Stokes radius - Revision history

en>Monkbot: Task 5: Fix CS1 deprecated coauthor parameter errors

2014-07-24T15:22:59Z

Task 5: Fix CS1 deprecated coauthor parameter errors

← Older revision		Revision as of 17:22, 24 July 2014
Line 1:		Line 1:
	~~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}}.~~		The name of the writer is Jayson. I am an invoicing officer and I'll be promoted soon. Mississippi is where her house is but her spouse wants them to move. I am really fond of to go to karaoke but I've been taking on new issues recently.<br><br>my web site online psychic chat ([http://www.publicpledge.com/blogs/post/7034 visit the following website])

	~~==Definition==~~

	~~Whitehead's original homomorphism is defined geometrically,~~ and ~~gives a homomorphism~~

	~~:<math>J \colon \pi_r (\mathrm{SO}(q)) \to \pi_{r+q}(S^q) \,\!</math>~~

	~~of abelian groups for integers ''q'', and ''r'' ≥ 2. (Hopf defined this for the special case ''q''=''r''+1.)~~

	~~The ''J''-homomorphism can be defined as follows.~~
	~~An element of the special orthogonal group SO(''q'~~'~~) can~~ be ~~regarded as a map~~
	~~:<math>S^{q-1}\rightarrow S^{q-1}</math>~~
	~~and the homotopy group π<sub>''r''</sub>(SO(''q'')) consists of [[homotopy]]-equivalence classes of maps from the ''r''-sphere to SO(''q'')~~.
	~~Thus an element of π<sub>''r''</sub>(SO(''q'')) can be represented by a map~~
	~~:<math>S^r\times S^{q-1}\rightarrow S^{q-1}</math>~~
	~~The [[suspension (topology)\|suspension]] of this map give a map~~
	~~:<math>S(S^r\times S^{q-1})\rightarrow S^{q}</math>~~
	~~On the other hand, there~~ is ~~a natural map~~
	~~:<math>S^{r+q}\rightarrow S^{r+q}/(S^r\times S^{q-1}) \cong S(S^r\times S^{q-1})</math>~~
	~~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~~
	~~:<math>S^{r+q}\rightarrow S^{q}</math>~~
	~~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.~~

	~~The stable ''J''-homomorphism in [[stable homotopy theory]] gives a homomorphism~~

	~~:<math> J \colon \pi_r(\mathrm{SO}) \to \pi_r^S , \,\!</math>~~

	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]]~~.

	~~==Image of the J-homomorphism==~~

	~~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.~~
	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~~.

	~~:{\| class="wikitable" style="text-align: center; background-color:white"~~
	\|-
	~~! style="text-align:right;width:10%" \| r~~
	~~! style="width:5%" \| 0~~
	~~! style="width:5%" \| 1~~
	~~! style="width:5%" \| 2~~
	~~! style="width:5%" \| 3~~
	~~! style="width:5%" \| 4~~
	~~! style="width:5%" \| 5~~
	~~! style="width:5%" \| 6~~
	~~! style="width:5%" \| 7~~
	~~! style="width:5%" \| 8~~
	~~! style="width:5%" \| 9~~
	~~! style="width:5%" \| 10~~
	~~! style="width:5%" \| 11~~
	~~! style="width:5%" \| 12~~
	~~! style="width:5%" \| 13~~
	~~! style="width:5%" \| 14~~
	~~! style="width:5%" \| 15~~
	~~! style="width:5%" \| 16~~
	~~! style="width:5%" \| 17~~
	\|-
	~~! style="text-align:right" \| π<sub>''r''</sub>(SO)~~
	~~\| 1 \|\| 2 \|\| 1 \|\| '''Z''' \|\| 1 \|\| 1 \|\| 1 \|\| '''Z''' \|\| 2 \|\| 2 \|\| 1 \|\| '''Z''' \|\| 1 \|\| 1 \|\| 1 \|\| '''Z''' \|\| 2 \|\| 2~~
	\|-
	~~! style="text-align:right" \| \|im(''J'')\|~~
	~~\| 1 \|\| 2 \|\| 1 \|\| 24 \|\| 1 \|\| 1 \|\| 1 \|\| 240 \|\| 2 \|\| 2 \|\| 1 \|\| 504 \|\| 1 \|\| 1 \|\| 1 \|\| 480 \|\| 2 \|\| 2~~
	\|-
	~~! style="text-align:right" \| π<sub>''r''</sub><sup>''S''</sup>~~
	~~\| '''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>~~
	\|-
	~~! style="text-align:right" \| ''B''<sub>2''n''~~<~~/sub~~>
	~~\| \|\| \|\| \|\|~~ <~~sup~~>~~1</sup>⁄<sub>6</sub> \|\| \|\| \|\| \|\| −<sup>1</sup>⁄<sub>30</sub> \|\| \|\| \|\| \|\| <sup>1</sup>⁄<sub>42</sub> \|\| \|\| \|\| \|\| −<sup>1</sup>⁄<sub>30</sub> \|\| \|\|~~
	\|}

