By Fred H. Croom

This article is meant as a one semester creation to algebraic topology on the undergraduate and starting graduate degrees. primarily, it covers simplicial homology conception, the basic team, masking areas, the better homotopy teams and introductory singular homology conception. The textual content follows a large historic define and makes use of the proofs of the discoverers of the $64000 theorems while this is often in line with the easy point of the path. this system of presentation is meant to lessen the summary nature of algebraic topology to a degree that's palatable for the start scholar and to supply motivation and team spirit which are usually missing in abstact remedies. The textual content emphasizes the geometric method of algebraic topology and makes an attempt to teach the significance of topological ideas via utilising 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 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 no longer spend time in the beginning engaged on the appendices, yet fairly 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 by way of university juniors of ordinary intelligence and doesn't require "mathematical adulthood" past the junior point.

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**Example text**

The pth Betti number indicates the number of "p-dimensional holes" in the polyhedron IKI. Definition. A rectilinear polyhedron in Euclidean 3-space 1R3 is a solid bounded by properly joined convex polygons. The bounding polygons are called faces, the intersections of the faces are called edges, and the intersections of the edges are called vertices. A simple polyhedron is a rectilinear polyhedron whose boundary is homeomorphic to the 2-sphere S2. A regular polyhedron is a rectilinear polyhedron whose faces are regular plane polygons and whose polyhedral angles are congruent.

Then H is continuous and H(x, 1) = ao = g(x), H(x,O) = I(x), XE IKI. This example illustrates one method by which homotopies will be defined in later applications. 5. Let both K and L be the I-skeleton of the closure of a 2-simplex Then the polyhedra IKI and ILl are both homeomorphic to the unit circle 81, so we may consider any function from IKI to ILl as a function from Sl to itself. For our function f, let us choose a rotation through a given angle a. Then, referring 8 1 to polar coordinates, f: S 1 --+ S 1 is defined by a 2• f(1,8) = (1,8 + a), (1, 8) E S1, 0 :::;; 8 :::;; 217.

Ap) represents a positively or negatively oriented p-simplex, then g.