Homology Theory

Abstract

In this chapter, the so-called homology groups will be defined for any topological space. An idea of constructing homology groups, as was already mentioned, goes back to Poincaré. The useful idea of “algebraization” of topological problems was carried out for the first time, by constructing homology groups and fundamental group. Homology theory still remains in a central position. In many cases, topological invariants are finally expressed in terms of homology groups and cohomology groups. This comes about because of a better computability of homology groups and cohomology groups, although their definitions are somehwat more complicated than, for example, the definition of homotopy groups.KeywordsSingular PointSimplicial ComplexChain ComplexHomology GroupHomology TheoryThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.