Editing Algebraic topology

{{short description|Branch of mathematics}}
{{For|the topology of pointwise convergence|Algebraic topology (object)}}
[[Image:Torus.png|right|thumb|250px|A [[torus]], one of the most frequently studied objects in algebraic topology]]
'''Algebraic topology''' is a branch of [[mathematics]] that 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]] [[homeomorphism]], though usually most classify up to [[Homotopy#Homotopy equivalence and null-homotopy|homotopy 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 any [[subgroup]] of a [[free group]] is again a free group.

==Main branches==
Below are some of the main areas studied in algebraic topology:

===Homotopy groups===
{{Main| Homotopy group}}
In mathematics, homotopy groups are used in algebraic topology to classify [[topological space]]s. The first and simplest homotopy group is the [[fundamental group]], which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space.

===Homology===
{{Main|Homology (mathematics)|l1=Homology}}
In algebraic topology and [[abstract algebra]], '''homology''' (in part from [[Greek language|Greek]] ὁμός ''homos'' "identical") is a certain general procedure to associate a [[sequence]] of [[abelian group]]s or [[module (mathematics)|modules]] with a given mathematical object such as a [[topological space]] or a [[group (mathematics)|group]].<ref>{{harvtxt|Fraleigh|1976|p=163}}</ref>

===Cohomology===
{{Main|Cohomology}}
In [[homology theory]] and algebraic topology, '''cohomology''' is a general term for a [[sequence]] of [[abelian group]]s defined from a [[chain complex|cochain complex]]. That is, cohomology is defined as the abstract study of '''cochains''', [[chain complex|cocycle]]s, and [[coboundary|coboundaries]]. Cohomology can be viewed as a method of assigning [[algebraic invariant]]s to a topological space that has a more refined [[algebraic structure]] than does [[homology (mathematics)|homology]]. Cohomology arises from the algebraic dualization of the construction of homology. In less abstract language, cochains in the fundamental sense should assign "quantities" to the ''[[chain (algebraic topology)|chains]]'' of homology theory.

===Manifolds===
{{Main|Manifold}}
A '''manifold''' is a [[topological space]] that near each point resembles [[Euclidean space]]. Examples include the [[Plane (geometry)|plane]], the [[sphere]], and the [[torus]], which can all be realized in three dimensions, but also the [[Klein bottle]] and [[real 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 example [[Poincaré duality]].

===Knot theory===
{{Main|Knot theory}}
'''Knot theory''' is the study of [[knot (mathematics)|mathematical knot]]s. 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 an [[embedding]] of a [[circle]] in three-dimensional [[Euclidean space]], <math>\mathbb{R}^3</math>. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of <math>\mathbb{R}^3</math> upon itself (known as an [[ambient isotopy]]); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.

===Complexes===
{{Main|Simplicial complex|CW complex}}
[[File:Simplicial complex example.svg|thumb|200px|A simplicial 3-complex.]]
A '''simplicial complex''' is a [[topological space]] of a certain kind, constructed by "gluing together" [[Point (geometry)|point]]s, [[line segment]]s, [[triangle]]s, and their [[Simplex|''n''-dimensional counterparts]] (see illustration). Simplicial complexes should not be confused with the more abstract notion of a [[simplicial set]] appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an [[abstract simplicial complex]].

A '''CW complex''' is a type of topological space introduced by [[J. H. C. Whitehead]] to meet the needs of [[homotopy theory]]. This class of spaces is broader and has some better [[category theory|categorical]] properties than [[simplicial complex]]es, but still retains a combinatorial nature that allows for computation (often with a much smaller complex).

