Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism. In Mathematics, an invariant is something that does not change under a set of transformations The property of being an invariant is invariance. In Mathematics, a classification theorem answers the classification problem "What are the objects of a given type up to some equivalence?" Topological equivalence redirects here see also Topological equivalence (dynamical systems. In many situations this is too much to hope for and it is more prudent to aim for a more modest goal, classification up to homotopy equivalence. In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical

Although algebraic topology primarily uses algebra to study topological problems, the converse, using topology to solve algebraic problems, is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. In Mathematics, a group G is called free if there is a Subset S of G such that any element of G can be

Contents 1 The method of algebraic invariants 2 Setting in category theory 3 Results on homology 4 Applications of algebraic topology 5 Notable algebraic topologists 6 Important theorems in algebraic topology 7 See also 8 References

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 modern standard tool for such construction is the CW-complex). In Mathematics, combinatorial topology was an older name for Algebraic topology, dating from the time when topological invariants of spaces (for example In Topology, a CW complex is a type of Topological space introduced by J 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 (or more general homotopy) of spaces. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element Topological equivalence redirects here see also Topological equivalence (dynamical systems. In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical This allows one to recast statements about topological spaces into statements about groups, which are often easier to prove.

Two major ways in which this can be done are through fundamental groups, or more generally homotopy theory, and through homology and cohomology groups. In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology. In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical In Mathematics (especially Algebraic topology and Abstract algebra) homology (in Greek ὁμός homos "identical" is In Mathematics, specifically in Algebraic topology, cohomology is a general term for a Sequence of Abelian groups defined from a Cochain 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. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the The fundamental group of a (finite) simplicial complex does have a finite presentation. In Mathematics, a simplicial complex is a Topological space of a particular kind constructed by "gluing together" points Line segments 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. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the 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 x 1

Setting in category theory

In general, all constructions of algebraic topology are functorial; the notions of category, functor and natural transformation originated here. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal Fundamental groups and 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, but their associated morphisms also correspond — 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. Topological equivalence redirects here see also Topological equivalence (dynamical systems.

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. In Algebraic topology, the Betti number of a Topological space is in intuitive terms a way of counting the maximum number of cuts that can be made without dividing In Mathematics, and more specifically in Algebraic topology and Polyhedral combinatorics, the Euler characteristic is a Topological invariant As another example, the top-dimensional integral homology 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. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be A surface S in the Euclidean space R 3 is orientable if a two-dimensional figure (for example) cannot be moved around the surface and back Thus, a great deal of topological information is encoded in the homology of a given topological space.

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. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In Mathematics, de Rham cohomology (after Georges de Rham) is a tool belonging both to Algebraic topology and to Differential topology, capable In Mathematics, sheaf cohomology is the aspect of Sheaf theory, concerned with sheaves of Abelian groups that applies Homological algebra to A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the 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. Georges de Rham ( 10 September 1903 &ndash 9 October 1990) was a Swiss Mathematician, known for his contributions to This was extended in the 1950s, when Eilenberg and Steenrod generalized this approach. They defined homology and cohomology as functors equipped with natural transformations subject to certain axioms (e. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal g. , a weak equivalence of spaces passes to an isomorphism of homology groups), verified that all existing (co)homology theories satisfied these axioms, and then proved that such an axiomatization uniquely characterized the theory. In Mathematics, a weak equivalence is a notion from Homotopy theory which in some sense identifies objects that have the same basic "shape"

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. In Mathematics, the Brouwer fixed point theorem is an important Fixed point theorem that applies to finite-dimensional spaces and which forms the basis for several In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output
• The n-sphere admits a nowhere-vanishing continuous unit vector field if and only if n is odd. In Mathematics, the discussion of vector fields on spheres was a classical problem of Differential topology, beginning with the Hairy ball theorem, and (For n=2, this is sometimes called the "hairy ball theorem". The hairy ball theorem of Algebraic topology states that there is no nonvanishing continuous Tangent vector field on the sphere )
• The Borsuk-Ulam theorem: any continuous map from the n-sphere to Euclidean n-space identifies at least one pair of antipodal points. The Borsuk–Ulam theorem states that any Continuous function from an ''n''-sphere into Euclidean ''n''-space maps some pair of Antipodal points
• Any subgroup of a free group is free. In Mathematics, a group G is called free if there is a Subset S of G such that any element of G can be 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. In Mathematics and Computer science, a graph is the basic object of study in Graph theory. 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. In Mathematics, a covering space is a Topological space C which "covers" another space X by a Surjective Local homeomorphism Therefore its fundamental group H is free.
• Topological combinatorics

Notable algebraic topologists

Important theorems in algebraic topology

See also

• Important publications in algebraic topology

References

• Bredon, Glen E. Saunders Mac Lane ( 4 August 1909, Taftville, Connecticut – 14 April 2005, San Francisco) was an American George William Whitehead Jr ( August 2[[ 918]] – April 12[[ 004]] was a professor of Mathematics at the Massachusetts Institute of Technology Not to be confused with Adolf Hurwitz. Witold Hurewicz ( June 29 1904 - September 6 1956) was a Egbert Rudolf van Kampen ( May 28, 1908, Berchem, Belgium – February 11, 1942, Baltimore, Maryland Daniel Gray ("Dan" Quillen (born June 22, 1940) is an American Mathematician and a Fields Medalist From 1984 to 2006 The Borsuk–Ulam theorem states that any Continuous function from an ''n''-sphere into Euclidean ''n''-space maps some pair of Antipodal points In Mathematics, the Brouwer fixed point theorem is an important Fixed point theorem that applies to finite-dimensional spaces and which forms the basis for several In Mathematics, specifically in Algebraic topology, the Eilenberg–Zilber theorem is an important result in establishing the link between the Homology In Mathematics, the Hurewicz theorem is a basic result of Algebraic topology, connecting Homotopy theory with Homology theory via a map known In Mathematics, especially in Homological algebra and Algebraic topology, a Künneth theorem is a statement relating the homology of two objects to the In Mathematics, the Poincaré duality theorem named after Henri Poincaré, is a basic result on the structure of the homology and Cohomology In Mathematics, the universal coefficient theorem in Algebraic topology establishes the relationship in Homology theory between the integral homology In Mathematics, the Seifert – van Kampen theorem of Algebraic topology, sometimes just called van Kampen's theorem, expresses the In Homotopy theory (a branch of Mathematics) the Whitehead theorem states that if a Continuous mapping f between Topological spaces Algebra Theory of equations Hisab (1993), Topology and Geometry, Graduate Texts in Mathematics 139, Springer, ISBN 0-387-97926-3, <http://books.google.com/books?id=G74V6UzL_PUC&printsec=frontcover&dq=bredon+topology+and+geometry&client=firefox-a&sig=4IMV0fFDS>. Retrieved on 1 April 2008 .
• Hatcher, Allen (2002), Algebraic Topology, Cambridge: Cambridge University Press, ISBN 0-521-79540-0, <http://www.math.cornell.edu/~hatcher/AT/ATpage.html> . A modern, geometrically flavored introduction to algebraic topology.
• Maunder, C. R. F. (1970), Algebraic Topology, London: Van Nostrand Reinhold, ISBN 0-486-69131-4 .

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