By Jean H Gallier; Dianna Xu

ISBN-10: 3642343643

ISBN-13: 9783642343643

This welcome boon for college students of algebraic topology cuts a much-needed critical course among different texts whose therapy of the category theorem for compact surfaces is both too formalized and intricate for these with no unique history wisdom, or too casual to find the money for scholars a entire perception into the topic. Its committed, student-centred procedure information a near-complete evidence of this theorem, commonly prominent for its efficacy and formal attractiveness. The authors current the technical instruments had to installation the strategy successfully in addition to demonstrating their use in a basically dependent, labored instance. learn more... The type Theorem: casual Presentation -- Surfaces -- Simplices, Complexes, and Triangulations -- the elemental staff, Orientability -- Homology teams -- The category Theorem for Compact Surfaces. The type Theorem: casual Presentation -- Surfaces -- Simplices, Complexes, and Triangulations -- the elemental team -- Homology teams -- The class Theorem for Compact Surfaces

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4. Thus, M is the disjoint union of two sets @M and Int M , where @M is the subset consisting of all points p 2 M that are mapped by some (in fact, all) coordinate map ' defined on p into @Hm , and where Int M D M @M . The set @M is called the boundary of M , and the set Int M is called the interior of M , even though this terminology clashes with some prior topological definitions. A good example of a surface with boundary is the M¨obius strip. The boundary of the M¨obius strip is a circle. The boundary @M of M may be empty but Int M is nonempty.

I , is called a coordinate map and its inverse, 'i 1 W ˝i ! Ui , is called a parametrization of Ui . U; '/, with 'W U ! ˝; ' 1 / is a parametrization of M at p. Ui ; 'i /i 2I , is often called an atlas for M . A (topological) surface is a connected 2-manifold. Remarks. 1. The terminology is not universally agreed upon. For example, some authors (including Fulton [4]) call the maps 'i 1 W ˝i ! Ui charts! Always check the 24 2 Surfaces direction of the homeomorphisms involved in the definition of a manifold (from M to Rm or the other way around).

I /i 2I , of homeomorphisms, 'i W Ui ! ˝i , where each ˝i is some open subset of Rm . Ui ; 'i /, is called a coordinate system or chart (or local chart) of M , each homeomorphism, 'i W Ui ! ˝i , is called a coordinate map and its inverse, 'i 1 W ˝i ! Ui , is called a parametrization of Ui . U; '/, with 'W U ! ˝; ' 1 / is a parametrization of M at p. Ui ; 'i /i 2I , is often called an atlas for M . A (topological) surface is a connected 2-manifold. Remarks. 1. The terminology is not universally agreed upon.

### A guide to the classification theorem for compact surfaces by Jean H Gallier; Dianna Xu

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