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| In [[mathematics]], specifically in [[homotopy theory]], a '''classifying space''' ''BG'' of a [[topological group]] ''G'' is the quotient of a [[weakly contractible]] space ''EG'' (i.e. a topological space for which all its [[homotopy group]]s are trivial) by a [[free action]] of ''G''. It has the property that any ''G'' [[principal bundle]] over a [[paracompact]] manifold is isomorphic to a [[pullback bundle|pullback]] of the principal bundle ''EG'' → ''BG''.<ref>{{Citation | last1=Stasheff | first1=James D.|authorlink=Jim Stasheff | title=Algebraic topology (Proc. Sympos. Pure Math., Vol. XXII, Univ. Wisconsin, Madison, Wis., 1970) | publisher=[[American Mathematical Society]] | location=Providence, R.I. | year=1971 | chapter=''H''-spaces and classifying spaces: foundations and recent developments | pages=247–272}}, Theorem 2</ref>
| | It is time to address the slow computer issues whether or not we never learn how. Just because your computer is working thus slow or keeps freezing up; does not signify to not address the problem plus fix it. You may or can not be aware which any computer owner must recognize which there are certain things that a computer requires to keep the greatest performance. The sad fact is that numerous folks whom own a system have no idea which it requires routine repair merely like their cars.<br><br>Before really obtaining the software it's best to check on the firms that create the software. If you may find details found on the form of reputation every business has, maybe the risk of malicious programs can be reduced. Software from reputed firms have aided me, plus other consumers, to make my PC run quicker.. If the product description does not look good to you, does not include details regarding the software, does not include the scan functions, you need to go for another one that ensures you're paying for what you desire.<br><br>Perfect Optimizer also offers to remove junk files plus is totally Windows Vista compatible. Many registry product merely don't have the time plus money to analysis Windows Vista errors. Because perfect optimizer has a big customer base, they do have the time, money and reasons to help completely support Windows Vista.<br><br>Your computer was especially fast when you first bought it. Because a registry was especially clean plus free of mistakes. After time, your computer begins to run slow and freezes up today and then. Because there are errors accumulating inside it and certain info is rewritten or completely deleted by your wrong uninstall of programs, improper operations, malware or different details. That is the reason why the computer performance decreases gradually and become especially unstable.<br><br>In a word, to speed up windows XP, Vista startup, it's quite significant to disable some startup goods and clean plus optimize the registry. You are able to follow the procedures above to disable unwanted programs. To optimize the registry, I suggest you employ a [http://bestregistrycleanerfix.com/tune-up-utilities tuneup utilities] software. Because it is rather dangerous for we to edit the registry by oneself.<br><br>Another key element when you compare registry cleaners is having a facility to manage the start-up tasks. This merely signifies that you can select what programs you want to commence whenever you commence a PC. If you have unwanted programs starting whenever you boot up a PC this might cause a slow running computer.<br><br>To speed up the computer, you merely have to be capable to get rid of all these junk files, allowing a computer to obtain what it wants, whenever it wants. Luckily, there's a tool that allows you to do this conveniently and instantly. It's a tool called a 'registry cleaner'.<br><br>A program and registry cleaner is downloaded from the web. It's convenient to use plus the process refuses to take long. All it does is scan plus then whenever it finds mistakes, it may fix and clean those errors. An error free registry may protect the computer from errors plus provide we a slow PC fix. |
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| For a [[discrete group]] ''G'', ''BG'' is, roughly speaking, a [[connected space|path-connected]] [[topological space]] ''X'' such that the [[fundamental group]] of ''X'' is isomorphic to ''G'' and the higher [[homotopy groups]] of ''X'' are [[trivial group|trivial]], that is, ''BG'' is an [[Eilenberg-Maclane space]], or a ''K(G,1)''.
