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In [[mathematics]], an '''Eilenberg–MacLane space'''<ref group="note">[[Saunders Mac Lane]] originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name. (See e.g. {{MR|13312}} In this context it is therefore conventional to write the name without a space.</ref> is a special kind of [[topological space]] that can be regarded as a building block for [[homotopy theory]]. These spaces are important in many contexts in [[algebraic topology]], including constructions of spaces, computations of [[homotopy group]]s of spheres, and definition of [[cohomology operation]]s. The name is for [[Samuel Eilenberg]] and [[Saunders Mac Lane]], who introduced such spaces in the late 1940s.
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Let ''G'' be a group and ''n'' a positive integer. A connected topological space ''X'' is called an Eilenberg–MacLane space of type ''K''(''G'', ''n''), if it has ''n''-th [[homotopy group]] π<sub>''n''</sub>(''X'') isomorphic to ''G'' and all other homotopy groups trivial. If ''n'' > 1 then ''G'' must be abelian. Then an Eilenberg–MacLane space exists, as a [[CW-complex]], and is unique up to a [[weak homotopy equivalence]]. By abuse of language, any such space is often called just ''K''(''G'', ''n'').
 
==Examples==
* The [[unit circle]] '''S'''<sup>1</sup> is a ''K''('''Z''',1).
* The infinite-dimensional [[complex projective space]] '''P'''<sup>∞</sup>('''C''') is a model of ''K''('''Z''',2). This is one of the rare examples of classifying spaces admitting a [[manifold]] model, and is also the [[topological space]] the [[homotopy group]]s of which satisfy π<sub>''i''</sub>&nbsp;=&nbsp;0 for ''i''&nbsp;=&nbsp;1 and ''i''&nbsp;>&nbsp;2, while π<sub>2</sub> = '''Z'''. Its [[cohomology ring]] is '''Z'''[''x''], namely the free polynomial ring on a single 2-dimensional generator ''x''&nbsp;∈&nbsp;''H''<sup>2</sup>.  The generator can be represented in [[de Rham cohomology]] by the [[Fubini–Study]] [[2-form]]. An application of ''K''('''Z''',2) is described at [[Abstract nonsense]].
* The infinite-dimensional [[real projective space]] '''P'''<sup>∞</sup>('''R''') is a ''K''('''Z'''<sub>2</sub>, 1).
* The [[wedge sum]] of ''k'' [[unit circle]]s <math>\textstyle\bigvee_{i=1}^k\mathbf{S}^1</math> is a ''K''(''G'', ''1'') for ''G'' the [[free group]] on ''k'' generators.
* The complement to any knot in a 3-dimensional sphere '''S'''<sup>3</sup> is of type ''K''(''G'', 1); this is called the "[[Aspherical space|asphericity]] of knots", and is a 1957 theorem of [[Christos Papakyriakopoulos]].<ref>{{Harv|Papakyriakopoulos|1957}}</ref>
 
Some further elementary examples can be constructed from these by using the obvious fact that the product ''K''(''G'', ''n'') × ''K''(''H'', ''n'') is ''K''(''G'' × ''H'', ''n'').
 
A ''K''(''G'', ''n'') can be constructed stage-by-stage, as a [[CW complex]], starting with a [[wedge sum|wedge]] of ''n''-[[sphere]]s, one for each generator of the group ''G'', and adding cells in (possibly infinite number of) higher dimensions so as to kill all extra homotopy.
 
==Properties of Eilenberg–MacLane spaces==
An important property of ''K''(''G'', ''n'') is that, for any abelian group ''G'', and any CW-complex ''X'', the set
 
:[''X'', ''K''(''G'', ''n'')]
 
of homotopy classes of maps from ''X'' to ''K''(''G'', ''n'') is in natural bijection with the ''n''-th [[singular homology|singular cohomology group]]
 
:''H''<sup>''n''</sup>(''X''; ''G'')
 
of the space ''X''. Thus one says that the ''K''(''G'', ''n'') are [[representing space]]s for cohomology with coefficients in ''G''. Since
 
:<math>H^n(K(G,n);G) = \mathrm{Hom}(H_n(K(G,n);\mathbf{Z}), G) = \mathrm{Hom}(\pi_n(K(G,n)), G) = \mathrm{Hom}(G,G),</math>
 
there is a distinguished element <math>u \in H^n(K(G,n);G)</math> corresponding to the identity. The above bijection is given by pullback of that element — <math> f \mapsto f^*u </math>.
 
Another version of this result, due to Peter J. Huber, establishes a bijection with the ''n''-th [[Čech cohomology|&#268;ech cohomology group]] when ''X'' is [[Hausdorff space|Hausdorff]] and [[paracompact]] and ''G'' is countable, or when ''X'' is Hausdorff, paracompact and [[compactly generated space|compactly generated]] and ''G'' is arbitrary. A further result of Morita establishes a bijection with the ''n''-th [[Čech cohomology|numerable &#268;ech cohomology group]] for an arbitrary topological space ''X'' and ''G'' an arbitrary abelian group.
 
It follows from the [[universal coefficient theorem]] for cohomology that the Eilenberg MacLane space is a ''quasi-functor'' of the group; that is, for each positive integer <math>n</math> if <math> a: G \to G' </math> is any homomorphism of Abelian groups, then there is a non-empty set  
 
<math>K(a,n) = \{[f]: f: K(G,n) \to K(G',n), H_n(f) = a\},</math>
 
satisfying <math>K(a \circ b,n) \supset K(a,n) \circ K(b,n) \mbox{  and  } 1 \in K(1,n), </math>
where <math>[f]</math> denotes the homotopy class of a continuous map <math>f</math> and <math> S \circ T := \{s \circ t: s \in S, t \in T \}.</math>
 
 
 
Every CW-complex possesses a [[Postnikov tower]], that is, it is homotopy equivalent to an iterated fibration with fibers the Eilenberg–MacLane spaces.  
 
