By Jean H Gallier; Dianna Xu

This welcome boon for college students of algebraic topology cuts a much-needed primary course among different texts whose therapy of the class theorem for compact surfaces is both too formalized and complicated for these with out designated historical past wisdom, or too casual to come up with the money for scholars a entire perception into the topic. Its devoted, student-centred procedure information a near-complete evidence of this theorem, extensively well-liked 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 sincerely based, labored instance. learn more... The type Theorem: casual Presentation -- Surfaces -- Simplices, Complexes, and Triangulations -- the basic team, Orientability -- Homology teams -- The type Theorem for Compact Surfaces. The category Theorem: casual Presentation -- Surfaces -- Simplices, Complexes, and Triangulations -- the elemental crew -- Homology teams -- The category Theorem for Compact Surfaces

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**Additional info for A guide to the classification theorem for compact surfaces**

**Example text**

Given any two paths 1 W Œ0; 1 ! E and 2 W Œ0; 1 ! 0/, the concatenation 1 2 of 1 and 2 is the path given by ( . 2t 1/ 1 2 Ä t Ä 1: of a path W Œ0; 1 ! 1 t/; 0 Ä t Ä 1: 0 0 0 0 It is easily verified that if 1 1 and 2 2 , then 1 2 1 2 , and that 0 1 1 1 ; see Massey [6] or Munkres [8]. Thus, it makes sense to define the composition and the inverse of homotopy classes. 4. E; a/, at the base point a is the set of homotopy classes of closed paths, W Œ0; 1 ! 1/ D a, under the multiplication operation, Œ 1 Œ 2 D Œ 1 2 , induced by the composition of closed paths based at a.

E/. E; a/. n However, we won’t have any use for the more general homotopy groups. E; a/, is reduced to the trivial group, f1g, consisting of the identity element. E; a/, are trivial for all a 2 E. This is an important case, which motivates the following definition. 5. E; a/ is the trivial one-element group. For example, the plane and the sphere are simply connected, but the torus is not simply connected (due to its hole). We now show that a continuous map between topological spaces (with base points) induces a homomorphism of fundamental groups.

C; d /. The geometric realization is a tetrahedron. 7 shows a triangulation of a surface called a torus. c; a/. 8 shows a triangulation of a surface called the projective plane. c; d /. This time, the gluing requires a “twist”, since the paired edges have opposite orientation. Visualizing this surface in A3 is actually nontrivial. 9 shows a triangulation of a surface called the Klein bottle. c; a/. Again, some of the gluing requires a “twist”, since 34 3 Simplices, Complexes, and Triangulations Fig.