By Kaczynski , Mischaikow , Mrozek
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However, as the following example indicates di erent graphs can give rise to the same subset. 44 CHAPTER 2. 13 Consider the family of graphs de ned by Gn = ffj=ng j j = 0 : : : ng f j=n (j + 1)=n] j j = 0 : : : n ; 1g Observe that as a subset of R each graph describes 0 1]. Thus, the same topological space has many di erent representations as a graph. In our motivation of homology we will use abstract graphs. Thus to prove topological invariance we would have to show that given any two abstract graphs that are associated to nite graphs that in turn represent homeomorphic spaces the corresponding homology is the same.
These problems have their origins in topology (not surprising), computer graphics, dynamical systems, parallel computing, and numerics. 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.
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 . The corresponding open edge is e:= e n fv g: The second observation, is that we used the edges to de ne the images of the vertices.
Algebraic Topology A Computational Approach by Kaczynski , Mischaikow , Mrozek