By Fred H. Croom

This article is meant as a one semester creation to algebraic topology on the undergraduate and starting graduate degrees. essentially, it covers simplicial homology concept, the elemental crew, masking areas, the better homotopy teams and introductory singular homology concept. The textual content follows a large old define and makes use of the proofs of the discoverers of the $64000 theorems whilst this is often in line with the basic 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 harmony 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 thoughts by way of making use of them to difficulties of geometry and research. the necessities for this path 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 strongly recommended that the reader now not spend time firstly engaged on the appendices, yet particularly that he learn from the start of the textual content, touching on the appendices as his reminiscence wishes fresh. 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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For p = 0, ... , n - I, let b~ = 8(d~+l). The set {b~} forms a basis for Bp. Let {z~}, i = I, ... , Rp, be a maximal set of p-cycles linearly independent mod Bp. These cycles span a subspace Gp of Zp, and O:$p:$n-l. Thus dimZp since Rp = = dim Gp + dim Bp = Rp + dim Bp dim Gp. Then I:$p:$n-l. 4 The Euler-Poincare Theorem Observe that Bp is spanned by the boundaries of elementary chains o(1·af+l) = 2: TJtlp)·af where (TJlip)) = TJ(p) is the pth incidence matrix. Thus dim Bp = rank TJ(p). Since the number of dJ + 1 is the same as the number of b~, then dim Dp+l = dim Bp = rank TJ(p), O:=:;p:=:;n-l.

8? 26. Show that the projective plane P is the quotient space of the 2-sphere obtained by identifying each pair x, -x of diametrically opposite points. 27. References [9] and [2] may be helpful for (b) and (c). (a) Define a I-dimensional complex Kin 1R3 for which IKI is not homeomorphic to a subspace of 1R2. (b) Prove that if K is a complex of dimension n, then IKI can be rectilinearly imbedded in 1R2n + 1. (c) Prove that every triangulation of an n-manifold is an n-pseudomanifold. 1 Introduction We turn now to the problem of comparing polyhedra by means of their associated homology groups.

3 Induced Homomorphisms on the Homology Groups polyhedron IKI induces the identity isomorphism i:: Hp(K) --+ Hp(K) in each dimension p. Then (hh- 1 )! h;l*: Hp(sn) --+ Hp(sn), (h-1h)! : Hism) --+ Hp(sm) are identity isomorphisms in each dimension, so h! is an isomorphism between Hp(sm) and Hp(sn). 9) reveals that this is impossible since m 1= n. Hence sm and sn are not homeomorphic when m 1= n. (b) Recall from point-set topology that sn is the one point compactification of ~n. Thus if ~m and ~n are homeomorphic, it must be true that their one point compactifications sm and sn are homeomorphic too.