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

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

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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 . 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.

1 Approximating Maps on an Interval To keep the technicalities to an absolute minimum, we begin our discussion with maps of the form f : a b] ! c d]. We do this for two reasons. First, each interval can be represented by a graph and so using the types of arguments employed in the previous section we can compute the homology. Second, we can actually draw pictures of the functions. This latter point is to help us develop our intuition, in practice we will want to apply these ideas to problems where it is not feasible to visualize the maps, either because the map is too complicated or because the dimension is too high.

This is important because it means that it can be stored and manipulated by the computer. The multivalued map F that we constructed above is fairly coarse. If we want a better approximation, then one approach is to use ner graphs to describe X and Y . 8. 8. Observe that this is a better approximation to the function than 62 CHAPTER 2. 8: Edges and Vertices for the graphs of X = ;2 2] and Y = ;2 4]. what was obtained with intervals of unit length. In fact, one can obtain as good an approximation as one likes by choosing the edge lengths su ciently small.

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