangulated. Form their connected sum by removing from each the interior of a triangle, and then identifying edges and vertices of the boundaries of the removed triangles.
Ary of a disc. 14 2 2 A triangulation of a torus. 3 1 18 / CHAPTER ONE Two-Dimensional Manifolds with the Opposite sides identiﬁed. There are 9 vertices, and the following 18 triangles: 124 245 235 356 361 146 457 578 658 689 649 479 187 128 289 239 379 137 We conclude our discussion of triangulations by noting that any triangulation of a compact surface satisﬁes the following two conditions: (1) Each edge is an edge of exactly two triangles. (2) Let 2) be a vertex of a triangulation. Then we may arrange the set of all triangles with v as a vertex in cyclic order, T0, T1, T2, .
It is clear that we can also work the process described above backwards; whenever there are three pairs of the second kind, we can replace them by one pair of the second kind and two pairs of the ﬁrst kind. 1 to any connected sum of which three or more of the summands are projective planes. 1, which may be preferable in some cases, results. 2 Any compact, orientable surface is homeomorphic to a sphere or a connected sum of tori. Any compact, nonorientable surface is homeomorphic to the connected sum of either a projective plane or Klein Bottle and a compact, orientable surface.
Algebraic Topology: An Introduction by William S. Massey