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| 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 [[loop (topology)|loop]]s in a [[mathematical space|space]]. Intuitively, homotopy groups record information about the basic shape, or ''holes'', of a topological space.
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| To define the ''n''-th homotopy group, the base point preserving maps from an ''n''-dimensional [[sphere]] (with base point) into a given space (with base point) are collected into [[equivalence class]]es, called '''[[homotopy class]]es.''' Two mappings are '''homotopic''' if one can be continuously deformed into the other. These homotopy classes form a [[group (mathematics)|group]], called the''' ''n''-th homotopy group''', π<sub>''n''</sub>(''X''), of the given space ''X'' with base point. Topological spaces with differing homotopy groups are never equivalent ([[homeomorphic]]), but the converse is not true.
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| The notion of homotopy of [[path (topology)|path]]s was introduced by [[Camille Jordan]].<ref>{{Citation|title=Marie Ennemond Camille Jordan|url=http://www-history.mcs.st-and.ac.uk/~history/Biographies/Jordan.html}}</ref>
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| ==Introduction==
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| In modern mathematics it is common to study a [[category (mathematics)|category]] by [[functor|associating]] to every object of this category a simpler object which still retains a sufficient amount of information about the object in question. Homotopy groups are such a way of associating [[group (mathematics)|group]]s to topological spaces.
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| [[Image:Torus.png|right|thumb|250px|A torus]]
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| [[Image:2sphere 2.png|left|thumb|150px|A [[sphere]]]]
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| That link between topology and groups lets mathematicians apply insights from [[group theory]] to [[topology]]. For example, if two topological objects have different homotopy groups, they can't have the same topological structure—a fact which may be difficult to prove using only topological means. For example, the [[torus]] is different from the [[sphere]]: the torus has a "hole"; the sphere doesn't. However, since continuity (the basic notion of topology) only deals with the local structure, it can be difficult to formally define the obvious global difference. The homotopy groups, however, carry information about the global structure.
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| As for the example: the first homotopy group of the torus ''T'' is
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| :π<sub>1</sub>(''T'')='''Z'''<sup>2</sup>,
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| because the [[universal cover]] of the torus is the [[complex numbers|complex]] plane '''C''', mapping to the torus ''T'' ≅ '''C''' / '''Z'''<sup>2</sup>. Here the quotient is in the category of topological spaces, rather than groups or rings. On the other hand the sphere ''S''<sup>2</sup> satisfies
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| :π<sub>1</sub>(''S''<sup>2</sup>)=0,
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| because every loop can be contracted to a constant map (see [[homotopy groups of spheres]] for this and more complicated examples of homotopy groups).
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| Hence the torus is not [[homeomorphic]] to the sphere.
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| ==Definition==
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| In the [[hypersphere|''n''-sphere]] ''S''<sup>''n''</sup> we choose a base point ''a''. For a space ''X'' with base point ''b'', we define π<sub>''n''</sub>(''X'') to be the set of homotopy classes of maps
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| :''f'' : ''S''<sup>''n''</sup> → ''X''
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| that map the base point ''a'' to the base point ''b''. In particular, the equivalence classes are given by homotopies that are constant on the basepoint of the sphere. Equivalently, we can define π<sub>''n''</sub>(X) to be the group of homotopy classes of maps ''g'' : [0,1]<sup>''n''</sup> → ''X'' from the [[hypercube|''n''-cube]] to ''X'' that take the boundary of the ''n''-cube to ''b''.
