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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. military in the course of international warfare II. After the struggle he bought his Ph.D. from Princeton college and spent extra years there as a post-doctoral examine 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 learn articles on algebraic topology and similar issues. This ebook constructed from lecture notes of classes taught to Yale undergraduate and graduate scholars over a interval of a number of years.

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In these more general circumstances, the type of proof we have suggested is not possible. (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.

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;). We make these identi- fications for each of the edges 62, 63, . , en. Let D denote the resulting quotient space of T’. It is clear that the map go : T’ ——> S induces a map w of D onto S, and that S has the quotient topology induced by w (because D is compact and S is Hausdorff, up is a closed map).

In the second stage we then connect the boundaries of these two holes with a tube that is the remainder of the torus or Klein Bottle. The difference between the two cases depends on whether we connect the boundaries so they will have the same or opposite orientations. 23, where S is a Mobius strip. , the connected sum of a Mobius strip with a torus and a Klein Bottle, reSpectively) are homeomorphic. To see this, imagine that we out each of these t0pological Spaces along the lines AB. 23 along the line AB.

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