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By William S. Massey

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

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Because T’ is compact and S is a Hausdorff space, go is a closed map, and hence S has the quotient topology determined by go (see Section 1 of Appendix A). This is a rigorous mathematical statement of our intuitive idea that S is obtained by gluing the triangles T1, T2, . . together along the appropriate edges. The polygon we desire will be constructed as a quotient space of T’. Consider any of the edges 65, 2 g 2' _S_ 11.. - and one other triangle T,-, for which 1 g j < i. -) consists of an edge of the triangle T:- and an edge of the triangle T} We identify these two edges of the triangles T; and T;- by identifying points which map onto the same point of e,- (speaking intuitively, we glue together the triangles T:- and T;).

Pairs of the first kind. 21). We shall show that we can transform the polygon so that the four sides in question are consecutive around the perimeter of the polygon. 20 A pair of edges of the first kind. 21(b). Then, out along d and paste together along a to obtain (0), as desired. 21(c). 1. (c) 26 / CHAPTER ONE Two-Dimensional Manifolds second kind, this leads to the desired result because, in that case, the symbol must be of the form alblaI‘bflazbzaglbz‘l . . a;‘b;1 and the surface is the connected sum of n tori.

Bk denote the components of the boundary. , then T has exactly two edges which have one vertex in B,- but do not lie in 8;. Similarly, if e is an edge that has a vertex in B,- but does lie in B;, then e is an edge of two triangles, both of which meet 8;. It follows that the edges and triangles that meet B,- but do not lie in B,- can be arranged in one or more cycles of alternating edges and triangles, T1, 81, T2, 82, . , Tn, 8n, Tn+1 = T1, such that each e,- is an edge of T,- and Ti“, whereas each T,- has ej_1 and e,- as edges.

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