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Homotopy groups of spheres
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===Framed cobordism=== Homotopy groups of spheres are closely related to [[cobordism]] classes of manifolds. In 1938 [[Lev Pontryagin]] established an isomorphism between the homotopy group {{math|Ο<sub>''n''+''k''</sub>(''S''<sup>''n''</sup>)}} and the group {{math|Ξ©{{su|lh=1|b=''k''|p=framed}}(''S''<sup>''n''+''k''</sup>)}} of cobordism classes of [[Differentiable manifold|differentiable]] {{mvar|k}}-submanifolds of {{math|''S''<sup>''n''+''k''</sup>}} which are "framed", i.e. have a trivialized [[normal bundle]]. Every map {{math|''f'' : ''S''<sup>''n''+''k''</sup> β ''S''<sup>''n''</sup>}} is homotopic to a differentiable map with {{math|1=''M''<sup>''k''</sup> = ''f''<sup>−1</sup>(1, 0, ..., 0) β ''S''<sup>''n''+''k''</sup>}} a framed {{mvar|k}}-dimensional submanifold. For example, {{math|Ο<sub>''n''</sub>(''S''<sup>''n''</sup>) {{=}} Z}} is the cobordism group of framed 0-dimensional submanifolds of {{math|''S''<sup>''n''</sup>}}, computed by the algebraic sum of their points, corresponding to the [[degree of a map|degree]] of maps {{math|''f'' : ''S''<sup>''n''</sup> β ''S''<sup>''n''</sup>}}. The projection of the [[Hopf fibration]] {{math|''S''<sup>3</sup> β ''S''<sup>2</sup>}} represents a generator of {{math|Ο<sub>3</sub>(''S''<sup>2</sup>) {{=}} Ξ©{{su|lh=1|b=1|p=framed}}(''S''<sup>3</sup>) {{=}} Z}} which corresponds to the framed 1-dimensional submanifold of {{math|''S''<sup>3</sup>}} defined by the standard embedding {{math|''S''<sup>1</sup> β ''S''<sup>3</sup>}} with a nonstandard trivialization of the normal 2-plane bundle. Until the advent of more sophisticated algebraic methods in the early 1950s (Serre) the Pontrjagin isomorphism was the main tool for computing the homotopy groups of spheres. In 1954 the Pontrjagin isomorphism was generalized by [[RenΓ© Thom]] to an isomorphism expressing other groups of cobordism classes (e.g. of all manifolds) as [[homotopy group]]s of spaces and [[Spectrum (homotopy theory)|spectra]]. In more recent work the argument is usually reversed, with cobordism groups computed in terms of homotopy groups.{{sfn|Scorpan|2005}}
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