By Fred H. Croom
This article is meant as a one semester advent to algebraic topology on the undergraduate and starting graduate degrees. primarily, it covers simplicial homology concept, the elemental crew, overlaying areas, the better homotopy teams and introductory singular homology conception. The textual content follows a extensive ancient define and makes use of the proofs of the discoverers of the real theorems whilst this is often in step with the uncomplicated 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 pupil and to supply motivation and unity 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 recommendations by way of employing them to difficulties of geometry and research. the necessities for this direction are calculus on the sophomore point, a one semester creation 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 before everything engaged on the appendices, yet really that he learn from the start of the textual content, concerning the appendices as his reminiscence wishes clean. The textual content is designed to be used by way of collage juniors of ordinary intelligence and doesn't require "mathematical adulthood" past the junior point.
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Extra info for Basic Concepts of Algebraic Topology
F(ost(p» c ost(g(p». To see that this vertex map g is simplicial, suppose that Vo, ... , Vn are vertices of a simplex in K. According to the lemma, this is equivalent to saying that the intersection nf=o ost(VI) is not empty. Hence o =I f (Q ost(VI») c Qf( ost(Vi)) c Q ost(g(VI», so nf=o ost(g(vl» is not empty. The lemma thus insures that g(vo), ... , g(v n ) are vertices of a simplex in L. Then g is a simplicial vertex map and has an extension to a simplicial map g: IKI -+ ILl. Let x E IKI and let a be the simplex of K of smallest dimension which contains x.
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. 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.
PP_Io(g·aP) = O. p collapses aP. pp}O' is a chain mapping. 2 was given under the assumption that VI < ... < vp. Why is it sufficient to consider only this orientation? aP and its faces af have orientations induced by the ordering Vo < Definition. P be a simplicial mapping from the vertices of K into the vertices of L. p: IKI-+ ILl as follows: If x E IKI, there is a simplex ar =
Basic Concepts of Algebraic Topology by Fred H. Croom