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By M.M. Cohen

Cohen M.M. A direction in simple-homotopy concept (Springer, [1973)(ISBN 3540900551)

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Let the cycle z be equal to zero on all the other edges of triangulation incident to the vertex A of the triangle ,12 and different from the edges T and g. Then necessarily b = c and the cycle z is homologous to another cycle z' which on the edges t, T, (J assumes respectively the values a + b, 0, O. Given this, the values of the cycles z and z' coincide on all the other edges of triangulation. In other words, the cycle z can be taken from the sum of the edges T and (J, and its value b can be directed along the edge t.

Ps are sometimes referred to as torsion coefficients of the simplicial complex X in dimension k. 6 Examples of Calculations of Homology Groups. Homologies of Two-dimensional Surfaces Zero-dimensional homology groups Linearly connected polyhedron Any two vertices of a connected polyhedron are homologous Fig. 5 1) Zero-dimensional homology groups. To begin with, we shall calculate a zero-dimensional homology group Ho(X) of a linearly (arcwise) connected polyhedron X. A polyhedron (a simplicial complex) is called linearly connected if any pair of its points can be joined by a continuous arc lying wholly in the polyhedron.

A. Bulgakov "Master and Margaret". About 40 illustrations were created by the authors many years ago for the first Russian edition of Bulgakov's novel. But this book was published without any pictures. Now the reader of the present mathematical book can see not only the geometrical content of these drawings but also can try to connect them with remarkable story about Jesus Christ and Pontius Pilate (from Bulgakov's novel): Christ is delivered to Pilate, story of Judas, cruciftxion of Christ, story of Matthew.

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