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By Kaczynski , Mischaikow , Mrozek

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1 If f : a b] ! R is a continuous function and if f (a) > 0 and f (b) < 0, then there exists c 2 a b] such that f (c) = 0. This is a model topological theorem. e. the values of the end points are given, and yet one is still able to draw a conclusion concerning the behavior of the function on its domain. Homology provides us with a variety of algebraic tools for determining if there exists a point c such that f (c) = 0. But this process of going from topology to algebra loses information. This should not be surprising.

14 A point x of a graph G is called a regular point of G if a su ciently small ball in G around x is homeomorphic to an open interval. A point which is not a regular point is called an extreme point of G. The set of all extreme points of G is called the geometric boundary of G and denoted by bd G. Let us now think of 0 1] and ;1 as graphs. 2. 0 1] is represented by a graph consisting of four intervals a b], b c], c d] and d e]. 1. 2: Finite graphs and corresponding abstract nite graphs for 0 1] and ;1 We mentioned earlier that the boundary points of 0 1] are where we can see a di erence in local topology.

2. 2: Topology and algebra of boundaries in ;1. e. algebraic objects whose boundaries add up to zero, are topologically nontrivial. This is almost true. To see how this fails, observe that ;1 C 2, and in fact ;1 = bd C 2. 1. 3: Topology and algebra of boundaries in C 2. how the nontrivial algebra in ;1 becomes trivialized. To do this we need to go beyond graphs into cubical complexes which will be de ned later. 3. The new aspect is the square C 2. This is coded in the combinatorial information as the element fC 2g.

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