Algebraic Topology

The algebraic topology , in the past called combinative Topology, is a branch of the Mathématiques applying the tools of the Algèbre in the study of the topological spaces. More exactly, she seeks to associate in a natural way invariants with the associated topological structures. The naturality means that these invariants check properties of fonctoriality within the meaning of the Théorie of the categories.

Algebraic invariants

The fundamental idea is to be able to associate with any topological space algebraic objects (Nombre, group, vector Space…), so that with two homeomorphic spaces two isomorphous structures are associated. Such objects are called algebraic invariants. In erudite terms, it is a question of studying functors since the category of topological spaces on an algebraic category, like the categories of groups, algebras, groupoïdes, etc Of the results of topology pass then by the more accessible demonstration of algebraic properties.

Among the notable invariants, let us quote:

  • the fundamental Group of a topological space X in a point X : the whole of the classes of homotopy of the laces of X basic X , the internal law of composition being concatenation of the laces.
  • the groups of higher homotopy of a topological space X in a point X .
  • groups of homology or cohomology of a topological space X .
  • the classes characteristic of a fiber vectorial reality, complex, Euclidean or square.

Applications

  • the theorem of Brouwer: any continuous application of the disc unit of \ mathbb {R} ^n in itself admits a fixed point.
  • the theorem of the hairy ball: any continuous-current field of tangent vectors to a sphere of even size is cancelled in at least a point.
  • the Theorem of Borsuk-Ulam: any application S n in \ mathbb {R} ^n takes the same value in two antipodean points. For example, at any moment, there exist two points on the surface of the Earth diametrically opposite having even temperature and even pressure.

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