20/09/03 17:22:26.78 k0Z0EEBv.net
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つづき
Framed cobordism
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.
Homotopy groups of spheres are closely related to cobordism classes of manifolds. In 1938 Lev Pontryagin established an isomorphism between the homotopy group πn+k(Sn) and the group Ωframed
k(Sn+k) of cobordism classes of differentiable k-submanifolds of Sn+k which are "framed", i.e. have a trivialized normal 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 Rene Thom to an isomorphism expressing other groups of cobordism classes (e.g. of all manifolds) as homotopy groups of spaces and spectra. In more recent work the argument is usually reversed, with cobordism groups computed in terms of homotopy groups (Scorpan 2005).
Finiteness and torsion
In 1951, Jean-Pierre Serre showed that homotopy groups of spheres are all finite except for those of the form πn(Sn) or π4n?1(S2n) (for positive n), when the group is the product of the infinite cyclic group with a finite abelian group (Serre 1951). In particular the homotopy groups are determined by their p-components for all primes p. The 2-components are hardest to calculate, and in several ways behave differently from the p-components for odd primes.
In the same paper, Serre found the first place that p-torsion occurs in the homotopy groups of n dimensional spheres, by showing that πn+k(Sn) has no p-torsion if k < 2p ? 3, and has a unique subgroup of order p if n ? 3 and k = 2p ? 3.
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