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| In [[mathematics]], the '''''J''-homomorphism''' is a mapping from the [[homotopy group]]s of the [[special orthogonal group]]s to the [[homotopy groups of spheres]]. It was defined by {{harvs|txt|authorlink=George W. Whitehead|first=George W.|last=Whitehead|year=1942}}, extending a construction of {{harvtxt|Hopf|1935}}.
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| ==Definition==
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| Whitehead's original homomorphism is defined geometrically, and gives a homomorphism
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| :<math>J \colon \pi_r (\mathrm{SO}(q)) \to \pi_{r+q}(S^q) \,\!</math>
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| of abelian groups for integers ''q'', and ''r'' ≥ 2. (Hopf defined this for the special case ''q''=''r''+1.)
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| The ''J''-homomorphism can be defined as follows.
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| An element of the special orthogonal group SO(''q'') can be regarded as a map
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| :<math>S^{q-1}\rightarrow S^{q-1}</math>
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| and the homotopy group π<sub>''r''</sub>(SO(''q'')) consists of [[homotopy]]-equivalence classes of maps from the ''r''-sphere to SO(''q'').
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| Thus an element of π<sub>''r''</sub>(SO(''q'')) can be represented by a map
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| :<math>S^r\times S^{q-1}\rightarrow S^{q-1}</math>
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| The [[suspension (topology)|suspension]] of this map give a map
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| :<math>S(S^r\times S^{q-1})\rightarrow S^{q}</math>
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| On the other hand, there is a natural map
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| :<math>S^{r+q}\rightarrow S^{r+q}/(S^r\times S^{q-1}) \cong S(S^r\times S^{q-1})</math>
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| which contracts a copy of <math>S^r\times S^{q-1}</math> inside <math>S^{r+q}</math> to a point, and composing these gives a map
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| :<math>S^{r+q}\rightarrow S^{q}</math>
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| in π<sub>''r''+''q''</sub>(''S''<sup>''q''</sup>), which Whitehead defined as the image of the element of π<sub>''r''</sub>(SO(''q'')) under the J-homomorphism.
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| The stable ''J''-homomorphism in [[stable homotopy theory]] gives a homomorphism
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| :<math> J \colon \pi_r(\mathrm{SO}) \to \pi_r^S , \,\!</math>
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| where SO is the infinite [[special orthogonal group]], and the right-hand side is the ''r''-th [[stable stem]] of the [[stable homotopy groups of spheres]]. | |
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| ==Image of the J-homomorphism==
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| The image of the ''J''-homomorphism was described by {{harvtxt|Adams|1966}}, assuming the '''Adams conjecture''' of {{harvtxt|Adams|1963}} which was proved by {{harvtxt|Quillen|1971}}, as follows.
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| The group π<sub>''r''</sub>(SO) is given by [[Bott periodicity]]. It is always cyclic; and if ''r'' is positive, it is of order 2 if ''r'' is 0 or 1 mod 8, infinite if ''r'' is 3 mod 4, and order 1 otherwise {{harv|Switzer|1975|p=488}}. In particular the image of the stable ''J''-homomorphism is cyclic. The stable homotopy groups π<sub>''r''</sub><sup>''S''</sup> are the direct sum of the (cyclic) image of the ''J''-homomorphism, and the kernel of the Adams e-invariant {{harv|Adams|1966}}, a homomorphism from the stable homotopy groups to '''Q'''/'''Z'''. The order of the image is 2 if ''r'' is 0 or 1 mod 8 and positive (so in this case the ''J''-homomorphism is injective). If ''r'' = 4''n''−1 is 3 mod 4 and positive the image is a cyclic group of order equal to the denominator of ''B''<sub>2''n''</sub>/4''n'', where ''B''<sub>2''n''</sub> is a [[Bernoulli number]]. In the remaining cases where ''r'' is 2, 4, 5, or 6 mod 8 the image is trivial because π<sub>''r''</sub>(SO) is trivial.