	~~==Applications==~~

	~~{{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.~~

	~~The [[cokernel]] of the ''J''-homomorphism appears in the group of [[exotic sphere]]s ({{harvtxt\|Kosinski \|1992}}).~~

	~~==References==~~
	*{{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}}
	*{{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 }}~~
	*{{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 }}
	*{{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 }}
	*{{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 }}
	*{{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}}~~
	*{{Citation \|author=Kosinski, Antoni A. \|title=Differential Manifolds\|publisher=Academic Press \|location=San Diego, CA \|year=1992 \|pages= 195ff\|isbn=0-12-421850-4}}
	*{{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}}~~
	*{{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}}
	*{{citation\|first=Robert M. \|last=Switzer \|title=Algebraic Topology—Homotopy and Homology \|publisher=[[Springer-Verlag]] \|year=1975 \|isbn=978-0-387-06758-2}}
	*{{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}}
	* {{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 }}

	~~[[Category:Homotopy theory]]~~
	~~[[Category:Topology of Lie groups]~~]

en>BG19bot: WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB (9774)

2013-12-10T07:26:54Z

WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB (9774)

← Older revision		Revision as of 09:26, 10 December 2013
Line 1:		Line 1:
	~~The author~~ is ~~called Wilber Pegues~~. ~~Since he~~ was ~~18 he~~'s ~~been working~~ as ~~an info officer but he plans on changing it~~. ~~For~~ a ~~while I~~'~~ve been~~ in ~~Alaska but I will have to move~~ in a yr or ~~two~~. It'~~s not~~ a ~~typical factor but what I like doing~~ is to ~~climb but I don~~'~~t have~~ the ~~time lately~~.<br><br>~~My web blog~~ :: ~~psychic love readings~~ ([http://~~mybrandcp~~.~~com~~/xe/~~board_XmDx25~~/~~107997 mybrandcp~~.~~com~~])		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}}.

			==Definition==

			Whitehead's original homomorphism is defined geometrically, and gives a homomorphism

			:<math>J \colon \pi_r (\mathrm{SO}(q)) \to \pi_{r+q}(S^q) \,\!</math>

			of abelian groups for integers ''q'', and ''r'' ≥ 2. (Hopf defined this for the special case ''q''=''r''+1.)

			The ''J''-homomorphism can be defined as follows.
			An element of the special orthogonal group SO(''q'') can be regarded as a map
			:<math>S^{q-1}\rightarrow S^{q-1}</math>
			and the homotopy group π<sub>''r''</sub>(SO(''q'')) consists of [[homotopy]]-equivalence classes of maps from the ''r''-sphere to SO(''q'').
			Thus an element of π<sub>''r''</sub>(SO(''q'')) can be represented by a map
			:<math>S^r\times S^{q-1}\rightarrow S^{q-1}</math>
			The [[suspension (topology)\|suspension]] of this map give a map
			:<math>S(S^r\times S^{q-1})\rightarrow S^{q}</math>
			On the other hand, there is a natural map
			:<math>S^{r+q}\rightarrow S^{r+q}/(S^r\times S^{q-1}) \cong S(S^r\times S^{q-1})</math>
			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
			:<math>S^{r+q}\rightarrow S^{q}</math>
			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.

