# Algebraic topology

﻿
Algebraic topology

Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism. 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.

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.

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). 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. 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. 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 groups are completely classified and are particularly easy to work with.

etting in category theory

In general, all constructions of algebraic topology are functorial; the notions of category, functor and natural transformation originated here. 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.

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 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. 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. 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. 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.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.

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.
* Topological combinatorics

Notable algebraic topologists

*Karol Borsuk
*Luitzen Egbertus Jan Brouwer
*Otto Hermann Künneth
*Samuel Eilenberg
*J.A. Zilber
*Heinz Hopf
*Saunders Mac Lane
*Witold Hurewicz
*Egbert van Kampen
*Daniel Quillen
*Dennis Sullivan

Important theorems in algebraic topology

*Borsuk-Ulam theorem
*Brouwer fixed point theorem
*Cellular approximation theorem
*Eilenberg–Zilber theorem
*Hurewicz theorem
*Kunneth theorem
*Poincaré duality theorem
*Universal coefficient theorem
*Van Kampen's theorem

ee also

* Important publications in algebraic topology

*citation |last= May |first=J. P. |title=A Concise Course in Algebraic Topology |year=1999 |publisher=U. Chicago Press, Chicago |url=http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf |accessdate=2008-09-27.

References

*citation |last=Bredon |first=Glen E. |title=Topology and Geometry |year=1993 |publisher=Springer |series=Graduate Texts in Mathematics 139 |url=http://books.google.com/books?id=G74V6UzL_PUC&printsec=frontcover&dq=bredon+topology+and+geometry&client=firefox-a&sig=4IMV0fFDS |accessdate=2008-04-01 |isbn=0-387-97926-3.
*citation| last=Hatcher |first= Allen |title=Algebraic Topology |url=http://www.math.cornell.edu/~hatcher/AT/ATpage.html |year= 2002 |publisher=Cambridge University Press |place=Cambridge |isbn=0-521-79540-0. A modern, geometrically flavored introduction to algebraic topology.
*citation| last=Maunder |first=C.R.F. |title=Algebraic Topology |year=1970 |publisher= Van Nostrand Reinhold |place=London |isbn=0-486-69131-4.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• algebraic topology — Math. the branch of mathematics that deals with the application of algebraic methods to topology, esp. the study of homology and homotopy. * * * Field of mathematics that uses algebraic structures to study transformations of geometric objects. It …   Universalium

• algebraic topology — Math. the branch of mathematics that deals with the application of algebraic methods to topology, esp. the study of homology and homotopy …   Useful english dictionary

• algebraic topology — noun That branch of topology that associates objects from abstract algebra to topological spaces …   Wiktionary

• Algebraic topology (object) — In mathematics, the algebraic topology on the set of group representations from G to a topological group H is the topology of pointwise convergence, i.e. pi converges to p if the limit of pi ( g ) = p ( g ) for every g in G .This terminology is… …   Wikipedia

• Directed algebraic topology — In mathematics, directed algebraic topology is a form of algebraic topology that studies topological spaces equipped with a family of directed paths, closed under some operations. The term d space is applied to these spaces. Directed algebraic… …   Wikipedia

• List of algebraic topology topics — This is a list of algebraic topology topics, by Wikipedia page. See also: topology glossary List of topology topics List of general topology topics List of geometric topology topics Publications in topology Topological property Contents 1… …   Wikipedia

• Chain (algebraic topology) — This article is about algebraic topology. For the term chain in order theory, see chain (order theory). In algebraic topology, a simplicial k chain is a formal linear combination of k simplices. Integration on chains Integration is defined on… …   Wikipedia

• Moore space (algebraic topology) — See also Moore space for other meanings in mathematics. In algebraic topology, a branch of mathematics, Moore space is the name given to a particular type of topological space that is the homology analogue of the Eilenberg–Maclane spaces of… …   Wikipedia

• Induced homomorphism (algebraic topology) — In mathematics, especially in the area of topology known as algebraic topology, an induced homomorphism is a way of relating the algebraic invariants of topological spaces which are already related by a continuous function. Such homomorphism… …   Wikipedia

• Topology — (Greek topos , place, and logos , study ) is the branch of mathematics that studies the properties of a space that are preserved under continuous deformations. Topology grew out of geometry, but unlike geometry, topology is not concerned with… …   Wikipedia