Kernel (linear algebra)

In mathematics, an Eilenberg–MacLane space 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 groups of spheres, and definition of cohomology operations. The name is for Samuel Eilenberg and Saunders Mac Lane, who introduced such spaces in the late 1940s.

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_n(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 S¹ is a K(Z,1).
• The infinite-dimensional complex projective space P^∞(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 groups of which satisfy π_i = 0 for i = 1 and i > 2, while π₂ = Z. Its cohomology ring is Z[x], namely the free polynomial ring on a single 2-dimensional generator x ∈ H². 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^∞(R) is a K(Z₂, 1).
• The wedge sum of k unit circles ${\bigvee }_{i=1}^k{\mathbf{S}}^1$ is a K(G, 1) for G the free group on k generators.
• The complement to any knot in a 3-dimensional sphere S³ is of type K(G, 1); this is called the "asphericity of knots", and is a 1957 theorem of Christos Papakyriakopoulos.

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 of n-spheres, 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 cohomology group

Hⁿ(X; G)

of the space X. Thus one says that the K(G, n) are representing spaces for cohomology with coefficients in G. Since

$H^n(K(G,n);G)=\mathrm{H}\mathrm{o}\mathrm{m}(H_n(K(G,n);\mathbf{Z}),G)=\mathrm{H}\mathrm{o}\mathrm{m}({\pi }_n(K(G,n)),G)=\mathrm{H}\mathrm{o}\mathrm{m}(G,G),$

there is a distinguished element $u\in H^n(K(G,n);G)$ corresponding to the identity. The above bijection is given by pullback of that element — $f\mapsto f^{*}u$.

Another version of this result, due to Peter J. Huber, establishes a bijection with the n-th Čech cohomology group when X is Hausdorff and paracompact and G is countable, or when X is Hausdorff, paracompact and compactly generated and G is arbitrary. A further result of Morita establishes a bijection with the n-th numerable Čech 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 $n$ if $a:G\to G^{\prime }$ is any homomorphism of Abelian groups, then there is a non-empty set

$K(a,n)=\{[f]:f:K(G,n)\to K(G^{\prime },n),H_n(f)=a\},$

satisfying $K(a\circ b,n)\supset K(a,n)\circ K(b,n)\text{ and }1\in K(1,n),$ where $[f]$ denotes the homotopy class of a continuous map $f$ and $S\circ T:=\{s\circ t:s\in S,t\in T\}.$

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 operations.

See also

• Brown representability theorem, regarding representation spaces
• Moore space, the homology analogue.

Notes

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References

• S. Eilenberg, S. MacLane, Relations between homology and homotopy groups of spaces Ann. of Math. 46 (1945) pp. 480–509
• S. Eilenberg, S. MacLane, Relations between homology and homotopy groups of spaces. II Ann. of Math. 51 (1950) pp. 514–533
• Peter J. Huber (1961), Homotopical cohomology and Čech cohomology, Mathematische Annalen 144 , 73–76.
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  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
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  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
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