			The stable ''J''-homomorphism in [[stable homotopy theory]] gives a homomorphism

			:<math> J \colon \pi_r(\mathrm{SO}) \to \pi_r^S , \,\!</math>

			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]].

			==Image of the J-homomorphism==

			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.
			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.

			:{\| class="wikitable" style="text-align: center; background-color:white"
			\|-
			! style="text-align:right;width:10%" \| r
			! style="width:5%" \| 0
			! style="width:5%" \| 1
			! style="width:5%" \| 2
			! style="width:5%" \| 3
			! style="width:5%" \| 4
			! style="width:5%" \| 5
			! style="width:5%" \| 6
			! style="width:5%" \| 7
			! style="width:5%" \| 8
			! style="width:5%" \| 9
			! style="width:5%" \| 10
			! style="width:5%" \| 11
			! style="width:5%" \| 12
			! style="width:5%" \| 13
			! style="width:5%" \| 14
			! style="width:5%" \| 15
			! style="width:5%" \| 16
			! style="width:5%" \| 17
			\|-
			! style="text-align:right" \| π<sub>''r''</sub>(SO)
			\| 1 \|\| 2 \|\| 1 \|\| '''Z''' \|\| 1 \|\| 1 \|\| 1 \|\| '''Z''' \|\| 2 \|\| 2 \|\| 1 \|\| '''Z''' \|\| 1 \|\| 1 \|\| 1 \|\| '''Z''' \|\| 2 \|\| 2
			\|-
			! style="text-align:right" \| \|im(''J'')\|
			\| 1 \|\| 2 \|\| 1 \|\| 24 \|\| 1 \|\| 1 \|\| 1 \|\| 240 \|\| 2 \|\| 2 \|\| 1 \|\| 504 \|\| 1 \|\| 1 \|\| 1 \|\| 480 \|\| 2 \|\| 2
			\|-
			! style="text-align:right" \| π<sub>''r''</sub><sup>''S''</sup>
			\| '''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>
			\|-
			! style="text-align:right" \| ''B''<sub>2''n''</sub>
			\| \|\| \|\| \|\| <sup>1</sup>⁄<sub>6</sub> \|\| \|\| \|\| \|\| −<sup>1</sup>⁄<sub>30</sub> \|\| \|\| \|\| \|\| <sup>1</sup>⁄<sub>42</sub> \|\| \|\| \|\| \|\| −<sup>1</sup>⁄<sub>30</sub> \|\| \|\|
			\|}

			==Applications==

			{{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.

			The [[cokernel]] of the ''J''-homomorphism appears in the group of [[exotic sphere]]s ({{harvtxt\|Kosinski \|1992}}).

			==References==
			*{{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}}
			*{{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 }}
			*{{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 }}
			*{{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 }}
			*{{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 }}
			*{{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}}
			*{{Citation \|author=Kosinski, Antoni A. \|title=Differential Manifolds\|publisher=Academic Press \|location=San Diego, CA \|year=1992 \|pages= 195ff\|isbn=0-12-421850-4}}
			*{{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}}
			*{{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}}
			*{{citation\|first=Robert M. \|last=Switzer \|title=Algebraic Topology—Homotopy and Homology \|publisher=[[Springer-Verlag]] \|year=1975 \|isbn=978-0-387-06758-2}}
			*{{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}}
			* {{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 }}

			[[Category:Homotopy theory]]
			[[Category:Topology of Lie groups]]

130.63.133.30: Fixed grammar.