There is a method due to [[Jean-Pierre Serre]] which allows one, at least theoretically, to compute homotopy groups of spaces using a [[spectral sequence]] for special fibrations with Eilenberg–MacLane spaces for fibers.
 
The cohomology groups of Eilenberg–MacLane spaces can be used to classify all [[cohomology operation]]s.
 
==See also==
*[[Brown representability theorem]], regarding representation spaces
*[[Moore space (algebraic topology)|Moore space]], the homology analogue.
 
==Notes==
{{reflist|group="note"}}
{{reflist}}
 
==References==
*S. Eilenberg,   S. MacLane,   Relations between homology and homotopy groups of spaces  Ann. of Math.  46  (1945)  pp.&nbsp;480–509
*S. Eilenberg,  S. MacLane,  Relations between homology and homotopy groups of spaces. II  Ann. of Math.  51  (1950)  pp.&nbsp;514–533
*Peter J. Huber (1961), Homotopical cohomology and Čech cohomology, ''[[Mathematische Annalen]]'' '''144 ''', 73&ndash;76.
*{{cite journal | last1 = Morita | first1 = Kiiti | author-separator =, | author-name-separator= | year = 1975 | title = &#268;ech cohomology and covering dimension for topological spaces | url = | journal = [[Fundamenta Mathematicae]] | volume = 87 | issue = | pages = 31–52 }}
* {{cite journal
  | last = Papakyriakopoulos
  | first = C. D.
  | authorlink = Christos Papakyriakopoulos
  | coauthors =
  | title = On Dehn's Lemma and the Asphericity of Knots
  | journal = Proc. Nat. Acad. Sci. USA
  | volume = 43
  | issue = 1
  | pages = 169–172
  | publisher =
  | year = 1957
  | pmid = 16589993 
  | pmc = 528404
  | doi = 10.1073/pnas.43.1.169
  }}
* {{cite journal
  | last = Papakyriakopoulos
  | first = C. D.
  | authorlink = Christos Papakyriakopoulos
  | coauthors =
  | title = On Dehn's Lemma and the Asphericity of Knots
  | journal = [[Annals of Mathematics|Ann. Math.]]
  | volume = 66
  | issue = 1
  | pages = 1–26
  | publisher =
  | year = 1957
  | jstor = 1970113
  | doi = 10.2307/1970113 
}}
*{{springer|title=Eilenberg−MacLane space|id=E/e035200|first=Yu.B.|last= Rudyak}}
 
{{DEFAULTSORT:Eilenberg-MacLane space}}
[[Category:Algebraic topology]]
[[Category:Homotopy theory]]

Latest revision as of 12:24, 9 January 2015

SINGAPORE, 28 APRIL 2014 – PropertyGuru , Singapore's #1 real property portal by most variety of page views and property listings, is happy to supply its latest opinion piece. This market view is the mixed output effort of PropertyGuru's research workforce culling the newest developments from it knowledge-box repository.

All 2000s-designed flats are designed to supply full privacy (besides rental blocks constructed since 2007). Punggol is the primary town with NO hall-going through flats Last flats with rooms facing to corridor have been leased in 2005. First BTO challenge was completed in 2005 and acquired lease from 2006, so all BTOs supply full privacy. If you want floorplan of a sure block , attempt a Google search with road or precinct name (reasonably than block number). Look on www.renotalk.com or other forums, the place individuals submit their flooring plans asking for renovation ideas. Sengkang, blk 297-299 (Compassvale Inexperienced, 2001), Premium Executive Apartment, one of the craziest-proportioned HDB flats, ENORMOUS lounge and tiny bedrooms, no hallways, contrasting with Bukit Batok ones.

Listed under are the Singapore property measures that the government have introduced since 1996. The Government's goal for these cooling measures is to make sure a stable and sustainable property market where costs move in keeping with economic fundamentals. New Singapore Property Guidelines - 10 February 2014 - Rapid Impact Exemption from TDSR Threshold for Refinancing of Proprietor-Occupied Residential Properties MAS has decided to broaden the existing exemption from the TDSR threshold of 60 per cent for such loans to ease the debt servicing burden of these borrowers.

The place else are you able to watch international sport events such because the System One Singapore Grand Prix or the Volvo Ocean Race practically from your doorstep when you get pleasure from Sentosa Island's ' many tourist points of interest and the beach , resorts and motels and a on line casino. Together with your office set within the new monetary district in Marina Bay commuting time is reduced to a few minutes across the bay and a short drive let's you to set off from Singapore's Changi Airport to any destination on the earth.

Almost each business runs via contacts and networks. For an agent the requirement of his enterprise is to be able to set up and preserve these contacts to grasp which of the areas are at present thriving with properties to be offered or rented out. An agent would possibly be able to break a take care of other brokers as well to have the ability to shut a sale. Narrowing down on the property that fits your finances is just not an easy activity. Except the property secured is in a primary location, the amount of appreciation might be lesser.

To Serve Singapore Actual Estate Market with all our hearts to do our highest for all clients. Stop Throwing Away your laborious earn cash on rental to landlord! Come seek for an incredible worth among the many least expensive PropertyCondominiumHouses at Singapore New Launch. Return to Batam Property web page from Batam Property for Sale page PERMANENT residents (PRs) now face unprecedented limits on their capability to buy property in Singapore. To steer, help buyers to seek out their dream houses, their investments of selection in new launch property in Singapore and the areas." Roger Seah 谢富帆 (sixty five) 9620 1223 Offer buyers larger selection and choice when it comes time to buying a brand new rental or property developers in singapore list What is restricted residential property under the Residential Property Act Property pricing.

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