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 remedy of the type theorem for compact surfaces is both too formalized and intricate for these with out specified historical past wisdom, or too casual to find the money for scholars a finished perception into the topic. Its devoted, student-centred procedure info a near-complete evidence of this theorem, largely in demand for its efficacy and formal good looks. 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 basic team, Orientability -- Homology teams -- The category Theorem for Compact Surfaces. The type Theorem: casual Presentation -- Surfaces -- Simplices, Complexes, and Triangulations -- the elemental crew -- Homology teams -- The category Theorem for Compact Surfaces
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Additional resources for A guide to the classification theorem for compact surfaces
1. For every plane closed path, W Œ0; 1 ! A2 , for every z0 not on , the index n. ; z0 / is continuous on the complement of in A2 , and in fact, constant in each connected component of the complement of . We have n. ; z0 / D 0 in the unbounded component of the complement of . Proof. t/ z00 j > for all t. We can then use the same subdivision to define n. ; z0 / and n. ; z00 /. vi / with new argument Âi0 is connected to Âi by Âi0 D Âi C ˛i C1 2 < ˛i < 2. The ˛i C k 2 ; P P and the resulting estimate jkj < 1 implies k D 0.
1 46 4 The Fundamental Group, Orientability 1 and each arc k ık kC1 is homotopic either to ˛, or ˛ 1 , or 1. Thus, ˛ m , for some integer m 2 Z. It remains to prove that ˛ m is not homotopic to 1 for m 6D 0. 2. 1; 0/ D 0, and thus ˛ m and 1 are not homotopic when m 6D 0. But then, we have shown that the homotopy classes are in bijection with the set of integers. t u The above proof also applies to a circular annulus, closed or open, and to a circle. In particular, the circle is not simply connected.
The following proposition shows that topologically closed, bounded, convex sets in An are equivalent to closed balls. We will need this proposition in dealing with triangulations. 2. If C is any nonempty bounded and convex open set in An , for any point a 2 C , any ray emanating from a intersects @C D C C in exactly one point. Furthermore, there is a homeomorphism of C onto the (closed) unit ball B n , which maps @C onto the n-sphere S n 1 . 4. Remark. It is useful to note that the second part of the proposition proves that if C is a bounded convex open subset of An , then any homeomorphism gW S n 1 !