By Fred H. Croom

ISBN-10: 0387902880

ISBN-13: 9780387902883

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Extra resources for Basic Concepts of Algebraic Topology

Sample text

Let K be a geometric complex with two orientations, and let K l , K2 denote the resulting oriented geometric complexes. Then the homology groups Hp(Kl) and HiK2) are isomorphic for each dimension p. PROOF. For a p-simplex aP of K, let jaP denote the positive orientation of aP in the complex Kb i = 1,2. Then there is a function a defined on the simplexes of K such that a( aP) is ± I and laP = a(aP)2aP. laf represents a p-chain on K l . K = a(aP)g a(aP-l)ga(aP-1)a(aP)[2aP, 2aP-l]. 2aP-l L [2 aP, 2aP-l].

As mentioned earlier, he did not define the homology groups. The proof of the EulerPoincare Theorem given in the text is essentially Poincare's original one. Complexes (in slightly different form) and incidence numbers were defined in Complement a l'Analysis Situs [50] in 1899. The Betti numbers were named for Enrico Betti (1823-1892) and generalize the connectivity numbers that he used in studying curves and surfaces. Poincare assumed, but did not prove, that the Betti numbers are topological invariants.

Definition. An n-pseudomanifold is a complex K with the following properties: (a) Each simplex of K is a face of some n-simplex of K. (b) Each (n - 1)-simplex is a face of exactly two n-simplexes of K. (c) Given a pair (]~ and (]~ of n-simplexes of K, there is a sequence of nsimplexes beginning with (]~ and ending with (]~ such that any two successive terms of the sequence have a common (n - 1)-face. 4. 5) is a 2-pseudomanifold and is a triangulation of the 2-sphere S2. 2 is a 2-pseudomanifold.