Framed bordism
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== Introduction == | == Introduction == | ||
<wikitex>; | <wikitex>; | ||
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 [[wikipedia:Homotopy_groups_of_spheres#Stable_and_unstable_groups|stable homotopy groups of spheres]] $\pi_n^{s}$. | 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 [[wikipedia:Homotopy_groups_of_spheres#Stable_and_unstable_groups|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 $p$.) | (As an illustration: there is $p$-torsion in $\Omega_*^{fr}$ for all primes $p$.) | ||
</wikitex> | </wikitex> | ||
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* $\Omega_0^{fr}=\Zz$, generated by a point. | * $\Omega_0^{fr}=\Zz$, generated by a point. | ||
− | * $\Omega_1^{fr}=\Zz_2$, generated by $S^1$. | + | * $\Omega_1^{fr}=\Zz_2$, generated by $S^1$ with the Lie group framing. |
* $\Omega_2^{fr}=\Zz_2$, | * $\Omega_2^{fr}=\Zz_2$, | ||
− | * $\Omega_3^{fr}=\Zz_{24}$, generated by | + | * $\Omega_3^{fr}=\Zz_{24}$, generated by $S^3=SU(2)$ with the Lie group framing of |
* $\Omega_4^{fr}=\Omega_5^{fr}=0$. | * $\Omega_4^{fr}=\Omega_5^{fr}=0$. | ||
− | See [[ | + | * $\Omega_6^{fr} = \Zz_2$, 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 $\pi_7(O) \cong \Zz$. | ||
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+ | See also: | ||
+ | *[[Wikipedia:Homotopy_groups_of_spheres#Table_of_stable_homotopy_groups|this table]] from the Wikipedia article on homotopy groups of spheres for more values. | ||
+ | *[http://www.math.cornell.edu/~hatcher/stemfigs/stems.html this table] from Allen Hatchers home page. | ||
Serre {{cite|Serre1951}} proved that $\Omega_n^{fr}$ is a finite abelian group for $n>0$. | Serre {{cite|Serre1951}} proved that $\Omega_n^{fr}$ is a finite abelian group for $n>0$. | ||
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<wikitex>; | <wikitex>; | ||
Degree of a map $S^n\to S^n$. Since stably framed manifolds have stably trivial tangent bundles, all other characteristic numbers are zero. | Degree of a map $S^n\to S^n$. Since stably framed manifolds have stably trivial tangent bundles, all other characteristic numbers are zero. | ||
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</wikitex> | </wikitex> | ||
== Classification == | == Classification == | ||
<wikitex>; | <wikitex>; | ||
− | The case of framed bordism | + | The case of framed bordism is the original case of the [[B-Bordism#The Pontrjagin-Thom isomorphism|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 $$\Omega_n^{fr} \cong \pi_n(MPBO) \cong \pi_n^s.$$ |
Consequently most of the classification results use homotopy theory. | Consequently most of the classification results use homotopy theory. | ||
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== Further topics == | == Further topics == | ||
− | Kervaire invariant 1, Hopf invariant 1 problems, J-homomorphism, first $p$-torsion in degree | + | <wikitex>; |
+ | Kervaire invariant 1, Hopf invariant 1 problems, J-homomorphism, first $p$-torsion in degree. | ||
+ | </wikitex> | ||
== References == | == References == | ||
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[[Category:Manifolds]] | [[Category:Manifolds]] | ||
− | + | [[Category:Bordism]] |
Latest revision as of 11:40, 8 July 2011
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
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 .)
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 .
3 Invariants
Degree of a map . Since stably framed manifolds have stably trivial tangent bundles, all other characteristic numbers are zero.
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.
5 Further topics
Kervaire invariant 1, Hopf invariant 1 problems, J-homomorphism, first -torsion in degree.
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