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| [[Image:Homotopy group addition.svg|thumb|240px|Composition in the fundamental group]]
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| For ''n'' ≥ 1, the homotopy classes form a [[group (mathematics)|group]]. To define the group operation, recall that in the [[fundamental group]], the product ''f'' * ''g'' of two loops ''f'' and ''g'' is defined by setting
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| : <math>
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| f \ast g =
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| \begin{cases}
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| f(2t) & \text{if } t \in [0,1/2] \\
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| g(2t-1), & \text{if } t \in [1/2,1] | |
| \end{cases}
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| </math>
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| The idea of composition in the fundamental group is that of traveling the first path and the second in succession, or, equivalently, setting their two domains together. The concept of composition that we want for the ''n''-th homotopy group is the same, except that now the domains that we stick together are cubes, and we must glue them along a face. We therefore define the sum of maps ''f'', ''g'' : [0,1]<sup>''n''</sup> → ''X'' by the formula (''f'' + ''g'')(''t''<sub>1</sub>, ''t''<sub>2</sub>, ... ''t''<sub>''n''</sub>) = ''f''(2''t''<sub>1</sub>, ''t''<sub>2</sub>, ... ''t''<sub>''n''</sub>) for ''t''<sub>1</sub> in [0,1/2] and (''f'' + ''g'')(''t''<sub>1</sub>, ''t''<sub>2</sub>, ... ''t''<sub>''n''</sub>) = ''g''(2''t''<sub>1</sub> − 1, ''t''<sub>2</sub>, ... ''t''<sub>''n''</sub>) for ''t''<sub>1</sub> in [1/2,1]. For the corresponding definition in terms of spheres, define the sum ''f'' + ''g'' of maps ''f, g'' : ''S''<sup>''n''</sup> → ''X'' to be Ψ composed with ''h'', where Ψ is the map from ''S''<sup>''n''</sup> to the [[wedge sum]] of two ''n''-spheres that collapses the equator and ''h'' is the map from the wedge sum of two ''n''-spheres to ''X'' that is defined to be ''f'' on the first sphere and ''g'' on the second.
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| If ''n'' ≥ 2, then π<sub>''n''</sub> is [[abelian group|abelian]]. (For a proof of this, note that in two dimensions or greater, two homotopies can be "rotated" around each other. See [[Eckmann–Hilton argument]])
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| It is tempting to try to simplify the definition of homotopy groups by omitting the base points, but this does not usually work for spaces that are not [[Simply connected space|simply connected]], even for path connected spaces. The set of homotopy classes of maps from a sphere to a path connected space is not the homotopy group, but is essentially the set of orbits of the fundamental group on the homotopy group, and in general has no natural group structure.
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| A way out of these difficulties has been found by defining higher homotopy [[groupoids]] of filtered spaces and of ''n''-cubes of spaces. These are related to relative homotopy groups and to ''n''-adic homotopy groups respectively. A higher homotopy van Kampen theorem then enables one to derive some new information on homotopy groups and even on homotopy types. For more background and references, see [http://www.bangor.ac.uk/r.brown/hdaweb2.htm "Higher dimensional group theory"] and the references below.
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| ==Long exact sequence of a fibration==
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| Let ''p'': ''E'' → ''B'' be a basepoint-preserving [[Serre fibration]] with fiber ''F'', that is, a map possessing the [[homotopy lifting property]] with respect to [[CW complex]]es. Suppose that ''B'' is path-connected. Then there is a long [[exact sequence]] of homotopy groups
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| :... → π<sub>''n''</sub>(''F'') → π<sub>''n''</sub>(''E'') → π<sub>''n''</sub>(''B'') → π<sub>''n''−1</sub>(''F'') →... → π<sub>0</sub>(''E'') → 0.
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| Here the maps involving π<sub>0</sub> are not group [[homomorphism]]s because the π<sub>0</sub> are not groups, but they are exact in the sense that the image equals the kernel.
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| Example: the [[Hopf fibration]]. Let ''B'' equal ''S''<sup>2</sup> and ''E'' equal ''S''<sup>3</sup>. Let ''p'' be the [[Hopf fibration]], which has fiber ''S''<sup>1</sup>. From the long exact sequence
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| :⋯ → π<sub>''n''</sub>(''S''<sup>1</sup>) → π<sub>''n''</sub>(''S''<sup>3</sup>) → π<sub>''n''</sub>(''S''<sup>2</sup>) → π<sub>''n''−1</sub>(''S''<sup>1</sup>) → ⋯
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| and the fact that π<sub>''n''</sub>(''S''<sup>1</sup>) = 0 for ''n'' ≥ 2, we find that π<sub>''n''</sub>(''S''<sup>3</sup>) = π<sub>''n''</sub>(''S''<sup>2</sup>) for ''n'' ≥ 3. In particular, π<sub>3</sub>(''S''<sup>2</sup>) = π<sub>3</sub>(''S''<sup>3</sup>) = '''Z'''.
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| In the case of a cover space, when the fiber is discrete, we have that π<sub>''n''</sub>(E) is isomorphic to π<sub>''n''</sub>(B) for all n greater than 1, that π<sub>''n''</sub>(E) embeds injectively into π<sub>''n''</sub>(B) for all positive ''n'', and that the subgroup of π<sub>1</sub>(B) that corresponds to the embedding of π<sub>1</sub>(E) has cosets in bijection with the elements of the fiber.
