Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...

The method of algebraic invariants

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, by mapping them, for example, to groups which have a great deal of manageable structure in a way that respects the relation of homeomorphism of spaces. This allows one to recast statements about topological spaces into statements about groups, which are often easier to prove. In mathematics, combinatorial topology was an older name for algebraic topology, dating from the time when topological invariants of spaces (for example the Betti numbers) were regarded as derived from combinatorial decompositions such as simplicial complexes. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ...

Two major ways in which this can be done are through fundamental groups, or more generally 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. In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ... An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ... In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homos = identical) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ... In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. ... In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ... In mathematics, a simplicial complex is a topological space of a particular kind, built up of points, line segments, triangles, and their n-dimensional counterparts. ... In mathematics, one method of defining a group is by a presentation. ...

Homology and cohomology groups, on the other hand, are abelian and in many important cases finitely generated. Finitely generated abelian groups are completely classified and are particularly easy to work with. In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x1,...,xs in G such that every x in G can be written in the form x = n1x1 + n2x2 + ... + nsxs with integers n1,...,ns. ...

Results on homology

Several useful results follow immediately from working with finitely generated abelian groups. The free rank of the n-th homology group of a simplicial complex is equal to the n-th Betti number, so one can use the homology groups of a simplicial complex to calculate its Euler-Poincaré characteristic. As another example, the top-dimensional integral cohomology group of a closed manifold detects orientability: this group is isomorphic to either the integers or 0, according as the manifold is orientable or not. Thus, a great deal of topological information is encoded in the homology of a given topological space. In algebraic topology, the Betti numbers of a topological space X are a sequence b0, b1, ... of topological invariants. ... In algebraic topology, the Euler characteristic is a topological invariant (in fact, homotopy invariant) defined for a broad class of topological spaces. ... A manifold is a mathematical space which is constructed, like a patchwork, by gluing and bending together copies of simple spaces. ... This article or section should be merged with Orientable manifold. ...

Beyond simplicial homology, which is defined only for simplicial complexes, one can use the differential structure of smooth manifolds via de Rham cohomology, or Čech or sheaf 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, for a closed, oriented manifold, the Betti numbers derived through simplicial homology were the same Betti numbers as those derived through de Rham cohomology. A manifold is a mathematical space which is constructed, like a patchwork, by gluing and bending together copies of simple spaces. ... In 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 representation of cohomology classes. ... In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F. This is the main step, in numerous areas, from sheaf theory as a description of a geometric... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... Georges de Rham (10 September 1903-9 October 1990) was a Swiss mathematician, known for his contributions to differential topology. ...

Setting in category theory

In general, all constructions of algebraic topology are functorial: the notions of category, functor and natural transformation originated here. Fundamental groups, homology and cohomology groups are not only invariants of the underlying topological space, in the sense that two topological spaces which are homeomorphic have the same associated groups; a continuous mapping of spaces induces a group homomorphism on the associated groups, and these homomorphisms can be used to show non-existence (or, much more deeply, existence) of mappings. Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... Look up category in Wiktionary, the free dictionary. ... Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ... In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ... This word should not be confused with homomorphism. ...

Applications of algebraic topology

Classic applications of algebraic topology include:

• The Brouwer fixed point theorem: every continuous map from the unit n-disk to itself has a fixed point.
• The n-sphere admits a nowhere-vanishing continuous unit vector field if and only if n is odd. (For n=2, this is sometimes called the "hairy ball theorem".)
• The Borsuk-Ulam theorem: any continuous map from the n-sphere to Euclidean n-space identifies at least one pair of antipodal points.
• Any subgroup of a free group is free. This result is quite interesting, because the statement is purely algebraic yet the simplest proof is topological. Namely, any free group G may be realized as the fundamental group of a graph X. The main theorem on covering spaces tells us that every subgroup H of G is the fundamental group of some covering space Y of X; but every such Y is again a graph. Therefore its fundamental group H is free.

The most celebrated geometric open problem in algebraic topology is the Poincaré conjecture, which may have been resolved by Grigori Perelman. The field of homotopy theory contains many mysteries, most famously the right way to describe the homotopy groups of spheres. In mathematics, the Brouwer fixed point theorem states that every continuous function from the closed unit ball D n to itself has a fixed point. ... In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In mathematics, the discussion of vector fields on spheres was a classical problem of differential topology, beginning with the hairy ball theorem, and early work on the classification of division algebras. ... The Borsuk-Ulam theorem states that any continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point. ... The Cayley graph of the free group on two generators a and b In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many... In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint open... In mathematics, the Poincaré conjecture (see Henri Poincaré for pronunciation) is a conjecture about the characterisation of the three-dimensional sphere amongst 3-manifolds. ... Grigori Grisha Yakovlevich Perelman (Russian: Григорий Яковлевич Перельман) (born 13 June 1966) is a Russian Jewish mathematician who is an expert on Ricci flow. ... An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ... In mathematics, the homotopy groups of spheres are the groups πk(Sn) in algebraic topology, more specifically homotopy theory, where πk(.) for k ≥ 1 denotes the homotopy group and Sn the n-sphere. ...

See also

• Important publications in algebraic topology

This is a list of important publications in mathematics, organized by field. ...

References

• Allen Hatcher, Algebraic Topology , Cambridge University Press, Cambridge, 2002. ISBN 0-521-79540-0. A modern, geometrically flavored introduction to algebraic topology. The book is available free in PDF and PostScript formats on the author's homepage.
• C. R. F. Maunder, Algebraic Topology (1970) Van Nostrand Reinhold, London ISBN 73-105346.