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| :{| class="wikitable" style="text-align: center; background-color:white"
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| ! style="text-align:right" | π<sub>''r''</sub>(SO)
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| | 1 || 2 || 1 || '''Z''' || 1 || 1 || 1 || '''Z''' || 2 || 2 || 1 || '''Z''' || 1 || 1 || 1 || '''Z''' || 2 || 2
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| |-
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| ! style="text-align:right" | |im(''J'')|
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| | 1 || 2 || 1 || 24 || 1 || 1 || 1 || 240 || 2 || 2 || 1 || 504 || 1 || 1 || 1 || 480 || 2 || 2
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| ! style="text-align:right" | π<sub>''r''</sub><sup>''S''</sup>
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| | '''Z''' || 2 || 2 || 24 || 1 || 1 || 2 || 240 || 2<sup>2</sup> || 2<sup>3</sup> || 6 || 504 || 1 || 3 || 2<sup>2</sup> || 480×2 || 2<sup>2</sup> || 2<sup>4</sup>
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| ! style="text-align:right" | ''B''<sub>2''n''</sub>
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| | || || || <sup>1</sup>⁄<sub>6</sub> || || || || −<sup>1</sup>⁄<sub>30</sub> || || || || <sup>1</sup>⁄<sub>42</sub> || || || || −<sup>1</sup>⁄<sub>30</sub> || ||
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| |}
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| ==Applications==
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| {{harvtxt|Atiyah|1961}} introduced the group ''J''(''X'') of a space ''X'', which for ''X'' a sphere is the image of the ''J''-homomorphism in a suitable dimension.
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| The [[cokernel]] of the ''J''-homomorphism appears in the group of [[exotic sphere]]s ({{harvtxt|Kosinski |1992}}).
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| ==References==
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| *{{Citation | last1=Atiyah | first1=Michael Francis | author1-link=Michael Atiyah | title=Thom complexes | doi=10.1112/plms/s3-11.1.291 | mr=0131880 | year=1961 | journal=Proceedings of the London Mathematical Society. Third Series | issn=0024-6115 | volume=11 | pages=291–310}}
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| *{{citation|first=J. F. |last=Adams|title=On the groups J(X) I|journal= Topology |volume=2|year=1963|doi=10.1016/0040-9383(63)90001-6|pages=181|issue=3 }}
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| *{{citation|first=J. F. |last=Adams|title=On the groups J(X) II|journal= Topology |volume=3|year=1965a|doi=10.1016/0040-9383(65)90040-6|pages=137|issue=2 }}
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| *{{citation|first=J. F. |last=Adams|title=On the groups J(X) III|journal= Topology |volume=3|year=1965b|doi=10.1016/0040-9383(65)90054-6|pages=193|issue=3 }}
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| *{{citation|first=J. F. |last=Adams|title=On the groups J(X) IV|journal= Topology |volume=5|year=1966|doi= 10.1016/0040-9383(66)90004-8|pages=21 }} {{citation|title= Correction|journal= Topology |volume=7|year=1968|doi= 10.1016/0040-9383(68)90010-4|author= Adams, J|pages= 331|issue= 3 }}
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| *{{Citation | last1=Hopf | first1=Heinz | author1-link=Heinz Hopf | title=Über die Abbildungen von Sphären auf Sphäre niedrigerer Dimension | url=http://matwbn.icm.edu.pl/tresc.php?wyd=1&tom=25 | year=1935 | journal=[[Fundamenta Mathematicae]] | issn=0016-2736 | volume=25 | pages=427–440}}
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| *{{Citation |author=Kosinski, Antoni A. |title=Differential Manifolds|publisher=Academic Press |location=San Diego, CA |year=1992 |pages= 195ff|isbn=0-12-421850-4}}
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| *{{citation|first=John W.|last= Milnor |title=Differential topology forty-six years later|journal= [[Notices of the American Mathematical Society]] |volume=58|year=2011|issue= 6 |pages=804–809|url=http://www.ams.org/notices/201106/rtx110600804p.pdf}}
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| *{{Citation | last1=Quillen | first1=Daniel | author1-link=Daniel Quillen | title=The Adams conjecture | doi=10.1016/0040-9383(71)90018-8 | mr=0279804 | year=1971 | journal=[[Topology (journal)|Topology. an International Journal of Mathematics]] | issn=0040-9383 | volume=10 | pages=67–80}}
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| *{{citation|first=Robert M. |last=Switzer |title=Algebraic Topology—Homotopy and Homology |publisher=[[Springer-Verlag]] |year=1975 |isbn=978-0-387-06758-2}}
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| *{{Citation | last1=Whitehead | first1=George W. | title=On the homotopy groups of spheres and rotation groups | jstor=1968956 | mr=0007107 | year=1942 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=43 | pages=634–640 | issue=4 | doi=10.2307/1968956}}
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| * {{Citation |author=Whitehead, George W. |title=Elements of homotopy theory |publisher=Springer |location=Berlin |year=1978 |pages= |isbn=0-387-90336-4 |doi=|mr= 0516508 }}
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| [[Category:Homotopy theory]]
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| [[Category:Topology of Lie groups]]
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