2012-07-24T20:29:08Z

Fixed grammar.

New page

The author is called Wilber Pegues. Since he was 18 he's been working as an info officer but he plans on changing it. For a while I've been in Alaska but I will have to move in a yr or two. It's not a typical factor but what I like doing is to climb but I don't have the time lately.<br><br>My web blog :: psychic love readings ([http://mybrandcp.com/xe/board_XmDx25/107997 mybrandcp.com])

← Older revision		Revision as of 17:22, 24 July 2014
Line 1:		Line 1:
	~~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}}.~~		The name of the writer is Jayson. I am an invoicing officer and I'll be promoted soon. Mississippi is where her house is but her spouse wants them to move. I am really fond of to go to karaoke but I've been taking on new issues recently.<br><br>my web site online psychic chat ([http://www.publicpledge.com/blogs/post/7034 visit the following website])

	~~==Definition==~~

	~~Whitehead's original homomorphism is defined geometrically,~~ and ~~gives a homomorphism~~

	~~:<math>J \colon \pi_r (\mathrm{SO}(q)) \to \pi_{r+q}(S^q) \,\!</math>~~

	~~of abelian groups for integers ''q'', and ''r'' ≥ 2. (Hopf defined this for the special case ''q''=''r''+1.)~~

	~~The ''J''-homomorphism can be defined as follows.~~
	~~An element of the special orthogonal group SO(''q'~~'~~) can~~ be ~~regarded as a map~~
	~~:<math>S^{q-1}\rightarrow S^{q-1}</math>~~
	~~and the homotopy group π<sub>''r''</sub>(SO(''q'')) consists of [[homotopy]]-equivalence classes of maps from the ''r''-sphere to SO(''q'')~~.
	~~Thus an element of π<sub>''r''</sub>(SO(''q'')) can be represented by a map~~
	~~:<math>S^r\times S^{q-1}\rightarrow S^{q-1}</math>~~
	~~The [[suspension (topology)\|suspension]] of this map give a map~~
	~~:<math>S(S^r\times S^{q-1})\rightarrow S^{q}</math>~~
	~~On the other hand, there~~ is ~~a natural map~~
	~~:<math>S^{r+q}\rightarrow S^{r+q}/(S^r\times S^{q-1}) \cong S(S^r\times S^{q-1})</math>~~
	~~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~~
	~~:<math>S^{r+q}\rightarrow S^{q}</math>~~
	~~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.~~

	~~The stable ''J''-homomorphism in [[stable homotopy theory]] gives a homomorphism~~

	~~:<math> J \colon \pi_r(\mathrm{SO}) \to \pi_r^S , \,\!</math>~~

	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]]~~.

	~~==Image of the J-homomorphism==~~

	~~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.~~
	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~~.

