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In [[mathematics]], the '''cohomology operation''' concept became central to [[algebraic topology]], particularly [[homotopy theory]], from the 1950s onwards, in the shape of the simple definition that if ''F'' is a [[functor]] defining a [[cohomology theory]], then a cohomology operation should be a [[natural transformation]] from ''F'' to itself. Throughout there have been two basic points: | |||
#the operations can be studied by combinatorial means; and | |||
#the effect of the operations is to yield an interesting [[bicommutant]] theory. | |||
The origin of these studies was the work of Pontryagin, Postnikov, and [[Norman Steenrod]], who first defined the [[Pontryagin square]], [[Postnikov square]], and [[Steenrod square]] operations for [[singular cohomology]], in the case of mod 2 coefficients. The combinatorial aspect there arises as a formulation of the failure of a [[natural diagonal]] map, at [[cochain]] level. The general theory of the [[Steenrod algebra]] of operations has been brought into close relation with that of the [[symmetric group]]. | |||
In the [[Adams spectral sequence]] the ''bicommutant'' aspect is implicit in the use of [[Ext functor]]s, the [[derived functor]]s of Hom-functors; if there is a bicommutant aspect, taken over the Steenrod algebra acting, it is only at a ''derived'' level. The convergence is to groups in [[stable homotopy theory]], about which information is hard to come by. This connection established the deep interest of the cohomology operations for [[homotopy theory]], and has been a research topic ever since. An [[extraordinary cohomology theory]] has its own cohomology operations, and these may exhibit a richer set on constraints. | |||
== | ==Formal definition== | ||
A '''cohomology operation''' <math>\theta</math> of type | |||
:<math>(n,q,\pi,G)\,</math> | |||
is a [[natural transformation]] of functors | |||
:<math>\theta:H^{n}(-,\pi)\to H^{q}(-,G)\,</math> | |||
defined on [[CW complex]]es. | |||
==Relation to Eilenberg-MacLane spaces== | |||
Cohomology of CW complexes is [[representable functor|representable]] by an [[Eilenberg-MacLane space]], so by the [[Yoneda lemma]] a cohomology operation of type <math>(n,q,\pi,G)</math> is given by a [[homotopy]] class of maps <math>K(\pi,n) \to K(G,q)</math>. Using [[representable functor|representability]] once again, the cohomology operation is given by an element of <math>H^{q}(K(\pi,n),G)</math>. | |||
Symbolically, letting <math>[A,B]</math> denote the set of homotopy classes of maps from <math>A</math> to <math>B</math>, | |||
::<math>\displaystyle\mathrm{Nat}(H^n(-,\pi),H^q(-,G)) = \mathrm{Nat}([-,K(\pi,n)],[-,K(G,q)]) = [K(\pi,n),K(G,q)] = H^q(K(\pi,n);G)</math> | |||
==See also== | |||
*[[Secondary cohomology operation]] | |||
==References== | |||
*{{Citation | last1=Mosher | first1=Robert E. | last2=Tangora | first2=Martin C. | title=Cohomology operations and applications in homotopy theory | origyear=1968 | url=http://books.google.com/books?id=FFCaPwAACAAJ | publisher=[[Dover Publications]] | location=New York | isbn=978-0-486-46664-4 | mr=0226634 | year=2008}} | |||
*{{Citation | last1=Steenrod | first1=N. E. | editor1-last=Epstein | editor1-first=D. B. A. | editor1-link=David B. A. Epstein | title=Cohomology operations | url=http://books.google.com/books?id=CF3bt4oYZ2oC | publisher=[[Princeton University Press]] | series= Annals of Mathematics Studies | isbn=978-0-691-07924-0 | mr=0145525 | year=1962 | volume=50}} | |||
{{DEFAULTSORT:Cohomology Operation}} | |||
[[Category:Algebraic topology]] | |||
Revision as of 15:13, 24 November 2013
In mathematics, the cohomology operation concept became central to algebraic topology, particularly homotopy theory, from the 1950s onwards, in the shape of the simple definition that if F is a functor defining a cohomology theory, then a cohomology operation should be a natural transformation from F to itself. Throughout there have been two basic points:
- the operations can be studied by combinatorial means; and
- the effect of the operations is to yield an interesting bicommutant theory.
The origin of these studies was the work of Pontryagin, Postnikov, and Norman Steenrod, who first defined the Pontryagin square, Postnikov square, and Steenrod square operations for singular cohomology, in the case of mod 2 coefficients. The combinatorial aspect there arises as a formulation of the failure of a natural diagonal map, at cochain level. The general theory of the Steenrod algebra of operations has been brought into close relation with that of the symmetric group.
In the Adams spectral sequence the bicommutant aspect is implicit in the use of Ext functors, the derived functors of Hom-functors; if there is a bicommutant aspect, taken over the Steenrod algebra acting, it is only at a derived level. The convergence is to groups in stable homotopy theory, about which information is hard to come by. This connection established the deep interest of the cohomology operations for homotopy theory, and has been a research topic ever since. An extraordinary cohomology theory has its own cohomology operations, and these may exhibit a richer set on constraints.
Formal definition
A cohomology operation of type
is a natural transformation of functors
defined on CW complexes.
Relation to Eilenberg-MacLane spaces
Cohomology of CW complexes is representable by an Eilenberg-MacLane space, so by the Yoneda lemma a cohomology operation of type is given by a homotopy class of maps . Using representability once again, the cohomology operation is given by an element of .
Symbolically, letting denote the set of homotopy classes of maps from to ,
See also
References
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Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
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15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
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