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| ==Methods of calculation==
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| Calculation of homotopy groups is in general much more difficult than some of the other homotopy [[invariant (mathematics)|invariants]] learned in algebraic topology. Unlike the [[Seifert–van Kampen theorem]] for the fundamental group and the [[Excision theorem]] for [[singular homology]] and [[cohomology]], there is no simple known way to calculate the homotopy groups of a space by breaking it up into smaller spaces. However, methods developed in the 1980s involving a van Kampen type theorem for higher homotopy groupoids have allowed new calculations on homotopy types and so on homotopy groups. See for a sample result the 2008 [http://xxx.soton.ac.uk/abs/0804.3581 paper by Ellis and Mikhailov] listed below.
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| For some spaces, such as [[Torus|tori]], all higher homotopy groups (that is, second and higher homotopy groups) are trivial. These are the so-called [[aspherical space]]s. However, despite intense research in calculating the homotopy groups of spheres, even in two dimensions a complete list is not known. To calculate even the fourth homotopy group of '''S'''<sup>2</sup> one needs much more advanced techniques than the definitions might suggest. In particular the [[Serre spectral sequence]] was constructed for just this purpose.
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| Certain Homotopy groups of [[n-connected]] spaces can be calculated by comparison with [[homology group]]s via the [[Hurewicz theorem]].
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| ==A list of methods for calculating homotopy groups==
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| * The long exact sequence of homotopy groups of a fibration.
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| * [[Hurewicz theorem]], which has several versions.
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| * [[Blakers–Massey theorem]], also known as excision for homotopy groups.
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| * [[Freudenthal suspension theorem]], a corollary of excision for homotopy groups.
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| ==Relative homotopy groups==
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| There are also relative homotopy groups π<sub>''n''</sub>(''X'',''A'') for a pair (''X'',''A''), where ''A'' is a subspace of ''X.'' The elements of such a group are homotopy classes of based maps ''D<sup>n</sup> → X'' which carry the boundary ''S''<sup>''n''−1</sup> into A. Two maps ''f, g'' are called homotopic '''relative to''' ''A'' if they are homotopic by a basepoint-preserving homotopy ''F'' : ''D<sup>n</sup>'' × [0,1] → ''X'' such that, for each ''p'' in ''S''<sup>''n''−1</sup> and ''t'' in [0,1], the element ''F''(''p'',''t'') is in ''A''. The ordinary homotopy groups are the special case in which ''A'' is the base point.
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| These groups are abelian for ''n'' ≥ 3 but for n = 2 form the top group of a [[crossed module]] with bottom group π<sub>''1''</sub>(''A'').
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| There is a long exact sequence of relative homotopy groups.
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| ==Related notions==
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| The homotopy groups are fundamental to [[homotopy theory]], which in turn stimulated the development of [[model category|model categories]]. It is possible to define abstract homotopy groups for [[simplicial set]]s.
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| ==See also==
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| *[[Knot theory]]
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| *[[Homotopy class]]
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| *[[Homotopy groups of spheres]]
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| *[[Topological invariant]]
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| *[[Homotopy group with coefficients]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * {{Citation | last1=Hatcher | first1=Allen | title=Algebraic topology | url=http://www.math.cornell.edu/~hatcher/AT/ATpage.html | publisher=[[Cambridge University Press]] | isbn=978-0-521-79540-1 | year=2002}}
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| * {{springer|title=Homotopy group|id=p/h047930}}
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| * Ronald Brown, `[[Groupoids]] and crossed objects in algebraic topology', [http://www.intlpress.com/HHA//v1/n1/a1/ Homology, homotopy and applications], 1 (1999) 1–78.
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| * G.J. Ellis and R. Mikhailov, `A colimit of classifying spaces', [http://xxx.soton.ac.uk/abs/0804.3581 arXiv:0804.3581v1 [math.GR] ]
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| * R. Brown, P.J. Higgins, R. Sivera, [http://pages.bangor.ac.uk/~mas010/nonab-a-t.html Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids], EMS Tracts in Mathematics Vol. 15, 703 pages. (August 2011).
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| [[Category:Homotopy theory]]
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| [[cs:Homotopická grupa]]
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