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 international warfare II. After the struggle 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 school 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 themes. This ebook 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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**Example text**

We have now obtained a polygon D whose edges have to be identiﬁed in pairs to obtain the given surface S. 16, the identiﬁcations are described by aa‘lfbb“lf—‘e‘lgcc—lg—ldd—le. If the letter designating a certain pair of edges occurs with both exponents, +1 and —1, in the symbol, then we will call that pair of edges a pair of the ﬁrst kind; otherwise, the pair is of the second kind. 16, all seven pairs are of the ﬁrst kind. We wish to show that an adjacent pair of edges of the ﬁrst kind can be eliminated, provided there are at least four edges in all.

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, . - and Ti+1 have an edge in common for 0 g 2' < n — 1. The truth of (1) follows from the fact that each point on the edge in question must have an Open neighborhood homeomorphic to the Open disc U2. If an edge were an edge of only one triangle or more than two triangles, this would not be possible. The rigorous proof of this last assertion would take us rather far aﬁeld; however, its plausibility cannot be diSputed.

CkBkcfl. (b) Normal form for the connected sum of n tori with k holes. 29(a) and (b) show how to proceed when n = 2 and k = 4. It is entirely analogous to the case of a sphere with holes cut in it. The result is a polygon with 4n + 3k sides, which must be identiﬁed in accordance with the following symbol: — — alblallbll — — — anbnanlbnlclBlcl1 ckBkck—l . (c) Normal form for the connected sum of n projective planes with k holes. We leave it to the reader to see that in this case we obtain a polygon with 2n + 3k sides, which are identiﬁed by the symbol a1a1 .