By William S. Massey

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William S. Massey Professor Massey, born in Illinois in 1920, acquired 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 conflict II. After the warfare he bought 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 collage, and moved to his current place at Yale in 1960. he's the writer of diverse examine articles on algebraic topology and comparable themes. This e-book 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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29 Orientable surface of genus 2 with four holes. surface, every edge is an edge of exactly two triangles. However, if a bordered surface is triangulated, some edges will be edges of only one triangle. Such edges will be contained in the boundary. It is a theorem, which we will assume without proof, that every compact bordered surface may be triangulated (for a proof, see Ahlfors and Sario [1], Chapter I, Section 8). Let M be a compact bordered surface, with a given triangulation. 30 Two-Dimensional Manifolds Barycentric subdivision of a triangle.

We can regard a triangulated surface as having been constructed by gluing together the various triangles in a certain way, much as we put together a jigsaw puzzle or build a wall of bricks. Because two different triangles cannot have the same vertices we can specify completely a triangulation of a surface by numbering the vertices, and then listing which triples of vertices are vertices of a triangle. Such a list of triangles completely determines the surface together with the given triangulation up to homeomorphism.

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. (b) Prove that any surface (orientable or nonorientable) of Euler characteristic g -—2 can be obtained by suitably identifying in pairs the sides of a regular octagon. , it is a hexagon) and such that distinct regions have no more than one side in common. 6 Let SI be a surface that is the sum of m tori, m g 1, and let S, be a surface that is the sum of n projective planes, n _2_ 1.