==Method of algebraic invariants==
An older name for the subject was [[combinatorial topology]], implying an emphasis on how a space X was constructed from simpler ones<ref>{{citation|title=Invitation to Combinatorial Topology| first1=Maurice|last1=Fréchet|author-link1=Maurice René Fréchet| first2=Ky|last2=Fan|author-link2=Ky Fan|publisher=Courier Dover Publications|year=2012|isbn=9780486147888|page=101|url=https://books.google.com/books?id=dfLSzs0vHNQC&pg=PA101}}.</ref> (the modern standard tool for such construction is the [[CW complex]]). In the 1920s and 1930s, there was growing emphasis on investigating topological spaces by finding correspondences from them to algebraic [[group (mathematics)|groups]], which led to the change of name to algebraic topology.<ref>{{citation|title=A Combinatorial Introduction to Topology|first=Michael|last=Henle|publisher=Courier Dover Publications|year=1994|isbn=9780486679662|page=221|url=https://books.google.com/books?id=S0RJYkBxowEC&pg=PA221}}.</ref> The combinatorial topology name is still sometimes used to emphasize an algorithmic approach based on decomposition of spaces.<ref>{{citation|title=Blowups, slicings and permutation groups in combinatorial topology|first=Jonathan|last=Spreer|publisher=Logos Verlag Berlin GmbH|year=2011|isbn=9783832529833|page=23|url=https://books.google.com/books?id=vdt5c07k0M4C&pg=PA23}}.</ref>

In the algebraic approach, one finds a correspondence between spaces and [[group (mathematics)|groups]] 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 have a great deal of manageable structure, often making these statements easier to prove. 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]].

Homology and cohomology groups, on the other hand, are abelian and in many important cases finitely generated. [[Finitely generated abelian group]]s are completely classified and are particularly easy to work with.

==Setting in category theory==
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.

One of the first mathematicians to work with different types of cohomology was [[Georges de Rham]]. One can use the differential structure of [[differentiable manifold|smooth manifolds]] via [[de Rham cohomology]], or [[Čech cohomology|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 [[Samuel Eilenberg]] and [[Norman Steenrod]] generalized this approach. They defined homology and cohomology as [[functor]]s equipped with [[natural transformation]]s subject to certain axioms (e.g., a [[weak equivalence (homotopy theory)|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==
Classic applications of algebraic topology include:
* The [[Brouwer fixed point theorem]]: every [[continuous function|continuous]] map from the unit [[Disk (mathematics)|''n''-disk]] to itself has a fixed point.
* The free rank of the ''n''th homology group of a [[simplicial complex]] is the ''n''th [[Betti number]], which allows one to calculate the [[Euler–Poincaré characteristic]].
* One can use the differential structure of [[differentiable manifold|smooth manifolds]] via [[de Rham cohomology]], or [[Čech cohomology|Čech]] or [[sheaf cohomology]] to investigate the solvability of [[differential equation]]s defined on the manifold in question.
* A manifold is [[orientable]] when the top-dimensional integral homology group is the integers, and is non-orientable when it is 0.
* The [[n-sphere|''n''-sphere]] admits a nowhere-vanishing continuous unit [[vector fields on spheres|vector field]] if and only if ''n'' is odd. (For ''n''&nbsp;=&nbsp;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 known proof is topological. Namely, any free group ''G'' may be realized as the fundamental group of a [[Graph (discrete 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. On the other hand, this type of application is also handled more simply by the use of covering morphisms of [[groupoids]], and that technique has yielded subgroup theorems not yet proved by methods of algebraic topology; see {{harvtxt|Higgins|1971}}.
* [[Topological combinatorics]].

==Notable people==
{{columns-list|colwidth=20em|style=width: 600px;|
*[[Frank Adams]]
*[[Michael Atiyah]]
*[[Enrico Betti]]
*[[Armand Borel]]
*[[Karol Borsuk]]
*[[Raoul Bott]]
*[[Luitzen Egbertus Jan Brouwer]]
*[[William Browder (mathematician)|William Browder]]
*[[Ronald Brown (mathematician)|Ronald Brown]]
*[[Henri Cartan]]
*[[Shiing-Shen Chern]]
*[[Albrecht Dold]]
*[[Charles Ehresmann]]
*[[Samuel Eilenberg]]
*[[Hans Freudenthal]]
*[[Peter Freyd]]
*[[Pierre Gabriel]]
*[[Israel Gelfand]]
*[[Alexander Grothendieck]]
*[[Allen Hatcher]]
*[[Friedrich Hirzebruch]]
*[[Heinz Hopf]]
*[[Michael J. Hopkins]]
*[[Witold Hurewicz]]
*[[Egbert van Kampen]]
*[[Daniel Kan]]
*[[Hermann Künneth]]
*[[Ruth Lawrence]]
*[[Solomon Lefschetz]]
*[[Jean Leray]]
*[[Saunders Mac Lane]]
*[[Mark Mahowald]]
*[[J. Peter May]]
*[[Barry Mazur]]
*[[John Milnor]]
*[[John Coleman Moore]]
*[[Jack Morava]]
*[[Joseph Neisendorfer]]
*[[Emmy Noether]]
*[[Sergei Novikov (mathematician)|Sergei Novikov]]
*[[Grigori Perelman]]
*[[Henri Poincaré]]
*[[Lev Pontryagin]]
*[[Nicolae Popescu]]
*[[Mikhail Postnikov]]
*[[Daniel Quillen]]
*[[Jean-Pierre Serre]]
*[[Isadore Singer]]
*[[Stephen Smale]]
*[[Edwin Spanier]]
*[[Norman Steenrod]]
*[[Dennis Sullivan]]
*[[René Thom]]
*[[Hiroshi Toda]]
*[[Leopold Vietoris]]
*[[Hassler Whitney]]
*[[J. H. C. Whitehead]]
*[[Gordon Thomas Whyburn]]
}}