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| ==Motivation==
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| An example for ''G'' [[infinite cyclic]] is the [[circle]] as ''X''. When ''G'' is a [[discrete group]], another way to specify the condition on ''X'' is that the [[universal cover]] ''Y'' of ''X'' is [[contractible]]. In that case the projection map
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| :<math>\pi: Y\longrightarrow X\ </math>
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| becomes a [[fiber bundle]] with structure group ''G'', in fact a [[principal bundle]] for ''G''. The interest in the classifying space concept really arises from the fact that in this case ''Y'' has a [[universal property]] with respect to principal ''G''-bundles, in the [[homotopy category]]. This is actually more basic than the condition that the higher homotopy groups vanish: the fundamental idea is, given ''G'', to find such a contractible space ''Y'' on which ''G'' acts ''[[group action|freely]]''. (The [[weak equivalence (homotopy theory)|weak equivalence]] idea of homotopy theory relates the two versions.) In the case of the circle example, what is being said is that we remark that an infinite cyclic group ''C'' acts freely on the [[real line]] ''R'', which is contractible. Taking ''X'' as the [[quotient space]] circle, we can regard the projection π from ''R'' = ''Y'' to ''X'' as a [[helix]] in geometrical terms, undergoing projection from three dimensions to the plane. What is being claimed is that π has a universal property amongst principal ''C''-bundles; that any principal ''C''-bundle in a definite way 'comes from' π.
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| ==Formalism==
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| A more formal statement takes into account that ''G'' may be a [[topological group]] (not simply a ''discrete group''), and that [[group action]]s of ''G'' are taken to be continuous; in the absence of continuous actions the classifying space concept can be dealt with, in homotopy terms, via the [[Eilenberg–MacLane space]] construction. In homotopy theory the definition of a topological space ''BG'', the '''classifying space''' for principal ''G''-bundles, is given, together with the space ''EG'' which is the '''total space''' of the [[universal bundle]] over ''BG''. That is, what is provided is in fact a [[continuous mapping]]
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| :<math>\pi: EG\longrightarrow BG.\ </math>
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| Assume that the homotopy category of [[CW complex]]es is the underlying category, from now on. The ''classifying'' property required of ''BG'' in fact relates to π. We must be able to say that given any principal ''G''-bundle
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| :<math>\gamma: Y\longrightarrow Z\ </math>
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| over a space ''Z'', there is a '''classifying map''' φ from ''Z'' to ''BG'', such that γ is the [[pullback of a bundle]] of π along φ. In less abstract terms, the construction of γ by 'twisting' should be reducible via φ to the twisting already expressed by the construction of π.
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| For this to be a useful concept, there evidently must be some reason to believe such spaces ''BG'' exist. In abstract terms (which are not those originally used around 1950 when the idea was first introduced) this is a question of whether the [[contravariant functor]] from the homotopy category to the [[category of sets]], defined by
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| :''h''(''Z'') = set of isomorphism classes of principal ''G''-bundles on ''Z''
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| is a [[representable functor]]. The abstract conditions being known for this ([[Brown's representability theorem]]) the result, as an [[existence theorem]], is affirmative and not too difficult.
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| ==Examples==
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| #The [[circle]] '''S'''<sup>1</sup> is a classifying space for the [[infinite cyclic group]] '''Z'''.
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| #The [[torus|''n''-torus]] <math>\mathbb T^n</math> is a classifying space for '''Z'''<sup>''n''</sup>, the [[free abelian group]] of rank ''n''.
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| #The wedge of ''n'' circles is a classifying space for the [[free group]] of rank ''n''.
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| #A [[closed manifold|closed]] (that is [[compact space|compact]] and without boundary) connected [[surface]] ''S'' of [[Genus (mathematics)|genus]] at least 1 is a classifying space for its [[fundamental group]] <math>\pi_1(S)</math>.
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| #The [[Real_projective_space#Infinite_real_projective_space|infinite-dimensional projective space]] <math>\mathbb {RP}^\infty</math> is a classifying space for '''Z'''/2'''Z'''.
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| #A [[closed manifold|closed]] (that is [[compact space|compact]] and without boundary) connected [[hyperbolic manifold]] ''M'' is a classifying space for its [[fundamental group]] <math>\pi_1(M)</math>.
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| #A finite connected locally [[CAT(0) space|CAT(0)]] [[cubical complex]] is a classifying space of its [[fundamental group]].
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| #<math>\mathbb{CP}^\infty</math> is the classifying space ''B'''''S'''<sup>1</sup> for the circle '''S'''<sup>1</sup> thought of as a compact topological group.
