Plücker formula - Revision history

en>Jumpow: "Double point" is not "Fixed point". Redirected to "Double points"

2014-12-20T17:29:12Z

"Double point" is not "Fixed point". Redirected to "Double points"

← Older revision		Revision as of 19:29, 20 December 2014
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	~~In mathematics,~~ the '~~''Whitehead product''' is a [[Graded Lie algebra\|graded]] [[quasi-Lie algebra]] structure on the [[homotopy group]]s of a space. It was defined by [[J. H. C. Whitehead]] in {{harv\|Whitehead\|1941}}~~.		Alyson Meagher is the title her mothers and fathers gave her but she doesn't like when people use her complete name. The favorite hobby for him and his children is to perform lacross and he would never give it up. Since he was 18 he's been operating as an info officer but he plans on altering it. Alaska is where I've always been residing.<br><br>My web blog - online psychic ([http://www.sirudang.com/siroo_Notice/2110 you can try here])

	~~== Definition ==~~
	~~Given elements <math>f \in \pi_k(X), g \in \pi_l(X)</math>, the '''Whitehead bracket'''~~

	~~:<math>[f,g] \in \pi_{k+l-1}(X) \, </math>~~

	~~is defined as follows:~~

	The ~~product <math>S^k \times S^l</math> can be obtained by attaching a <math>(k+l)</math>-cell to the [[wedge sum]]~~

	~~:<math>S^k \vee S^l</math>;~~

	~~the [[attaching map]]~~ is ~~a map~~

	~~:<math>S^{k+l-1} \~~to ~~S^k \vee S^l~~. ~~\,</math>~~

	~~Represent <math>f</math> and <math>g</math> by maps~~

	~~:<math>f\colon S^k \to X \, </math>~~

	~~and~~
	~~:<math>g\colon S^l \to X, \, </math>~~

	~~then compose their wedge with the attaching map,~~ as

	~~:<math>S^{k+l-1} \to S^k \vee S^l \to X \, </math>~~

	~~The [[homotopy class]] of the resulting map does not depend~~ on ~~the choices of representatives, and thus one obtains a well-defined element of~~

	~~:<math>\pi_{k+l-1}(X)~~. ~~\, </math>~~

	~~==Grading==~~
	~~Note that there~~ is ~~a shift of 1 in the grading (compared to the indexing of [[homotopy group]]s), so <math>\pi_k(X)</math> has degree <math>(k-1)</math>; equivalently, <math>L_k = \pi_{k+1}(X)</math> (setting~~ '~~'L'' to be the graded quasi-Lie algebra)~~. ~~Thus~~ <~~math~~>~~L_0 = \pi_1(X)~~<~~/math~~> ~~acts on each graded component.~~

	~~==Properties==~~
	The Whitehead product is bilinear, graded-symmetric, and satisfies the [[graded Lie algebra\|graded Jacobi identity]], and is thus a [[Graded Lie algebra\|graded]] [[quasi-Lie algebra]]; this is proven in {{harvtxt\|Uehara\|Massey\|1957}} via the [[Massey product\|Massey triple product]].
	~~<!-~~- (~~I don't know any example where <math>~~[~~f,f]\neq 0</math>.) -->~~

	~~If <math>f \in \pi_1(X)</math>, then the Whitehead bracket is related to the usual conjugation action of <math>\pi_1</math> on <math>\pi_k</math> by~~

	:~~<math>[f,g]=g^f-g, \, </math>~~

	~~where <math>g^f</math> denotes the [[Inner automorphism\|conjugation]] of <math>g<~~/~~math> by <math>f<~~/~~math>~~.
	~~For <math>k=1</math>, this reduces to~~

	~~:<math>[f,g]=fgf^{-1}g^{-1}, \,</math>~~

	~~which is the usual commutator.~~

	~~The relevant [[Mathematics Subject Classification\|MSC]] code is: 55Q15, Whitehead products and generalizations~~.

	~~==See also==~~
	* [[Generalised Whitehead product]]
	* [[Massey product]]
	* [[Toda bracket]]

	~~==References==~~
	*{{citation\|mr=0091473
	~~\|last=Uehara\|first= Hiroshi\|last2= Massey\|first2= W. S.~~
	~~\|authorlink2=William Massey~~
	\|chapter=The Jacobi identity for Whitehead products\|title= Algebraic geometry and topology. A symposium in honor of S. Lefschetz\|pages=361–377\|publisher= [[Princeton University Press]]\|publication-place= Princeton, N. J.,\|year= 1957}}
	* {{citation
	~~\|first=George W.~~
	~~\|last=Whitehead~~
	~~\|authorlink=George W. Whitehead~~
	~~\|title=On products in homotopy groups~~
	~~\|journal=[[Annals of Mathematics]]~~
	~~\|series=2~~
	~~\|volume=47~~
	~~\|issue=3~~
	~~\|date=July 1946~~
	~~\|pages=460–475~~
	~~\|doi=10.2307~~/~~1969085~~
	~~\|jstor=1969085}}~~
	* {{citation
	~~\|first=J. H. C.~~
	~~\|last=Whitehead~~
	~~\|authorlink=J. H. C. Whitehead~~
	~~\|title=On adding relations to homotopy groups~~
	~~\|journal=[[Annals of Mathematics]]~~
	~~\|series=2~~
	~~\|volume=42~~
	~~\|issue=2~~
	~~\|date=April 1941~~
	~~\|pages=409–428~~
	~~\|doi=10.2307~~/~~1968907~~
	~~\|jstor=1968907}}~~

