By William S. Massey

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William S. Massey Professor Massey, born in Illinois in 1920, got his bachelor's measure from the college of Chicago after which served for 4 years within the U.S. army in the course of global conflict II. After the conflict he acquired his Ph.D. from Princeton collage and spent extra years there as a post-doctoral learn 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 diverse examine articles on algebraic topology and similar issues. This e-book built from lecture notes of classes taught to Yale undergraduate and graduate scholars over a interval of a number of years.

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**Sample text**

No other triangulation of any surface can have this property. 13 A triangulation of the projective plane. 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.

Conversely, assume that M is a compact, bordered surface and that the boundary has 19 components 19 =_>_ 1. , a circle. It is clear that we obtain a compact surface M * if we take 19 closed discs and glue the boundary of the ith disc to the ith component of the boundary of M. The topological type of the resulting surface M * obviously depends only on the topological type of M. What is not so obvious is that a sort of converse statement is true: The topological type of the bordered surface M depends only on the number of its boundary components and the topological type of the surface M * obtained by gluing a disc onto each boundary component.

In succession. 4 of Appendix A [see application (a) of this lemma]. We now use these facts to prove that D is a disc as follows. T; and T; are topologically equivalent to discs. Therefore, the quotient space of T; U T; obtained by identifying points of ¢_l(62) is again a disc by (a). Form a quotient space of this disc and T; by making the identiﬁcations corresponding to the edge 63, etc. It is clear that S is obtained from D by identifying certain paired edges on the boundary of D. 15 shows an easily visualized example.