By Jean H Gallier; Dianna Xu

This welcome boon for college students of algebraic topology cuts a much-needed vital direction among different texts whose remedy of the type theorem for compact surfaces is both too formalized and complicated for these with no distinct heritage wisdom, or too casual to find the money for scholars a complete perception into the topic. Its committed, student-centred process information a near-complete evidence of this theorem, greatly trendy for its efficacy and formal attractiveness. The authors current the technical instruments had to set up the strategy successfully in addition to demonstrating their use in a sincerely dependent, labored instance. learn more... The type Theorem: casual Presentation -- Surfaces -- Simplices, Complexes, and Triangulations -- the elemental workforce, Orientability -- Homology teams -- The type Theorem for Compact Surfaces. The category Theorem: casual Presentation -- Surfaces -- Simplices, Complexes, and Triangulations -- the elemental staff -- Homology teams -- The type Theorem for Compact Surfaces

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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).

H. Gallier, Geometric methods and Applications For Computer Science and Engineering, TAM vol. 38, 2nd edn. (Springer, New York, 2011) 3. R. Munkres, Elements of Algebraic Topology, 1st edn. (Addison-Wesley, Redwood City, 1984) 4. J. Rotman, Introduction to Algebraic Topology, GTM No. 119, 1st edn. 1 The Fundamental Group If we want to somehow classify surfaces, we have to deal with the issue of deciding when we consider two surfaces to be equivalent. It seems reasonable to treat homeomorphic surfaces as equivalent, but this leads to the problem of deciding when two surfaces are not homeomorphic, which is a very difficult problem.

3 The Fundamental Group of the Punctured Plane First, we note that the fundamental group of An is the trivial group. Indeed, consider any closed path W Œ0; 1 ! 1/, take a as base point and as the origin in An , and let a be the constant closed path reduced to a. t; u/ 7! An ; a/ D f1g. The above reasoning also shows that the fundamental group of an open ball or a closed ball is trivial. However, the next proposition shows that the fundamental group of the punctured plane is the infinite cyclic group Z.