	~~[[Category:Homotopy theory]]~~
	~~[[Category:Lie algebras]~~]

en>ChrisGualtieri: General Fixes using AWB

2013-10-23T02:00:35Z

General Fixes using AWB

← Older revision		Revision as of 04:00, 23 October 2013
Line 1:		Line 1:
	The ~~writer~~ is ~~known as Wilber Pegues~~. ~~Some time ago he selected~~ to ~~reside in North Carolina~~ and ~~he doesn't strategy~~ on ~~altering it~~. ~~Office supervising~~ is ~~where my main earnings comes from but I~~'~~ve usually wanted my own business~~. ~~One of~~ the ~~issues she enjoys most~~ is ~~canoeing and she's been performing it for fairly~~ a ~~whilst~~.<~~br>~~<br>~~my weblog real psychics~~ [~~[http:~~/~~/bigpolis~~.~~com~~/~~blogs~~/~~post~~/~~6503 http~~://~~bigpolis~~.~~com~~/~~blogs~~/~~post~~/~~6503~~]]		In mathematics, the '''Whitehead product''' is a [[Graded Lie algebra\|graded]] [[quasi-Lie algebra]] structure on the [[homotopy group]]s of a space. It was defined by [[J. H. C. Whitehead]] in {{harv\|Whitehead\|1941}}.

			== Definition ==
			Given elements <math>f \in \pi_k(X), g \in \pi_l(X)</math>, the '''Whitehead bracket'''

			:<math>[f,g] \in \pi_{k+l-1}(X) \, </math>

			is defined as follows:

			The product <math>S^k \times S^l</math> can be obtained by attaching a <math>(k+l)</math>-cell to the [[wedge sum]]

			:<math>S^k \vee S^l</math>;

			the [[attaching map]] is a map

			:<math>S^{k+l-1} \to S^k \vee S^l. \,</math>

			Represent <math>f</math> and <math>g</math> by maps

			:<math>f\colon S^k \to X \, </math>

			and
			:<math>g\colon S^l \to X, \, </math>

			then compose their wedge with the attaching map, as

			:<math>S^{k+l-1} \to S^k \vee S^l \to X \, </math>

			The [[homotopy class]] of the resulting map does not depend on the choices of representatives, and thus one obtains a well-defined element of

			:<math>\pi_{k+l-1}(X). \, </math>

			==Grading==
			Note that there is a shift of 1 in the grading (compared to the indexing of [[homotopy group]]s), so <math>\pi_k(X)</math> has degree <math>(k-1)</math>; equivalently, <math>L_k = \pi_{k+1}(X)</math> (setting ''L'' to be the graded quasi-Lie algebra). Thus <math>L_0 = \pi_1(X)</math> acts on each graded component.

			==Properties==
			The Whitehead product is bilinear, graded-symmetric, and satisfies the [[graded Lie algebra\|graded Jacobi identity]], and is thus a [[Graded Lie algebra\|graded]] [[quasi-Lie algebra]]; this is proven in {{harvtxt\|Uehara\|Massey\|1957}} via the [[Massey product\|Massey triple product]].
			<!-- (I don't know any example where <math>[f,f]\neq 0</math>.) -->

			If <math>f \in \pi_1(X)</math>, then the Whitehead bracket is related to the usual conjugation action of <math>\pi_1</math> on <math>\pi_k</math> by

			:<math>[f,g]=g^f-g, \, </math>

			where <math>g^f</math> denotes the [[Inner automorphism\|conjugation]] of <math>g</math> by <math>f</math>.
			For <math>k=1</math>, this reduces to

			:<math>[f,g]=fgf^{-1}g^{-1}, \,</math>

			which is the usual commutator.

			The relevant [[Mathematics Subject Classification\|MSC]] code is: 55Q15, Whitehead products and generalizations.

			==See also==
			* [[Generalised Whitehead product]]
			* [[Massey product]]
			* [[Toda bracket]]

			==References==
			*{{citation\|mr=0091473
			\|last=Uehara\|first= Hiroshi\|last2= Massey\|first2= W. S.
			\|authorlink2=William Massey
			\|chapter=The Jacobi identity for Whitehead products\|title= Algebraic geometry and topology. A symposium in honor of S. Lefschetz\|pages=361–377\|publisher= [[Princeton University Press]]\|publication-place= Princeton, N. J.,\|year= 1957}}
			* {{citation
			\|first=George W.
			\|last=Whitehead
			\|authorlink=George W. Whitehead
			\|title=On products in homotopy groups
			\|journal=[[Annals of Mathematics]]
			\|series=2
			\|volume=47
			\|issue=3
			\|date=July 1946
			\|pages=460–475
			\|doi=10.2307/1969085
			\|jstor=1969085}}
			* {{citation
			\|first=J. H. C.
			\|last=Whitehead
			\|authorlink=J. H. C. Whitehead
			\|title=On adding relations to homotopy groups
			\|journal=[[Annals of Mathematics]]
			\|series=2
			\|volume=42
			\|issue=2
			\|date=April 1941
			\|pages=409–428
			\|doi=10.2307/1968907
			\|jstor=1968907}}

			[[Category:Homotopy theory]]
			[[Category:Lie algebras]]

en>Dirk math at 04:19, 5 August 2012

2012-08-05T04:19:11Z

New page

The writer is known as Wilber Pegues. Some time ago he selected to reside in North Carolina and he doesn't strategy on altering it. Office supervising is where my main earnings comes from but I've usually wanted my own business. One of the issues she enjoys most is canoeing and she's been performing it for fairly a whilst.<br><br>my weblog real psychics [[http://bigpolis.com/blogs/post/6503 http://bigpolis.com/blogs/post/6503]]