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 warfare 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 school of Brown collage, and moved to his current place at Yale in 1960. he's the writer of various study articles on algebraic topology and similar themes. 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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**Example text**

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). However, if there were more than one such subset, then the requirement that 22 have a neighborhood homeomorphic to U2 would be violated. We shall not attempt a rigorous proof of this last assertion. 1 Let S be a compact surface. 1 by prov- ing that S is homeomorphic to a polygon with the edges identiﬁed in pairs as indicated by one of the symbols listed at the end of Section 5.

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. 34 for k = 4. An orientable bordered surface of Euler characteristic 2 — 15 whose boundary has 15 components results. Note that the Euler characteristic is the maximum possible for the given number of boundary components.

We can state this in another way: If we start with a compact surface M * and construct a bordered surface by removing the interiors of k closed discs, which are pairwise disjoint, then the location of the discs that are to be removed does not matter. The resulting manifold with boundary will be topologically the same no matter how the position of the discs is chosen. 1 Let M1 and M2 be compact bordered surfaces; assume that their boundaries have the same number of components. Then, M1 and M 2 are homeomorphic if and only if the surfaces M1" and M;* (obtained by gluing a disc to each boundary component) are homeomorphic.