Algebraic topology: Difference between revisions
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{{for|the topology of pointwise convergence|Algebraic topology (object)}} |
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'''Algebraic topology''' is a branch of [[mathematics]] which uses tools from [[abstract algebra]] to study [[topological space]]s. The basic goal is to find algebraic [[invariant (mathematics)|invariants]] that [[classification theorem|classify]] topological spaces up to [[homotopy equivalence]] (if you restrict to CW-complexes). 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]]. |
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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. |
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==The method of algebraic invariants== |
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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|CW-complex]]). The basic method now applied in algebraic topology is to investigate spaces via algebraic invariants by mapping them, for example, to [[group (mathematics)|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. |
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Two major ways in which this can be done are through [[fundamental group]]s, or more generally [[homotopy theory]], and through [[homology (mathematics)|homology]] and [[cohomology]] groups. The fundamental groups give us basic information about the structure of a topological space, but they are often [[abelian group|nonabelian]] and can be difficult to work with. The fundamental group of a (finite) [[simplicial complex]] does have a finite [[presentation of a group|presentation]]. |
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Homology and cohomology groups, on the other hand, are [[abelian group|abelian]] and in many important cases finitely generated. [[Finitely generated abelian group]]s are completely classified and are particularly easy to work with. |
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==Setting in category theory== |
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In general, all constructions of algebraic topology are [[category theory|functorial]]; the notions of [[Category (mathematics)|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. |
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==Results on homology== |
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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 [[orientable|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. |
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Beyond simplicial homology, which is defined only for simplicial complexes, one can use the differential structure of smooth [[manifold]]s via [[de Rham cohomology]], or Čech or [[sheaf cohomology]] to investigate the solvability of [[differential equation]]s 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. |
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==Applications of algebraic topology== |
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Classic applications of algebraic topology include: |
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* The [[Brouwer fixed point theorem]]: every [[continuous function|continuous]] map from the unit ''n''-disk to itself has a fixed point. |
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* The ''n''-sphere admits a nowhere-vanishing continuous unit [[vector fields on spheres|vector field]] if and only if ''n'' is odd. (For ''n''=2, this is sometimes called the "[[hairy ball theorem]]".) |
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* The [[Borsuk-Ulam theorem]]: any continuous map from the ''n''-sphere to Euclidean ''n''-space identifies at least one pair of antipodal points. |
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* 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 (mathematics)|graph]] ''X''. The main theorem on [[covering space]]s 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. |
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* [[Topological combinatorics]] |
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== Notable algebraic topologists == |
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<div style="-moz-column-count:3; column-count:3;"> |
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*[[Karol Borsuk]] |
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*[[Luitzen Egbertus Jan Brouwer]] |
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*[[Otto Hermann Künneth]] |
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*[[Samuel Eilenberg]] |
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*[[J.A. Zilber]] |
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*[[Heinz Hopf]] |
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*[[Saunders Mac Lane]] |
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*[[J. H. C. Whitehead]] |
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*[[Witold Hurewicz]] |
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*[[Egbert van Kampen]] |
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*[[Daniel Quillen]] |
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*[[Dennis Sullivan]] |
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</div> |
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== Important theorems in algebraic topology == |
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<div style="-moz-column-count:3; column-count:3;"> |
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*[[Borsuk-Ulam theorem]] |
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*[[Brouwer fixed point theorem]] |
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*[[Cellular_approximation|Cellular approximation theorem]] |
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*[[Eilenberg–Zilber theorem]] |
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*[[Hurewicz theorem]] |
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*[[Kunneth theorem]] |
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*[[Poincaré duality|Poincaré duality theorem]] |
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*[[Universal coefficient theorem]] |
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*[[Van Kampen's theorem]] |
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*[[Whitehead's theorem]] |
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</div> |
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==See also== |
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* [[List of publications in mathematics#Algebraic topology|Important publications in algebraic topology]] |
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== Further reading == |
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*{{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}}. |
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==References== |
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{{commonscat|Algebraic topology}} |
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*{{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}}. |
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*{{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. |
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*{{citation| last=Maunder |first=C.R.F. |title=Algebraic Topology |year=1970 |publisher= Van Nostrand Reinhold |place=London |isbn=0-486-69131-4}}. |
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{{Mathematics-footer}} |
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[[Category:Topology]] |
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[[Category:Algebraic topology| ]] |
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[[Category:Abstract algebra]] |
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[[ca:Topologia algebraica]] |
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[[de:Algebraische Topologie]] |
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[[es:Topología algebraica]] |
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[[eo:Algebra topologio]] |
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[[fa:توپولوژی جبری]] |
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[[fr:Topologie algébrique]] |
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[[it:Topologia algebrica]] |
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[[he:טופולוגיה אלגברית]] |
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[[ka:ალგებრული ტოპოლოგია]] |
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[[nl:Algebraïsche topologie]] |
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[[pl:Topologia algebraiczna]] |
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[[pt:Topologia algébrica]] |
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[[ru:Алгебраическая топология]] |
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[[sr:Алгебарска топологија]] |
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[[fi:Algebrallinen topologia]] |
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[[vi:Tô pô đại số]] |
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[[zh:代数拓扑]] |
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Revision as of 03:34, 27 March 2009
This branch of math does not exist.