==Important theorems==
{{columns-list|colwidth=20em|style=width: 600px;|
*[[Blakers–Massey theorem]]
*[[Borsuk–Ulam theorem]]
*[[Brouwer fixed point theorem]]
*[[Cellular approximation theorem]]
*[[Dold–Thom theorem]]
*[[Eilenberg–Ganea theorem]]
*[[Eilenberg–Zilber theorem]]
*[[Freudenthal suspension theorem]]
*[[Hurewicz theorem]]
*[[Künneth theorem]]
*[[Lefschetz fixed-point theorem]]
*[[Leray–Hirsch theorem]]
*[[Poincaré duality|Poincaré duality theorem]]
*[[Seifert–van Kampen theorem]]
*[[Universal coefficient theorem]]
*[[Whitehead theorem]]
}}

==See also==
{{columns-list|colwidth=20em|style=width: 600px;|
* [[Algebraic K-theory]]
* [[Exact sequence]]
* [[Glossary of algebraic topology]]
* [[Grothendieck topology]]
* [[Higher category theory]]
* [[Higher-dimensional algebra]]
* [[Homological algebra]]
* [[K-theory]]
* [[Lie algebroid]]
* [[Lie groupoid]]
* [[Serre spectral sequence]]
* [[Sheaf (mathematics)|Sheaf]]
* [[TQFT|Topological quantum field theory]]
}}

