# Download Basic Concepts of Algebraic Topology by Fred H. Croom PDF

By Fred H. Croom

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7 . 4 is a 2-manifold. The n-sphere sn, n ;;::: 1, is an n-manifold. Incidentally, this indicates why the unit sphere in ~"+ 1 is called the "n-sphere" and not the "(n + 1)-sphere". The integer n refers to the local dimension as a manifold and not to the dimension of the containing Euclidean space. Note that each point of a circle has a neighborhood homeomorphic to an open interval in ~; each point of S2 has a neighborhood homeomorphic to an open disk in ~2; and so on. 31 2 Simplicial Homology Groups The relation between manifold (a type of topological space) and pseudomanifold (a type of geometric complex) is simple to state: If X is a triangulable n-manifold, then each triangulation K of X is an n-pseudomanifold.

A regular polyhedron is a rectilinear polyhedron whose faces are regular plane polygons and whose polyhedral angles are congruent. In Exercise 6 at the end of the chapter, the reader will find that the Betti numbers of the 2-sphere S2 are Ro(S2) = 1, R1 (S2) = 0, R2(S2) = 1. Then S2 has Euler characteristic X(S2) = L: (_1)PRp(S2) = 1 p=o 2 0 + 1 = 2. 6 (Euler's Theorem). If S is a simple polyhedron with V vertices, E edges, and F faces, then V - E + F = 2. Things are complicated slightly here by the fact that the faces of S need not be triangular.

Hence any elementary O-chain g. (b) is homologous to one of the O-chains g. (a') or -g. (a'). It follows that any O-chain on K' is homologous to an elementary O-chain h· (a') where h is some integer. Applying this result to each combinatorial component K 1, ... , Kr of K, there is a vertex 01 of K j such that any O-cycle on K j is homologous to a O-chain of the form h j • (01) where h j is an integer. Then, given any O-cycle Co on K, there are integers hi> ... , hr such that Lh r Co '" Suppose that two such O-chains homology class.