By Jean H Gallier; Dianna Xu

This welcome boon for college kids of algebraic topology cuts a much-needed important direction among different texts whose remedy of the category theorem for compact surfaces is both too formalized and intricate for these with no distinctive heritage wisdom, or too casual to have the funds for scholars a finished perception into the topic. Its devoted, student-centred strategy information a near-complete facts of this theorem, extensively in demand for its efficacy and formal attractiveness. The authors current the technical instruments had to installation the tactic successfully in addition to demonstrating their use in a basically based, labored instance. learn more... The category Theorem: casual Presentation -- Surfaces -- Simplices, Complexes, and Triangulations -- the elemental staff, Orientability -- Homology teams -- The category Theorem for Compact Surfaces. The category Theorem: casual Presentation -- Surfaces -- Simplices, Complexes, and Triangulations -- the basic crew -- Homology teams -- The class Theorem for Compact Surfaces

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**Example text**

4. V; S / and of a map W S ! s2 /, for all s1 ; s2 2 S . s /, for every s Â s. s/ cover M . s/. K; / is a triangulation of M , we also refer to the map W S ! 2M as a triangulation of M and we also say that K is a triangulation W S ! 2M of M . K; / of a surface M , the geometric realization Kg of K is homeomorphic to the surface M , as shown by the following proposition. 5. Given any triangulation W S ! 2M of a surface M , there is a homeomorphism hW Kg ! s/. Proof. x/. If s is a 1-simplex, we define h on sg using any of the homeomorphisms whose existence is asserted by (C1).

Since an m-manifold, M , has an open cover of sets homeomorphic with open sets of Rm , an m-manifold is locally arcwise connected and locally compact. 1, the connected components of an m-manifold are arcwise connected and, in particular, a surface is arcwise connected. An open subset, U , on a surface, S , is called a Jordan region if its closure, U , can be mapped homeomorphically onto a closed disk of R2 in such a way that U is mapped onto the open disk, and thus, that the boundary of U is mapped homeomorphically onto the circle, the boundary of the open disk.

1). If 1 2 and 2 3 then we have some homotopies F1 W Œ0; 1 Œ0; 1 ! E between 1 and 2 , and F2 W Œ0; 1 Œ0; 1 ! E between 2 and 3 . If we define GW Œ0; 1 Œ0; 1 ! t; 0/; and one can easily check that G is a homotopy between 1 and 3. 40 4 The Fundamental Group, Orientability A simple example of path homotopy is given by reparameterizations. A continuous nondecreasing function W Œ0; 1 ! 1/ D 1 is called a reparameterization. Then, given a path W Œ0; 1 ! E, the path ı W Œ0; 1 ! E is homotopic to W Œ0; 1 !