131.188.166.167 at 16:25, 9 November 2014

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195.37.209.182 at 15:53, 8 November 2013

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	~~Friends contact her Claude Gulledge~~. ~~Playing crochet~~ is a ~~thing that I~~'~~m completely addicted~~ to. He ~~presently life~~ in ~~Idaho~~ and ~~his mothers~~ and ~~fathers live nearby~~. ~~Interviewing~~ is ~~what I do~~ in ~~my day job~~.<br><br>~~my web~~-~~site~~: [http://www.~~Wintervideos~~.nl/~~profile~~.~~php?u=RiWhitcomb www~~.~~Wintervideos~~.nl]		In [[mathematics]], the '''Thom space,''' '''Thom complex,''' or '''Pontryagin-Thom construction''' (named after [[René Thom]] and [[Lev Pontryagin]]) of [[algebraic topology]] and [[differential topology]] is a [[topological space]] associated to a [[vector bundle]], over any [[paracompact]] space.

			==Construction of the Thom space==

			One way to construct this space is as follows. Let

			:''p'' : ''E'' → ''B''

			be a rank ''k'' [[real number\|real]] vector bundle over the paracompact space ''B''. Then for each point ''b'' in ''B'', the fiber ''F''<sub>''b''</sub> is a ''k''-dimensional real [[vector space]]. We can form an associated [[Fiber bundle #Sphere bundles\|sphere bundle]] Sph(''E'') → ''B'' by taking the [[one-point compactification]] of each fiber separately. Finally, from the total space Sph(''E'') we obtain the '''Thom complex''' ''T''(''E'') by identifying all the new points to a single point <math>\infty</math>, which we take as the [[basepoint]] of ''T''(''E'').

			==The Thom isomorphism==

			The significance of this construction begins with the following result, which belongs to the subject of [[cohomology]] of fiber bundles. (We have stated the result in terms of '''Z'''<sub>2</sub> [[coefficients]] to avoid complications arising from [[orientability]].)

			Let ''B'', ''E'', and ''p'' be as above. Then there is an isomorphism, now called a '''Thom isomorphism'''
			:<math>\Phi \colon H^i(B; \mathbf{Z}_2) \to \tilde{H}^{i+k}(T(E); \mathbf{Z}_2),</math>
			for all ''i'' greater than or equal to 0, where the [[Left-hand side and right-hand side of an equation\|right hand side]] is [[reduced cohomology]].

			We can loosely interpret the theorem as being a generalization of the suspension isomorphism on (co)homology, because the Thom space of a trivial bundle on ''B'' of rank ''k'' is isomorphic to the ''k''th suspension of ''B+'', ''B'' with a disjoint point added.

			This theorem was formulated and proved by [[René Thom]] in his 1952 thesis.

			==The Thom class==

			The isomorphism of the theorem is explicitly known: there is a certain cohomology class, the '''Thom class''', in the ''k''th cohomology group of the Thom space. Denote this Thom class by ''U''. Then for a class ''b'' in the cohomology of the base, we can compute the Thom isomorphism via the pullback of the bundle projection and the cohomology [[cup product]]:
			:<math>\Phi(b) = p^*(b) \smile U.</math>
			In particular, the Thom isomorphism sends the [[identity (mathematics)\|identity]] element of ''H''*(''B'') to ''U''.

			==Significance of Thom's work==

			In his 1952 paper, Thom showed that the Thom class, the [[Stiefel-Whitney class]]es, and the [[Steenrod operation]]s were all related. He used these ideas to prove in the 1954 paper ''Quelques propriétés globales des variétés differentiables'' that the [[cobordism]] groups could be computed as the [[homotopy groups]] of certain Thom spaces ''MG''(''n''). The proof depends on and is intimately related to the [[transversality (mathematics)\|transversality]] properties of [[smooth manifolds]] -- see [[Thom transversality theorem]]. By reversing this construction, [[John Milnor]] and [[Sergei Novikov (mathematician)\|Sergei Novikov]] (among many others) were able to answer questions about the existence and uniqueness of high-dimensional manifolds: this is now known as [[surgery theory]]. In addition, the spaces ''MG(n)'' fit together to form [[spectrum (homotopy theory)\|spectra]] ''MG'' now known as ''Thom spectra'', and the cobordism groups are in fact [[stable homotopy theory\|stable]]. Thom's construction thus also unifies [[differential topology]] and stable homotopy theory, and is in particular integral to our knowledge of the [[stable homotopy groups of spheres]].

			If the Steenrod operations are available, we can use them and the isomorphism of the theorem to construct the Stiefel-Whitney classes. Recall that the Steenrod operations (mod 2) are [[natural transformation]]s
			:<math>Sq^i \colon H^m(-; \mathbf{Z}_2) \to H^{m+i}(-; \mathbf{Z}_2),</math>
			defined for all nonnegative integers ''m''. If ''i'' = ''m'', then ''Sq<sup>i</sup>'' coincides with the cup square. We can define the ''i''th Stiefel-Whitney class ''w''<sub>''i''</sub> (''p'') of the vector bundle ''p'' : ''E'' → ''B'' by:
			:<math>w_i(p) = \Phi^{-1}(Sq^i(\Phi(1))) = \Phi^{-1}(Sq^i(U)).\,</math>

			==Consequences for differentiable manifolds==

			If we take the bundle in the above to be the [[tangent bundle]] of a smooth manifold, the conclusion of the above is called the [[Wu formula]], and has the following strong consequence: since the Steenrod operations are invariant under homotopy equivalence, we conclude that the Stiefel-Whitney classes of a manifold are as well. This is an extraordinary result that does not generalize to other characteristic classes. There exists a similar famous and difficult result establishing topological invariance for rational [[Pontryagin classes]], due to [[Sergei Novikov (mathematician)\|Sergei Novikov]].

			==See also==
			* [[Fiber bundle]]
			* [[Characteristic classes]]
			* [[Cobordism]]
			* [[Cohomology operation]]
			* [[Steenrod problem]]

			==References==
			* [[Dennis Sullivan]], ''[http://www.ams.org/bull/2004-41-03/S0273-0979-04-01026-2/home.html René Thom's Work on Geometric Homology and Bordism].'' Bull. Am. Math. Soc. 41 (2004), pp. 341–350.
			* [[Raoul Bott\|R. Bott]], L. Tu ''Differential Forms in Algebraic Topology'': a classic reference for [[differential topology]], treating the link to [[Poincaré duality]] and the [[Euler class]] of [[Sphere bundle]]s
			* J.P. May, ''A Concise Course in Algebraic Topology.'' University of Chicago Press, 1999, pp. 183–198.
			* [http://mathoverflow.net/questions/7375/explanation-for-the-thom-pontryagin-construction-and-its-generalisations Explanation for the Pontryagin-Thom construction] on [[MathOverflow]]
			* René Thom, ''[[List of important publications in mathematics#Quelques propriétés globales des variétés differentiables\|Quelques propriétés globales des variétés différentiables]].'' Comm. Math. Helv. 28 (1954), pp. 17–86.

			==External links==
			* {{springer\|title=Thom space\|id=p/t092680}}

			[[Category:Algebraic topology]]
			[[Category:Characteristic classes]]

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Friends contact her Claude Gulledge. Playing crochet is a thing that I'm completely addicted to. He presently life in Idaho and his mothers and fathers live nearby. Interviewing is what I do in my day job.<br><br>my web-site: [http://www.Wintervideos.nl/profile.php?u=RiWhitcomb www.Wintervideos.nl]

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