By William S. Massey

ISBN-10: 0387902716

ISBN-13: 9780387902715

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William S. Massey Professor Massey, born in Illinois in 1920, acquired 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 battle he bought his Ph.D. from Princeton college and spent extra years there as a post-doctoral study 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 study articles on algebraic topology and similar themes. This ebook built from lecture notes of classes taught to Yale undergraduate and graduate scholars over a interval of numerous years.

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**Example text**

If we remove a small Open 3-dimensional disc from the interior of P2 X I, then we obtain a mani- fold with boundary such that the boundary has three components: P2 X {0}, P2 X {1}, and a 2-sphere which is the boundary of the removed disc. Thus, two of the boundary components are nonorientable, and one of them is orientable. 1 Prove that the product of a manifold and a manifold with boundary is a manifold with boundary. What is the boundary of the product? 2 Let P be a polygon. Assume that certain paired edges of P are identiﬁed, but not all edges of P are included among these pairs.

The reader should note that the terms “interior” and “boundary” were used in the preceding paragraphs in a sense different from that which is usual in point set topology. However, this will seldom lead to any confusion. Examples show that a manifold with boundary may be compact or noncompact, connected or not connected. A noncompact manifold with boundary may or may not have a countable basis of open sets. In any 36 / CHAPTER ONE Two-Dimensional Manifolds case, it is always locally compact. We should note that the boundary of a connected manifold may be disconnected; also, the boundary of a noncompact manifold may be compact.

It remains to treat the case in which there are pairs of both the ﬁrst and second kind at this stage. 1 The connected sum of a torus and a projective plane is homeomorphic to the connected sum of three projective planes. 3). Thus, we must prove that the connected sum of a projective plane and a torus is homeomorphic to the connected sum of a projective plane and a Klein Bottle. To do this it will be convenient to give an alternative construc- tion for a connected sum of any surface S with a torus or a Klein Bottle.

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