By William S. Massey

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William S. Massey Professor Massey, born in Illinois in 1920, obtained 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 battle II. After the warfare he obtained his Ph.D. from Princeton collage and spent extra years there as a post-doctoral examine assistant. He then taught for ten years at the school of Brown collage, and moved to his current place at Yale in 1960. he's the writer of various examine articles on algebraic topology and comparable themes. This e-book built from lecture notes of classes taught to Yale undergraduate and graduate scholars over a interval of numerous years.

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**Additional info for Algebraic Topology: An Introduction**

**Example text**

It is understood, of course, that the ends of the strips do not overlap on the boundary of the disc. 36 illustrate the procedure. We now assert that models of all compact bordered surfaces can be constructed in this way. The relatively simple proof is as follows. 33 Two-Dimensional Manifolds A triangulated strip. 33. We identify the edges a and b with two edges of the triangulation of the boundary of M, and count vertices, edges, and triangles before and after the identiﬁcation. Now, we shall show how to construct any compact, orientable bordered surface whose boundary has 15 components, 15 g 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. Condition (2) can be demonstrated as follows. The fact that the set of all the triangles with v as a vertex can be divided into several disjoint subsets, such that the triangles in each subset can be arranged in cyclic order as described, is an easy consequence of condition (1).

We can repeatedly attach such "crossed" pairs of strips and reduce the Euler characteristic by any even integer. 34) to obtain an orientable surface of Euler characteristic 2 — 15 whose boundary has 15 components. 36 we reduce the Euler characteristic by one, keep the number of boundary components the same, and have a nonorientable bordered surface. By attaching more such half-twisted strips, we can reduce the Euler characteristic by any desired amount. From these indications, it should be clear how to construct a model of any compact bordered surface by this method.