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In [[mathematics]], specifically [[algebraic topology]], the '''cohomology ring''' of a [[topological space]] ''X'' is a [[ring (mathematics)|ring]] formed from the [[cohomology]] groups of ''X'' together with the [[cup product]] serving as the ring multiplication. Here 'cohomology' is usually understood as [[singular cohomology]], but the ring structure is also present in other theories such as [[de Rham cohomology]]. It is also [[functorial]]: for a [[continuous mapping]] of spaces one obtains a [[ring homomorphism]] on cohomology rings, which is contravariant.
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Specifically, given a sequence of cohomology groups ''H''<sup>''k''</sup>(''X'';''R'') on ''X'' with coefficients in a [[commutative ring]] ''R'' (typically ''R'' is '''Z'''<sub>''n''</sub>, '''Z''', '''Q''', '''R''', or '''C''') one can define the [[cup product]], which takes the form
 
:<math>H^k(X;R) \times H^\ell(X;R) \to H^{k+\ell}(X; R).</math>
 
The cup product gives a multiplication on the [[direct sum of modules|direct sum]] of the cohomology groups
 
:<math>H^\bullet(X;R) = \bigoplus_{k\in\mathbb{N}} H^k(X; R).</math>
 
This multiplication turns ''H''<sup>•</sup>(''X'';''R'') into a ring. In fact, it is naturally an '''N'''-[[graded ring]] with the nonnegative integer ''k'' serving as the degree. The cup product respects this grading.
 
The cohomology ring is [[graded-commutative]] in the sense that the cup product commutes up to a sign determined by the grading. Specifically, for pure elements of degree ''k'' and ℓ; we have
 
:<math>(\alpha^k \smile \beta^\ell) = (-1)^{k\ell}(\beta^\ell \smile \alpha^k).</math>
 
A numerical invariant derived from the cohomology ring is the '''cup-length''', which means the maximum number of graded elements of degree ≥ 1 that when multiplied give a non-zero result. For example a [[complex projective space]] has cup-length equal to its [[complex dimension]].
 
== Examples ==
*<math>\operatorname{H}^*(\mathbb{R}P^n; \mathbb{F}_2) = \mathbb{F}_2[\alpha]/(\alpha^{n+1})</math> where <math>|\alpha|=1</math>.
*<math>\operatorname{H}^*(\mathbb{R}P^\infty; \mathbb{F}_2) = \mathbb{F}_2[\alpha]</math> where <math>|\alpha|=1</math>.
*By the Kunneth formula, the mod 2 cohomology ring of ''n'' products of <math>\mathbb{R}P^\infty</math> is a polynomial ring in ''n'' variables with coefficients in <math>\mathbb{F}_2</math>.
 
== See also ==
*[[Quantum cohomology]]
 
==References==
* {{Cite book| first= S. P. |last=Novikov|title=Topology I, General Survey|publisher= Springer-Verlag|year=1996|ISBN= 7-03-016673-6}}
* {{Hatcher AT}}
 
[[Category:Homology theory]]

Latest revision as of 20:42, 20 November 2014

Showcasing you the best up and coming new condo launches in Singapore where you may search and discover the best residence. We have new apartment that can even fulfill the discerning few, it doesn't matter what's your necessities or your lifestyle wants. You'll bound to find the dream apartment right here.

This freehold condominium that's within the centre of Johor Bharu is scheduled to be completed in 2017, and is shortly becoming the discuss of recent residential area within the era the place up-avenue households have gotten the status of the proper life-style amongst citizens of Malaysia and other South East Asian nations. A lifetime of luxury and class is what awaits you at the TriTower Residence @ Johor Bharu The only problem is, nobody is shopping for it. Every other model of pasta sauce on the grocery store has far more sugar and fats than my sauce, and they style way higher. To get people to change to my sauce, I'll have so as to add much more sweeteners (sugar) and taste enhancers (salt). Twenty Myths about Market by Palmer, Tom. 2009, p. 156.

New launch apartment Whitehaven @ Pasir Panjang (former Westvale Condominium website) is nestled in a paradise of greenery within the extremely wanted district of Pasir Panjang. Whitehaven @ Pasir Panjang puts you in close proximity to your next destination in Sentosa, Vivo City, One North, NUS, Worldwide Business Park and plenty of extra. With quick access to main expressways plus Kent Ridge or Haw Par Villa stations just minutes away, residents can easily zip into town or different components of Singapore property Listing with minimal problem. It provides an attention-grabbing constrast of idyllic getaway but superb conveniency. 2/three/4 bedrooms models accessible !

Not to fear, we'll maintain you in our VIP Precedence listing for future new launch VIP Preview. We'll contact you to establish your needs and suggest relevant tasks, both new launch or resale properties that doubtless match your criteria. When you're in search of resale property, resembling those few years previous, or just acquired Non permanent Occupation Allow (PRIME), you may click right here here for fast search and submit your shortlisted listings to us, we'll examine and call you for viewing.

The other precinct would be Woodlands Central the place a 30 hectares area around Woodlands MRT station and Causeway Level mall. It's planned to be a industrial centre and a pedestrian-pleasant retail hub. Turning Woodlands right into a Regional Centre, with the development of Thomson Line, North-South Expressway and more industrial clusters will due to this fact, enhance the provision and demand of houses here. Orange denotes Residential zone (numbers point out plot ratio), and blue denotes Business – the blended use retail/residential improvement that is slated to be built next to City Vista. Some blocks have sea views on the higher flooring! Please contact Benson at 83187083 for particulars. City Vista Price Freehold cluster landed nestled inside exclusive Seletar Hills!

Upon obtained your registration kind (or your transient SMS), we'll name to invite you to our New Launch sales gallery (or present flat) on web site for obligation-free presentation. You will be offered with the brand new venture's truth sheet, comparative market analysis of the close by tasks, current and future improvement surrounding. We understand when you are making a property choice, you are seemingly exploring various options and comparing tasks. We would like to pay attention from you, and match your wants accordingly with initiatives from new launches or resale properties Rest assured your interests shall be well taken care of with our clear strategy. Our aim is to search out you a great house on the proper price.

Now that you've got all these to take into accounts, it is a good suggestion to look up for real property brokers that can assist you in making the perfect motion of investing on the right property as soon as you might be ready. A dependable agent will just be sure you have the proper funds for a new condo launch investing. Without knowing the precise quantity of how much you might be prepared to spend, you may be brief in the long term since you have to make sure that the funds additionally covers the maintenance bills. These bills will regularly enhance 12 months by yr and in case you are investing with no monetary plan, the dream of having your own private residential unit will never be a reality.

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