==Notes==
{{Reflist}}

==References==
{{Commons category|Algebraic topology}}
{{Wikiquote}}
*{{citation |first=Dylan G. L. |last=Allegretti |url=http://www.math.uchicago.edu/~may/VIGRE/VIGREREU2008.html |title=Simplicial Sets and van Kampen's Theorem |year=2008}} ''(Discusses generalized versions of van Kampen's theorem applied to topological spaces and simplicial sets).''
*{{citation |last=Bredon |first=Glen E. |author-link=Glen Bredon|title=Topology and Geometry |year=1993 |publisher=Springer |series=Graduate Texts in Mathematics |volume=139 |url=https://books.google.com/books?id=G74V6UzL_PUC |isbn=0-387-97926-3}}.
*{{citation |first=R. |last=Brown |author-link=Ronald Brown (mathematician) |url=http://www.bangor.ac.uk/r.brown/hdaweb2.html |title=Higher dimensional group theory |year=2007 |access-date=2022-08-17 |archive-date=2016-05-14 |archive-url=http://arquivo.pt/wayback/20160514115207/http://www.bangor.ac.uk/r.brown/hdaweb2.html |url-status=dead }} ''(Gives a broad view of higher-dimensional van Kampen theorems involving multiple groupoids)''.
*{{citation |first1=R. |last1=Brown |first2=A. |last2=Razak |title=A van Kampen theorem for unions of non-connected spaces |journal=Arch. Math. |volume=42 |pages=85–88 |year=1984 |doi=10.1007/BF01198133 |s2cid=122228464 }}. "Gives a general theorem on the [[fundamental groupoid]] with a set of base points of a space which is the union of open sets."
*{{citation |first1=R. |last1=Brown |first2=K. |last2=Hardie |first3=H. |last3=Kamps |first4=T. |last4=Porter |title=The homotopy double groupoid of a Hausdorff space |journal=Theory Appl. Categories |volume=10 |issue=2 |pages=71–93 |year=2002 |url=http://www.emis.de/journals/TAC/volumes/10/2/10-02abs.html}}.
*{{citation |first1=R. |last1=Brown |first2=P.J. |last2=Higgins |title=On the connection between the second relative homotopy groups of some related spaces |journal=Proc. London Math. Soc. |volume=S3-36 |issue= 2|pages=193–212 |year=1978 |doi=10.1112/plms/s3-36.2.193 }}. "The first 2-dimensional version of van Kampen's theorem." 
*{{citation |first1=Ronald |last1=Brown |first2=Philip J. |last2=Higgins |first3=Rafael |last3=Sivera |title=Nonabelian Algebraic Topology: Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids |url=http://www.bangor.ac.uk/~mas010/nonab-a-t.html |year=2011 |publisher=European Mathematical Society |isbn=978-3-03719-083-8 |series=European Mathematical Society Tracts in Mathematics |volume=15|archive-url=https://web.archive.org/web/20090604050453/http://www.bangor.ac.uk/~mas010/nonab-a-t.html |archive-date=2009-06-04 |arxiv=math/0407275 }} This provides a homotopy theoretic approach to basic algebraic topology, without needing a basis in [[singular homology]], or the method of simplicial approximation. It contains a lot of material on [[crossed module]]s.
* {{ citation | first1 = John B. | last1 = Fraleigh | year = 1976 | isbn = 0-201-01984-1 | title = A First Course In Abstract Algebra | edition = 2nd | publisher = [[Addison-Wesley]] | location = Reading }}
*{{citation |author-link1=Marvin Greenberg |last1=Greenberg |first1=Marvin J. |first2=John R. |last2=Harper |title=Algebraic Topology: A First Course, Revised edition |year=1981 |publisher=Westview/Perseus |series=Mathematics Lecture Note Series |isbn=9780805335576 |url-access=registration |url=https://archive.org/details/algebraictopolog00gree_0 }}. A functorial, algebraic approach originally by Greenberg with geometric flavoring added by Harper. 
*{{citation| last=Hatcher |first= Allen |author-link=Allen Hatcher| title=Algebraic Topology |url=http://pi.math.cornell.edu/~hatcher/AT/ATpage.html |year= 2002 |publisher=Cambridge University Press |place=Cambridge |isbn=0-521-79540-0}}. A modern, geometrically flavoured introduction to algebraic topology.
*{{citation |first=Philip J. |last=Higgins |title=Notes on categories and groupoids |url=https://books.google.com/books?id=IqdIAAAAMAAJ |year=1971 |publisher=Van Nostrand Reinhold |isbn=9780442034061}}
*{{citation| last=Maunder |first=C. R. F. |title=Algebraic Topology |journal=Nature |year=1970 |volume=227 |issue=5259 |page=756 |publisher= Van Nostrand Reinhold |doi=10.1038/227756a0 |bibcode=1970Natur.227..756F |place=London |isbn=0-486-69131-4}}.
*{{citation |author-link=Tammo tom Dieck |first=Tammo |last=tom Dieck |title=Algebraic Topology |url=https://books.google.com/books?id=ruSqmB7LWOcC |year=2008 |publisher=European Mathematical Society |isbn=978-3-03719-048-7 |series=EMS Textbooks in Mathematics}}
* {{citation|first=Egbert|last= van Kampen|author-link=Egbert van Kampen| title=On the connection between the fundamental groups of some related spaces| journal=[[American Journal of Mathematics]]|volume= 55 |year=1933 |issue= 1|pages=261–7 |jstor=51000091}}

== Further reading ==
* {{cite book|first=Allen|last= Hatcher|author-link=Allen Hatcher| url=http://pi.math.cornell.edu/~hatcher/AT/ATpage.html |title=Algebraic topology|year=2002|publisher= [[Cambridge University Press]]|isbn=0-521-79160-X}} and {{ISBN|0-521-79540-0}}.
* {{Springer |title=Algebraic topology |id=p/a011700}}
* {{cite book |last=May |first=J. Peter |author-link=J. Peter May|title=A Concise Course in Algebraic Topology |year=1999 |publisher=[[University of Chicago Press]] |url=http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf |archive-date=2022-10-09 |url-status=live |access-date=2008-09-27 |name-list-style=vanc }} Section 2.7 provides a category-theoretic presentation of the theorem as a colimit in the category of groupoids.

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[[Category:Algebraic topology| ]]