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Stable homotopy groups of spheres are used to describe the group Θ n of h-cobordism classes of oriented homotopy n-spheres (for n ≠ 4, this is the group of smooth structures on n-spheres, up to orientation-preserving diffeomorphism; the non-trivial elements of this group are represented by exotic spheres). More precisely, there is an ...
In algebraic topology, a branch of mathematics, a homotopy sphere is an n-manifold that is homotopy equivalent to the n-sphere. It thus has the same homotopy groups and the same homology groups as the n -sphere, and so every homotopy sphere is necessarily a homology sphere .
For the corresponding definition in terms of spheres, define the sum + of maps ,: to be composed with h, where is the map from to the wedge sum of two n-spheres that collapses the equator and h is the map from the wedge sum of two n-spheres to X that is defined to be f on the first sphere and g on the second.
One of the most important problems in stable homotopy theory is the computation of stable homotopy groups of spheres. According to Freudenthal's theorem, in the stable range the homotopy groups of spheres depend not on the specific dimensions of the spheres in the domain and target, but on the difference in those dimensions.
Rational homotopy theory has realized much of that goal. In homotopy theory, spheres and Eilenberg–MacLane spaces are two very different types of basic spaces from which all spaces can be built. In rational homotopy theory, these two types of spaces become much closer.
The direct sum = of the stable homotopy groups of spheres is a supercommutative graded ring, where multiplication (called composition product) is given by composition of representing maps, and any element of non-zero degree is nilpotent (Nishida 1973).
The cases n = 1 and 2 have long been known by the classification of manifolds in those dimensions.. For a PL or smooth homotopy n-sphere, in 1960 Stephen Smale proved for that it was homeomorphic to the n-sphere and subsequently extended his proof to ; [3] he received a Fields Medal for his work in 1966.
Smale's classification of immersions of spheres shows that sphere eversions exist, which can be realized via this Morin surface. Stephen Smale classified the regular homotopy classes of a k -sphere immersed in R n {\displaystyle \mathbb {R} ^{n}} – they are classified by homotopy groups of Stiefel manifolds , which is a generalization of the ...
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