By Kaczynski T., Mischaikow K., Mrozek M.

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Using this language we can view our edge mapping as a multivalued map ! F : ;2 2]! ;2 4] de ned by 8 ;1 4] if x = ;2 > > ;1 4] if x 2 (;2 ;1) > > > ;1 1] if x = ;1 > > < ;2 1] if x 2 (;1 0) F (x) := > ;2 0] if x = 0 > ;2 0] if x 2 (0 1) > > ;2 0] if x = 1 > > > 2 2] if x 2 (1 2) > :; ;2 2] if x = 2 There are three observations to be made at this point. e. the edges without its endpoints. Since we will used this idea later let us introduce some notation and a de nition. 19 Let e be and edge with endpoints v .

Obviously for such a broad set of issues a single chapter cannot do any of the topics justice. They are included solely for the purpose of motivating the formidable algebraic machinery we are about to start developing. This chapter is meant to be enjoyed in the sense of an entertaining story. Don't sweat the details - try to get a feeling for the big picture. We will return to these topics throughout the rest of this book. 1 Topology The importance in linear algebra of the dimension of a vector space is that any two nite dimensional vector spaces (over the same eld) of the same dimension are isomorphic.

Given a topological space X we want to de ne an algebraic object H (X ), called the 37 38 CHAPTER 2. MOTIVATING EXAMPLES homology of X , which is a topologically invariant that is, if X and Y are homeomorphic then H (X ) and H (Y ) are isomorphic. 1 Homotopy Notice that we did not claim that homology classi es spaces up to homeomorphism. It is not true that if two spaces have the same homology, then they are homeomorphic. Unfortunately, the classi cation problem in topology is too di cult for any purely algebraic classi cation.

### Algebraic topology: A computational approach by Kaczynski T., Mischaikow K., Mrozek M.

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