	~~:{\| class="wikitable" style="text-align: center; background-color:white"~~
	\|-
	~~! style="text-align:right;width:10%" \| r~~
	~~! style="width:5%" \| 0~~
	~~! style="width:5%" \| 1~~
	~~! style="width:5%" \| 2~~
	~~! style="width:5%" \| 3~~
	~~! style="width:5%" \| 4~~
	~~! style="width:5%" \| 5~~
	~~! style="width:5%" \| 6~~
	~~! style="width:5%" \| 7~~
	~~! style="width:5%" \| 8~~
	~~! style="width:5%" \| 9~~
	~~! style="width:5%" \| 10~~
	~~! style="width:5%" \| 11~~
	~~! style="width:5%" \| 12~~
	~~! style="width:5%" \| 13~~
	~~! style="width:5%" \| 14~~
	~~! style="width:5%" \| 15~~
	~~! style="width:5%" \| 16~~
	~~! style="width:5%" \| 17~~
	\|-
	~~! style="text-align:right" \| π<sub>''r''</sub>(SO)~~
	~~\| 1 \|\| 2 \|\| 1 \|\| '''Z''' \|\| 1 \|\| 1 \|\| 1 \|\| '''Z''' \|\| 2 \|\| 2 \|\| 1 \|\| '''Z''' \|\| 1 \|\| 1 \|\| 1 \|\| '''Z''' \|\| 2 \|\| 2~~
	\|-
	~~! style="text-align:right" \| \|im(''J'')\|~~
	~~\| 1 \|\| 2 \|\| 1 \|\| 24 \|\| 1 \|\| 1 \|\| 1 \|\| 240 \|\| 2 \|\| 2 \|\| 1 \|\| 504 \|\| 1 \|\| 1 \|\| 1 \|\| 480 \|\| 2 \|\| 2~~
	\|-
	~~! style="text-align:right" \| π<sub>''r''</sub><sup>''S''</sup>~~
	~~\| '''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>~~
	\|-
	~~! style="text-align:right" \| ''B''<sub>2''n''~~<~~/sub~~>
	~~\| \|\| \|\| \|\|~~ <~~sup~~>~~1</sup>⁄<sub>6</sub> \|\| \|\| \|\| \|\| −<sup>1</sup>⁄<sub>30</sub> \|\| \|\| \|\| \|\| <sup>1</sup>⁄<sub>42</sub> \|\| \|\| \|\| \|\| −<sup>1</sup>⁄<sub>30</sub> \|\| \|\|~~
	\|}

	~~==Applications==~~

	~~{{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.~~

	~~The [[cokernel]] of the ''J''-homomorphism appears in the group of [[exotic sphere]]s ({{harvtxt\|Kosinski \|1992}}).~~

	~~==References==~~
	*{{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}}
	*{{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 }}~~
	*{{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 }}
	*{{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 }}
	*{{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 }}
	*{{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}}~~
	*{{Citation \|author=Kosinski, Antoni A. \|title=Differential Manifolds\|publisher=Academic Press \|location=San Diego, CA \|year=1992 \|pages= 195ff\|isbn=0-12-421850-4}}
	*{{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}}~~
	*{{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}}
	*{{citation\|first=Robert M. \|last=Switzer \|title=Algebraic Topology—Homotopy and Homology \|publisher=[[Springer-Verlag]] \|year=1975 \|isbn=978-0-387-06758-2}}
	*{{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}}
	* {{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 }}

	~~[[Category:Homotopy theory]]~~
	~~[[Category:Topology of Lie groups]~~]

← Older revision		Revision as of 09:26, 10 December 2013
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	~~The author~~ is ~~called Wilber Pegues~~. ~~Since he~~ was ~~18 he~~'s ~~been working~~ as ~~an info officer but he plans on changing it~~. ~~For~~ a ~~while I~~'~~ve been~~ in ~~Alaska but I will have to move~~ in a yr or ~~two~~. It'~~s not~~ a ~~typical factor but what I like doing~~ is to ~~climb but I don~~'~~t have~~ the ~~time lately~~.<br><br>~~My web blog~~ :: ~~psychic love readings~~ ([http://~~mybrandcp~~.~~com~~/xe/~~board_XmDx25~~/~~107997 mybrandcp~~.~~com~~])		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}}.

			==Definition==

			Whitehead's original homomorphism is defined geometrically, and gives a homomorphism

			:<math>J \colon \pi_r (\mathrm{SO}(q)) \to \pi_{r+q}(S^q) \,\!</math>

			of abelian groups for integers ''q'', and ''r'' ≥ 2. (Hopf defined this for the special case ''q''=''r''+1.)

