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 collage of Chicago after which served for 4 years within the U.S. army in the course of global struggle II. After the struggle he obtained 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 various learn articles on algebraic topology and comparable issues. This publication 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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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 .

1. Fourth step. How to make any pair of edges of the second kind adjacent. We wish to show that our surface can be transformed so that any pair of edges of the second kind are adjacent to each other. 19(a). Cut along the dotted line labeled a and paste together along b. 19(b), the two edges are now adjacent. Continue this process until all pairs of edges of the second kind are adjacent. If there are no pairs of the ﬁrst kind, we are ﬁnished, because the symbol of the polygon must then be of the form alalazag .

Use the fact that x(S') = 2. 2 For any triangulation of a compact surface, show that 3t 28 e = 3(v — x) v g %(7 + V49 — 24x). In the case of the sphere, projective plane, and torus, what are the minimum values of the numbers v, e, and t? 3 In how many pieces do n great circles, no three of which pass through a common point, dissect a sphere? 4 (a) The sides of a regular octagon are identiﬁed in pairs in such a way as to obtain a compact surface. Prove that the Euler characteristic of this surface is g —2.