Framed bordism
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[edit] 1 Introduction
The framed bordism groups of manifolds with a framing of the stable normal bundle (or equivalently the stable tangent bundle) are isomorphic to the stable homotopy groups of spheres . These groups are now completely known only in a range up to 62: They seem to be very complicated, and no general description is known. (As an illustration: there is -torsion in for all primes .)
[edit] 2 Generators
- , generated by a point.
- , generated by with the Lie group framing.
- ,
- , generated by with the Lie group framing of
- .
- , generated with the Lie group framing.
- , generated by with twisted framing defined by the generator of .
See also:
- this table from the Wikipedia article on homotopy groups of spheres for more values.
- this table from Allen Hatchers home page.
Serre [Serre1951] proved that is a finite abelian group for .
[edit] 3 Invariants
Degree of a map . Since stably framed manifolds have stably trivial tangent bundles, all other characteristic numbers are zero.
[edit] 4 Classification
Consequently most of the classification results use homotopy theory.
Adams spectral sequence and Novikov's generalization [Ravenel1986]. Toda brackets. Nishida [Nishida1973] proved that in the ring all elements of positive degree are nilpotent.
[edit] 5 Further topics
Kervaire invariant 1, Hopf invariant 1 problems, J-homomorphism, first -torsion in degree.
[edit] 6 References
- [Nishida1973] G. Nishida, The nilpotency of elements of the stable homotopy groups of spheres, J. Math. Soc. Japan 25 (1973), 707–732. MR0341485 (49 #6236) Zbl 0316.55014
- [Ravenel1986] D. C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Academic Press Inc., Orlando, FL, 1986. MR860042 (87j:55003) Zbl 1073.55001
- [Serre1951] J. Serre, Homologie singulière des espaces fibrès. Applications, Ann. of Math. (2) 54 (1951), 425–505. MR0045386 (13,574g) Zbl 0045.26003