			The ''J''-homomorphism can be defined as follows.
			An element of the special orthogonal group SO(''q'') can be regarded as a map
			:<math>S^{q-1}\rightarrow S^{q-1}</math>
			and the homotopy group π<sub>''r''</sub>(SO(''q'')) consists of [[homotopy]]-equivalence classes of maps from the ''r''-sphere to SO(''q'').
			Thus an element of π<sub>''r''</sub>(SO(''q'')) can be represented by a map
			:<math>S^r\times S^{q-1}\rightarrow S^{q-1}</math>
			The [[suspension (topology)\|suspension]] of this map give a map
			:<math>S(S^r\times S^{q-1})\rightarrow S^{q}</math>
			On the other hand, there is a natural map
			:<math>S^{r+q}\rightarrow S^{r+q}/(S^r\times S^{q-1}) \cong S(S^r\times S^{q-1})</math>
			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
			:<math>S^{r+q}\rightarrow S^{q}</math>
			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.

			The stable ''J''-homomorphism in [[stable homotopy theory]] gives a homomorphism

			:<math> J \colon \pi_r(\mathrm{SO}) \to \pi_r^S , \,\!</math>

			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]].

			==Image of the J-homomorphism==

			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.
			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.

			:{\| class="wikitable" style="text-align: center; background-color:white"
			\|-
			! style="text-align:right;width:10%" \| r
			! style="width:5%" \| 0
			! style="width:5%" \| 1
			! style="width:5%" \| 2
			! style="width:5%" \| 3
			! style="width:5%" \| 4
			! style="width:5%" \| 5
			! style="width:5%" \| 6
			! style="width:5%" \| 7
			! style="width:5%" \| 8
			! style="width:5%" \| 9
			! style="width:5%" \| 10
			! style="width:5%" \| 11
			! style="width:5%" \| 12
			! style="width:5%" \| 13
			! style="width:5%" \| 14
			! style="width:5%" \| 15
			! style="width:5%" \| 16
			! style="width:5%" \| 17
			\|-
			! style="text-align:right" \| π<sub>''r''</sub>(SO)
			\| 1 \|\| 2 \|\| 1 \|\| '''Z''' \|\| 1 \|\| 1 \|\| 1 \|\| '''Z''' \|\| 2 \|\| 2 \|\| 1 \|\| '''Z''' \|\| 1 \|\| 1 \|\| 1 \|\| '''Z''' \|\| 2 \|\| 2
			\|-
			! style="text-align:right" \| \|im(''J'')\|
			\| 1 \|\| 2 \|\| 1 \|\| 24 \|\| 1 \|\| 1 \|\| 1 \|\| 240 \|\| 2 \|\| 2 \|\| 1 \|\| 504 \|\| 1 \|\| 1 \|\| 1 \|\| 480 \|\| 2 \|\| 2
			\|-
			! style="text-align:right" \| π<sub>''r''</sub><sup>''S''</sup>
			\| '''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>
			\|-
			! style="text-align:right" \| ''B''<sub>2''n''</sub>
			\| \|\| \|\| \|\| <sup>1</sup>⁄<sub>6</sub> \|\| \|\| \|\| \|\| −<sup>1</sup>⁄<sub>30</sub> \|\| \|\| \|\| \|\| <sup>1</sup>⁄<sub>42</sub> \|\| \|\| \|\| \|\| −<sup>1</sup>⁄<sub>30</sub> \|\| \|\|
			\|}

			==Applications==

			{{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.

			The [[cokernel]] of the ''J''-homomorphism appears in the group of [[exotic sphere]]s ({{harvtxt\|Kosinski \|1992}}).

			==References==
			*{{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}}
			*{{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 }}
			*{{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 }}
			*{{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 }}
			*{{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 }}
			*{{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}}
			*{{Citation \|author=Kosinski, Antoni A. \|title=Differential Manifolds\|publisher=Academic Press \|location=San Diego, CA \|year=1992 \|pages= 195ff\|isbn=0-12-421850-4}}
			*{{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}}
			*{{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}}
			*{{citation\|first=Robert M. \|last=Switzer \|title=Algebraic Topology—Homotopy and Homology \|publisher=[[Springer-Verlag]] \|year=1975 \|isbn=978-0-387-06758-2}}
			*{{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}}
			* {{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 }}

			[[Category:Homotopy theory]]
			[[Category:Topology of Lie groups]]