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| ==Applications==
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| This still leaves the question of doing effective calculations with ''BG''; for example, the theory of [[characteristic class]]es is essentially the same as computing the [[cohomology group]]s of ''BG'', at least within the restrictive terms of homotopy theory, for interesting groups ''G'' such as [[Lie group]]s. As was shown by the [[Bott periodicity theorem]], the [[homotopy group]]s of ''BG'' are also of fundamental interest. The early work on classifying spaces introduced constructions (for example, the [[bar construction]]), that gave concrete descriptions as a [[simplicial complex]].
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| An example of a classifying space is that when ''G'' is cyclic of order two; then ''BG'' is [[real projective space]] of infinite dimension, corresponding to the observation that ''EG'' can be taken as the contractible space resulting from removing the origin in an infinite-dimensional [[Hilbert space]], with ''G'' acting via ''v'' going to −''v'', and allowing for [[homotopy equivalence]] in choosing ''BG''. This example shows that classifying spaces may be complicated.
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| In relation with [[differential geometry]] ([[Chern–Weil theory]]) and the theory of [[Grassmannian]]s, a much more hands-on approach to the theory is possible for cases such as the [[unitary group]]s that are of greatest interest. The construction of the [[Thom complex]] ''MG'' showed that the spaces ''BG'' were also implicated in [[cobordism theory]], so that they assumed a central place in geometric considerations coming out of [[algebraic topology]]. Since [[group cohomology]] can (in many cases) be defined by the use of classifying spaces, they can also be seen as foundational in much [[homological algebra]].
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| Generalizations include those for classifying [[foliation]]s, and the [[classifying topos]]es for logical theories of the predicate calculus in [[intuitionistic logic]] that take the place of a 'space of models'.
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| ==References==
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| <references />
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| * J.P. May, ''A concise course in algebraic topology''
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| == External links ==
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| *{{Springer|id=C/c022440|title=Classifying space}}
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| ==See also==
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| * [[Classifying space for O(n)]], ''B''O(''n'')
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| * [[Classifying space for U(n)]], ''B''U(''n'')
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| * [[Classifying stack]]
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| [[Category:Algebraic topology]]
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| [[Category:Homotopy theory]]
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| [[Category:Fiber bundles]]
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| [[Category:Representable functors]]
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It is time to address the slow computer issues whether or not we never learn how. Just because your computer is working thus slow or keeps freezing up; does not signify to not address the problem plus fix it. You may or can not be aware which any computer owner must recognize which there are certain things that a computer requires to keep the greatest performance. The sad fact is that numerous folks whom own a system have no idea which it requires routine repair merely like their cars.
Before really obtaining the software it's best to check on the firms that create the software. If you may find details found on the form of reputation every business has, maybe the risk of malicious programs can be reduced. Software from reputed firms have aided me, plus other consumers, to make my PC run quicker.. If the product description does not look good to you, does not include details regarding the software, does not include the scan functions, you need to go for another one that ensures you're paying for what you desire.
Perfect Optimizer also offers to remove junk files plus is totally Windows Vista compatible. Many registry product merely don't have the time plus money to analysis Windows Vista errors. Because perfect optimizer has a big customer base, they do have the time, money and reasons to help completely support Windows Vista.
Your computer was especially fast when you first bought it. Because a registry was especially clean plus free of mistakes. After time, your computer begins to run slow and freezes up today and then. Because there are errors accumulating inside it and certain info is rewritten or completely deleted by your wrong uninstall of programs, improper operations, malware or different details. That is the reason why the computer performance decreases gradually and become especially unstable.
In a word, to speed up windows XP, Vista startup, it's quite significant to disable some startup goods and clean plus optimize the registry. You are able to follow the procedures above to disable unwanted programs. To optimize the registry, I suggest you employ a tuneup utilities software. Because it is rather dangerous for we to edit the registry by oneself.
Another key element when you compare registry cleaners is having a facility to manage the start-up tasks. This merely signifies that you can select what programs you want to commence whenever you commence a PC. If you have unwanted programs starting whenever you boot up a PC this might cause a slow running computer.
To speed up the computer, you merely have to be capable to get rid of all these junk files, allowing a computer to obtain what it wants, whenever it wants. Luckily, there's a tool that allows you to do this conveniently and instantly. It's a tool called a 'registry cleaner'.
A program and registry cleaner is downloaded from the web. It's convenient to use plus the process refuses to take long. All it does is scan plus then whenever it finds mistakes, it may fix and clean those errors. An error free registry may protect the computer from errors plus provide we a slow PC fix.