By Kaczynski T., Mischaikow K., Mrozek M.
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Extra resources for Algebraic topology: A computational approach
De ne F : 0 1] ! f0g by F (x s) = (1 ; s)x. Homology has the property that if two spaces are homotopic then their homologies are the same. On the other hand, there are spaces with the same homologies which are not homotopic. Thus, the algebraic invariants that we will develop in this book are extremely crude measurements of the topology of the space. Still there are interesting problems to which one can apply homology theory. 7 Let ;n := fx 2 Rn+1 j jjxjj = 1g: There is no deformation retraction of ;n to a point.
In de ning the approximation, we started on the level of edges. In trying to generate the algebra we will start with the vertices. 2. 8: The graph of the multivalued approximation to f (x) = (x ; 2)(x + 1) with edges of length 0:5. vertices ff;2g f;1g f0g f1g f2gg and that C0( ;2 4]) is generated by the vertices ff;2g f;1g f0g f1g f2g f3g f4gg. We will begin by de ning a linear map f#0 : C0 ( ;2 2]) ! C0 ( ;2 4]): Of course, to de ne a linear map it is su cient to de ne how it acts on the basis elements.
Assume that by adjoining w to S , linear independence is lost. 3) 1 v1 + 2 v2 + : : : + n vn + w = 0: Since F is a eld, ;1 2 F . 3) as w = ;1( 1 v1 + 2 v2 + : : : + nvn) which contradicts the assumption that w is not in the subspace spanned by S. 55 we made crucial use of the fact that F was a eld. 55 need not hold in Z3. Let 82 3 2 3 2 39 2 3 > 0 <6 1 7 6 0 7 6 0 7> = 6 S = >4 0 5 4 1 5 4 0 5> and w = 4 0 75 : : 0 0 2 1 Observe that w is not in the span of S since 2;1 62 Z, but S fwg is not a linearly independent set.