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 thought, the basic crew, protecting areas, the better homotopy teams and introductory singular homology concept. The textual content follows a huge ancient define and makes use of the proofs of the discoverers of the real theorems while this is often in line with the user-friendly 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 pupil and to supply motivation and solidarity 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 via employing them to difficulties of geometry and research. the must haves for this path are calculus on the sophomore point, a one semester advent to the idea 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 strongly recommended that the reader now not spend time in the beginning engaged on the appendices, yet quite that he learn from the start of the textual content, pertaining to the appendices as his reminiscence wishes clean. The textual content is designed to be used through collage juniors of ordinary intelligence and doesn't require "mathematical adulthood" past the junior point.

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**Example text**

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.