By Fred H. Croom
This article is meant as a one semester advent to algebraic topology on the undergraduate and starting graduate degrees. essentially, it covers simplicial homology conception, the basic team, overlaying areas, the better homotopy teams and introductory singular homology conception. The textual content follows a huge historic define and makes use of the proofs of the discoverers of the $64000 theorems while this can be in keeping with the straightforward point of the path. this system of presentation is meant to minimize the summary nature of algebraic topology to a degree that's palatable for the start scholar and to supply motivation and harmony which are frequently missing in abstact remedies. The textual content emphasizes the geometric method of algebraic topology and makes an attempt to teach the significance of topological recommendations through using them to difficulties of geometry and research. the must haves for this direction are calculus on the sophomore point, a one semester advent to the speculation of teams, a one semester introduc- tion to point-set topology and a few familiarity with vector areas. Outlines of the prerequisite fabric are available within the appendices on the finish of the textual content. it is recommended that the reader no longer spend time at first engaged on the appendices, yet fairly that he learn from the start of the textual content, bearing on the appendices as his reminiscence wishes clean. The textual content is designed to be used by way of collage juniors of standard intelligence and doesn't require "mathematical adulthood" past the junior point.
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Extra resources for Basic Concepts of Algebraic Topology
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  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.
Basic Concepts of Algebraic Topology by Fred H. Croom