By Kaczynski , Mischaikow , Mrozek

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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. 1 the topological boundaries of each of the edges.

Let e1 be an edge in T . Label its vertices by v1; and v1+. Since T has no free vertices, there is an edge e2 with vertices v2 such that v1+ = v2;. Continuing in this manner we can label the edges by ei and the vertices by vi where vi; = vi+;1. Note since there are only a nite number of vertices, at some point vi+ = vj; for some i > j 1. Then fej ej+1 : : : eig forms a loop. This is a contradition. 11 Every edge is homotopic to a point. Proof: Let e be an edge with vertices v ; and v+. Since e is a line segment it is homeomorphic to 0 1].

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. 9 one sees the graph of the multivalued map when the lengths of the edges is 0:1. 2 Constructing Chain Maps In the previous section we considered the problem of approximating maps from one interval to another. Of course the goal of this course is to use such an approximation to reduce the analytic problem to an algebraic problem. 7 to construct a map f : H ( ;2 2]) !

### Algebraic Topology A Computational Approach by Kaczynski , Mischaikow , Mrozek

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