Algebraic topology is a branch ofmathematics that uses tools fromabstract algebra to studytopological spaces. The basic goal is to find algebraicinvariants thatclassify topological spacesup tohomeomorphism, though usually most classify up tohomotopy equivalence.
Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that anysubgroup of afree group is again a free group.
Below are some of the main areas studied in algebraic topology:
In mathematics, homotopy groups are used in algebraic topology to classifytopological spaces. The first and simplest homotopy group is thefundamental group, which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space.
In algebraic topology andabstract algebra,homology (in part fromGreek ὁμόςhomos "identical") is a certain general procedure to associate asequence ofabelian groups ormodules with a given mathematical object such as atopological space or agroup.[1]
Inhomology theory and algebraic topology,cohomology is a general term for asequence ofabelian groups defined from acochain complex. That is, cohomology is defined as the abstract study ofcochains,cocycles, andcoboundaries. Cohomology can be viewed as a method of assigningalgebraic invariants to a topological space that has a more refinedalgebraic structure than doeshomology. Cohomology arises from the algebraic dualization of the construction of homology. In less abstract language, cochains in the fundamental sense should assign "quantities" to thechains of homology theory.
Amanifold is atopological space that near each point resemblesEuclidean space. Examples include theplane, thesphere, and thetorus, which can all be realized in three dimensions, but also theKlein bottle andreal projective plane which cannot be embedded in three dimensions, but can be embedded in four dimensions. Typically, results in algebraic topology focus on global, non-differentiable aspects of manifolds; for examplePoincaré duality.
Knot theory is the study ofmathematical knots. While inspired by knots that appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined so that it cannot be undone. In precise mathematical language, a knot is anembedding of acircle in three-dimensionalEuclidean space,. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of upon itself (known as anambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.
Asimplicial complex is atopological space of a certain kind, constructed by "gluing together"points,line segments,triangles, and theirn-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of asimplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is anabstract simplicial complex.
ACW complex is a type of topological space introduced byJ. H. C. Whitehead to meet the needs ofhomotopy theory. This class of spaces is broader and has some bettercategorical properties thansimplicial complexes, but still retains a combinatorial nature that allows for computation (often with a much smaller complex).
An older name for the subject wascombinatorial topology, implying an emphasis on how a space X was constructed from simpler ones[2] (the modern standard tool for such construction is theCW complex). In the 1920s and 1930s, there was growing emphasis on investigating topological spaces by finding correspondences from them to algebraicgroups, which led to the change of name to algebraic topology.[3] The combinatorial topology name is still sometimes used to emphasize an algorithmic approach based on decomposition of spaces.[4]
In the algebraic approach, one finds a correspondence between spaces andgroups that respects the relation ofhomeomorphism (or more generalhomotopy) of spaces. This allows one to recast statements about topological spaces into statements about groups, which have a great deal of manageable structure, often making these statements easier to prove. Two major ways in which this can be done are throughfundamental groups, or more generallyhomotopy theory, and throughhomology andcohomology groups. The fundamental groups give us basic information about the structure of a topological space, but they are oftennonabelian and can be difficult to work with. The fundamental group of a (finite)simplicial complex does have a finitepresentation.
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 general, all constructions of algebraic topology arefunctorial; the notions ofcategory,functor andnatural transformation originated here. Fundamental groups and homology and cohomology groups are not onlyinvariants of the underlying topological space, in the sense that two topological spaces which arehomeomorphic have the same associated groups, but their associated morphisms also correspond—a continuous mapping of spaces induces agroup homomorphism on the associated groups, and these homomorphisms can be used to show non-existence (or, much more deeply, existence) of mappings.
One of the first mathematicians to work with different types of cohomology wasGeorges de Rham. One can use the differential structure ofsmooth manifolds viade Rham cohomology, orČech orsheaf cohomology to investigate the solvability ofdifferential 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, whenSamuel Eilenberg andNorman Steenrod generalized this approach. They defined homology and cohomology asfunctors equipped withnatural transformations subject to certain axioms (e.g., aweak 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.
Classic applications of algebraic topology include:
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