Algebraic topology

Algebraic topology is a branch of mathematics, the topological spaces (or positional relations in space such as in knot theory ) were investigated with the aid of algebraic structures. It is a branch of topology.

The basic idea is to assign certain topological spaces, for example, subsets of the view space as spheres, tori or their surfaces, some algebraic structures such as groups or vector spaces, and that intricate relationships on the part of topological spaces simplified in a way to find pages of algebraic structures and thus are amenable to treatment.

Task

A major goal of topology is to classify all topological spaces up to homeomorphism. This goal can not be achieved in such a comprehensive way, still is even certain classes of topological spaces can be classified searched for effective and reliable methods used to analyze certain rooms or.

Typically, simplicial complexes, cell complexes, manifolds are studied with and without boundary, ie rooms that are composed of topologically simple components. The considered maps between them can be continuous piecewise linear or differentiable maps. The aim is the considered spaces and the images to be classified between them by means of associated algebraic structures such as groups, rings, vector spaces and homomorphisms (structures) between them and resulting variables as far as possible, up to homeomorphism, or at least to the coarser homotopy equivalence. This does not require mengentopologischen properties such as separation axioms or Metrisierbarkeit be used, but more global properties such as " turns " or "holes" in space, terms that need to be clarified in the context of algebraic topology first.

Methodology

Some results of algebraic topology are of a negative nature, such impossibility statements. So, one can show that there is no continuous, surjective mapping of the sphere to the sphere surface, which makes the spherical surface firmly. Such a map would have to somehow the hole that is enclosed by the spherical surface, and produce that does not seem to be possible with a continuous map. A clarification of these ideas leads to homology theory. Such impossibility statements may well again have positive consequences to be formulated. Thus, for example, the Brouwer fixed point theorem, according to which every continuous mapping of the sphere into itself has a fixed point, a simple consequence, because one can show that with a fixed-point- free imaging of the ball in a figure of just excluded type could be constructed.

Another typical approach in algebraic topology is the preparation of the invariants for the classification of certain algebraic structures. If you want to, for example, closed continuous curves in the plane up to continuous deformation (which would still give details) classify, it is found that there is only one such class, because you can clearly draw any such closed curve to a circle apart and these then (with radius 1 around the origin ) deform to the unit circle. Each closed curve is thus deformation equal to the unit circle. Note that the curves are allowed to pass through it themselves; there are no nodes in the plane ( for nodes, which are also treated in algebraic topology, you need three dimensions).

Just the indicated situation changes when we replace the plane through the level without the zero point. Pulling apart the circle now works no longer always, since the curve in the course of the deformation process the zero point can no longer be deleted. A clarification of these ideas leads to the fundamental group and generalized to homotopy theory. One can consider that two closed curves of the same class if and only if the numbers of rounds overrule around the zero point (eg counterclockwise). Each curve is therefore a number of associated, namely their circulation number, and this number classifies the curves. If we restrict ourselves to curves that start at a fixed point, and end because of the closeness of the curves there again, it may be two curves in succession through by first going through the first corner, and then after you arrived back at the fixed starting point is the second. Here, the circulation numbers add up. The consecutively executed run of the curves at the topological side corresponds to the addition of integers of the algebraic side. Thus, the topological space level is not zero an algebraic structure, the group assigned to the closed curves in it are classified by an element of this group.

These considerations suggest the role of category theory in algebraic topology. The general idea is a topological situation, ie topological spaces and continuous maps between them, an algebraic situation, that is, groups, rings, or vector spaces and morphisms between them, assign and functorial in an invariant way, and to draw conclusions. Invariant means in this case that homeomorphic spaces or homotopieäquivalenten be assigned isomorphic algebraic structures.

Historical development

Although the Greek mathematicians of antiquity have been concerned with deformations of three-dimensional body ( shearing, stretching ) and also interested in the complexity of nodes, but the first precise concept formation, which would be attributable to the algebraic topology is introduced by Leonhard Euler Euler characteristic.

In the 19th century, Gauss discovered the linking number of two curves, which does not change with continuous deformation without interpenetration. The physicist Kelvin began to be interested in knots, Betti examined holes and handles on manifolds and came to the Betti numbers named after him. Towards the end of the 19th century classified Poincaré two dimensional manifolds (see Classification Theorem for 2 -manifolds ) and resulted in this context, a basic concept of the fundamental group.

The first outstanding results in the algebraic topology of the 20th century were the proof of the invariance of the topological dimension by Brouwer in 1913 and the invariance of homology, ie the Betti numbers, by Alexander in the 1920s. By Vietoris, Alexandrov and Čech homology theory was extended to public spaces. Following ideas of Poincaré and Riemann led a Cartan differential forms and a more advanced homology theory whose equivalence was demonstrated to the usual homology theory of de Rham his students in the 1930s. Hurewicz generalized the notion of the fundamental group of homotopy. Having established that the n - spheres have non-trivial higher homotopy groups whose determination was a central task.

In the late 1930s discovered Whitney, boots, Pontryagin and Chern different named after them topological invariants, so-called characteristic classes that act as barriers: certain things can only work or be present when these classes satisfy certain conditions, otherwise they represent the obstacle and will produce

In the 1940s, the Morse Theory and Eilenberg -established managed a rigorous proof of the homotopy invariance of singular homology. Any further algebraization the Poincaré duality led to the cohomology theory. Eilenberg and Mac Lane abstracted further to the so-called homological algebra and apply in this context as founders of category theory. These considerations resulted in the Eilenberg - Steenrod - uniqueness theorem.

A breakthrough in the already initiated by Poincaré classification of manifolds was the surgery theory of Browder, Novikov, Sullivan and Wall, with the exception of a classification of simply connected manifolds of Diffeomorphie dimension which are homotopy equivalent to a given manifold, succeeded.

Another significant advance in the algebraic methods of topology and homology theory were Grothendieck's work on the theorem of Riemann -Roch, who founded the K- theory. Here are the Bott periodicity and the Atiyah-Singer index theorem in the 1960s of the most important results.

The algebraic topology is still the subject of current research, with a comprehensible presentation of the results is becoming increasingly difficult. For further details, please refer to the below article by Novikov.

When already undertaken by Poincaré attempt of classification of three-dimensional manifolds, the problem appeared to show that every simply connected compact unberandete 3 -dimensional manifold is homeomorphic to the 3- sphere. This became known as the Poincaré conjecture problem was solved only in 2002 by Perelman.

Applications

Outside the topology there are many applications of algebraic topology. The above-mentioned winding number is an important factor for integration paths in function theory is like talking of course of zero homologous cycles. When studying Riemannian surfaces methods of cohomology theory play an important role.

If we identify a compact space with the algebra of continuous, complex-valued functions on what you should do after the Gelfand - Neumark, then translate the above concept formation in the ring theory and C * - theory, at least for commutative rings and C * algebras, since is commutative. Leaving the commutativity now falling, so that leads to the so-called non-commutative topology, for example, to going back to Kasparov KK - theory. Thus, for important impulses for the algebra and functional analysis.

In physics, algebraic topology plays an important role in topological quantum field theory TQFT.