Framed bordism

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This page has not been refereed. The information given here might be incomplete or provisional.


1 Introduction

The framed bordism groups \Omega_n^{fr} 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 \pi_n^{s}. 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 p-torsion in \Omega_*^{fr} for all primes p.)

2 Generators

  • \Omega_0^{fr}=\Zz, generated by a point.
  • \Omega_1^{fr}=\Zz_2, generated by S^1 with the Lie group framing.
  • \Omega_2^{fr}=\Zz_2,
  • \Omega_3^{fr}=\Zz_{24}, generated by S^3=SU(2) with the Lie group framing of
  • \Omega_4^{fr}=\Omega_5^{fr}=0.
  • \Omega_6^{fr} = \Zz_2, generated S^3 \times S^3 with the Lie group framing.
  • \Omega_7^{fr} \cong \Zz_{240} \cong \Zz_{16} \oplus \Zz_3 \oplus \Zz_5, generated by S^7 with twisted framing defined by the generator of \pi_7(O) \cong \Zz.

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 \Omega_n^{fr} is a finite abelian group for n>0.

3 Invariants

Degree of a map S^n\to S^n. Since stably framed manifolds have stably trivial tangent bundles, all other characteristic numbers are zero.

4 Classification

The case of framed bordism is the original case of the Pontrjagin-Thom isomorphism, discovered by Pontryagin. The Thom spectrum MPBO corresponding to the path fibration over BO is homotopy equivalent to the sphere spectrum S since the path space is contractible. Thus we get
\displaystyle \Omega_n^{fr} \cong \pi_n(MPBO) \cong \pi_n^s.

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 \Omega_*^{fr} all elements of positive degree are nilpotent.

5 Further topics

Kervaire invariant 1, Hopf invariant 1 problems, J-homomorphism, first p-torsion in degree.

6 References

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