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By Kaczynski , Mischaikow , Mrozek

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Let us begin by emphasizing that this is not an obvious task. Recall that homology is by de nition a quotient of cycles by boundaries, which in turn belong to subspace of the set of chains. Thus, it seems that the rst place to begin is on the level of chains. Furthermore, in order to be able to use our intuition from linear algebra we will consider homology with Z2 coe cients. 7 we will consider ;2 2] and ;2 4] to be the graphs made up of the edges with vertices having integer values. In de ning the approximation, we started on the level of edges.

9: The graph of the multivalued approximation to f (x) = (x ; 2)(x + 1) with edges of length 0:1. We have now de ned a linear map between the 0-chains of the two spaces. The next step is to \lift" the de nition of f#0 to obtain a linear map f#1 : C1( ;2 2]) ! C1( ;2 4]). Of course the basis of these spaces are given by the intervals. So consider the interval ;2 ;1] ;2 2]. How should we de ne f#1( ;2 ;1])? We know that f#0(f;2g) = f4g and f#0(f;1g) = f1g so it seems reasonable to de ne f#1 ( ;2 ;1]) = 1 2] + 2 3] + 3 4].

Observe that X = ;2 ;1] ;1 0] 0 1] 1 2]: Therefore, we will do our computations in terms of edges. From the combi- CHAPTER 2. 6: The function f (x) = (x ; 2)(x + 1) and a homotopic function g. natorial point of view, this suggests trying to map edges to edges. Since f (;2) = 3:41421356 : : :, f (;1) = 0, and f is monotone over the edge ;2 ;1], it is clear that f ( ;2 ;1]) 0 4] = 0 1] 1 2] 2 3] 3 4]: Thus we could think of de ning a map that takes the edge 0 1] to the collection of edges f 0 1] 1 2] 2 3] 3 4]g.

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