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Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces.
The goal is to take topological spaces, and further categorize or classify them. An older name for the subject was combinatorial topology, implying an emphasis on how a space X was constructed from simpler ones. The basic method now applied in algebraic topology is to investigate spaces via algebraic invariants: for example by mapping them to groups, which have a great deal of manageable structure, in a way that respects the relation of homeomorphism of spaces.
Two major ways in which this can be done are through fundamental groups, or more general homotopy theory, and through homology and cohomology groups. The fundamental groups give us basic information about the structure of a topological space; but they are often nonabelian and can be difficult to work with. The fundamental group of a (finite) simplicial complex does have a finite presentation.
Homology and cohomology groups, on the other hand, are abelian, and in many important cases finitely generated. Finitely generated abelian groupIn abstract algebra, an abelian group G +) is called finitely generated if there exist finitely many elements x . x in G such that every x in G can be written in the form x n x + n x +. n x with integers n . In this case, we say that the set x . x is a ges are completely classified and are particularly easy to work with.
Several useful results follow immediately from working with finitely generated abelian groupsIn abstract algebra, an abelian group G +) is called finitely generated if there exist finitely many elements x . x in G such that every x in G can be written in the form x n x + n x +. n x with integers n . In this case, we say that the set x . x is a ge. The free rank of the n-th homology group of a simplicial complex is equal to the n-th Betti numberIn algebraic topology, the Betti numbers of a topological space X are a sequence b b . of topological invariants. Each Betti number is a natural number, or infinity. For the most reasonable spaces (such as compact manifolds, finite simplicial complexes or, so one can use the homology groups of a simplicial complex to calculate its Euler-Poincaré characteristic. If an n-th homology group of a simplicial complex has torsion, then the complex is nonorientable (query this). Thus, a great deal of topological information is encoded in the homology of a given topological space.
Beyond simplicial homology, one can use the differential structure of smooth manifoldIn mathematics, a manifold ''M is a type of space, characterised in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. Therefors via de Rham cohomologyIn mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete repre, or Cech or sheafAlternate meanings: River Sheaf, King Sceaf, sheaf toss In mathematics, a sheaf ''F on a given topological space X gives a set or richer structure F ''U for each open set U of X''. The structures F ''U are compatible with the operations of restricting the cohomology to investigate the solvability of differential equations defined on the manifold in question. De Rham showed that all of these approaches were interrelated and that the Betti numbers derived through simplicial homology were the same Betti numbers as those derived through De Rham cohomology. {That would be a compact oriented manifold then, to use Poincaré duality.)