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

By Jean H Gallier; Dianna Xu

This welcome boon for college students of algebraic topology cuts a much-needed primary direction among different texts whose remedy of the type theorem for compact surfaces is both too formalized and intricate for these with no specified historical past wisdom, or too casual to come up with the money for scholars a finished perception into the topic. Its committed, student-centred procedure info a near-complete evidence of this theorem, largely well-known for its efficacy and formal good looks. The authors current the technical instruments had to set up the strategy successfully in addition to demonstrating their use in a sincerely established, labored instance. learn more... The category Theorem: casual Presentation -- Surfaces -- Simplices, Complexes, and Triangulations -- the basic staff, Orientability -- Homology teams -- The type Theorem for Compact Surfaces. The class Theorem: casual Presentation -- Surfaces -- Simplices, Complexes, and Triangulations -- the elemental team -- Homology teams -- The class Theorem for Compact Surfaces

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**Example text**

1 Simplices and Complexes As explained in Sect. 2, every surface can be triangulated. This is a key ingredient in the proof of the classification theorem. Informally, a triangulation is a collection of triangles satisfying certain adjacency conditions. To give a rigorous definition of a triangulation it is helpful to define the notion of a simplex and of a simplicial complex. It does no harm to define these notions in any dimension. We assume some familiarity with affine spaces. If not, the reader should consult Munkres [3] (Chap.

Ahlfors, L. Sario, Riemann Surfaces, Princeton Math. Series, No. 2. (Princeton University Press, Princeton, 1960) 2. A. Amstrong, Basic Topology, UTM, 1st edn. (Springer, New York, 1983) 3. P. do Carmo, Differential Geometry of Curves and Surfaces (Prentice Hall, New Jersey, 1976) 4. W. Fulton, Algebraic Topology, A first course, GTM vol. 153, 1st edn. (Springer, New York, 1995) 5. W. Hirsch, Differential Topology, GTM vol. 33, 1st edn. (Springer, New York, 1976) 6. C. Kinsey, Topology of Surfaces, UTM, 1st edn.

Indeed, the definition does not assume that a surface is a subspace of any given ambient space, say Rn , for some n. Perhaps, such surfaces should be called “abstract surfaces”. In fact, it can be shown that every surface is a smooth 2-manifold and that every smooth 2-manifold can be embedded in R4 (see Hirsch [5], Sect. 3). This is somewhat annoying since R4 is hard to visualize! Fortunately, all orientable surfaces can be embedded in R3 (see do Carmo [3]). Unfortunately, as we mentioned in Sect.