دانلود کتاب توپولوژی جبری خالص و کاربردی

For many people, Poincaré’s works [59] are the real foundation of the algebraic topology, when he defined for the first time in 1985 what is meant by homologous chains in a manifold. His definition was rather imprecise, but the notion he used covered exactly the current acceptance: two closed chains are homologous if they differ by an edge. By 1940, the homological algebra theory was well defined and contributed greatly to the emergence of many other concepts like categories and functors. Various generalisations have been imagined later, like cohomology of groups with many surprising geometrical connections, bounded cohomology and equivariant cohomology. This shows, how homological notions have become largely widespread in almost all mathematics areas, and sometimes even in theoretical physics. The main principle of algebraic topology is to associate, in a functorial way, to any topological object an algebraic object which is invariant under certain kinds of transformations like homeomorphisms, homotopisms, holomorphisms and isomorphisms. A constructive example is how to apply to a torus, two scissors to make it homeomorphic to a paper sheet. Topologically speaking, scissors here symbolise loops. Two cuts are said to be equivalent, when they have the same effect on the ambient space. In other words one can continuously switch from one to the other. The first cut goes around the central hole; and we get a crown. The second one is a radial cut on this crown and gives us a rectangle.