Framed bordism

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1 Introduction

The framed bordism groups $\Omega_n^{fr}$${{Stub}} == Introduction == <wikitex>; The framed bordism groups $\Omega_n^{fr}$ of manifolds with a framing of the stable normal bundle (or equivalently the stable tangent bundle) are isomorphic to the [[wikipedia:Homotopy_groups_of_spheres#Stable_and_unstable_groups|stable homotopy groups of spheres]] $\pi_n^{s}$. These groups are now completely known only in a range up to 62: They seem to be very complicated, and no general description is known. (As an illustration: there is $p$-torsion in $\Omega_*^{fr}$ for all primes $p$.) </wikitex> == Generators == <wikitex>; * $\Omega_0^{fr}=\Zz$, generated by a point. * $\Omega_1^{fr}=\Zz_2$, generated by $S^1$ with the Lie group framing. * $\Omega_2^{fr}=\Zz_2$, * $\Omega_3^{fr}=\Zz_{24}$, generated by $S^3=SU(2)$ with the Lie group framing of * $\Omega_4^{fr}=\Omega_5^{fr}=0$. * $\Omega_6^{fr} = \Zz_2$, generated $S^3 \times S^3$ with the Lie group framing. * $\Omega_7^{fr} \cong \Zz_{240} \cong \Zz_{16} \oplus \Zz_3 \oplus \Zz_5$, generated by $S^7$ with twisted framing defined by the generator of $\pi_7(O) \cong \Zz$. See also: *[[Wikipedia:Homotopy_groups_of_spheres#Table_of_stable_homotopy_groups|this table]] from the Wikipedia article on homotopy groups of spheres for more values. *[http://www.math.cornell.edu/~hatcher/stemfigs/stems.html this table] from Allen Hatchers home page. Serre {{cite|Serre1951}} proved that $\Omega_n^{fr}$ is a finite abelian group for $n>0$. </wikitex> == Invariants == <wikitex>; Degree of a map $S^n\to S^n$. Since stably framed manifolds have stably trivial tangent bundles, all other characteristic numbers are zero. </wikitex> == Classification == <wikitex>; The case of framed bordism is the original case of the [[B-Bordism#The Pontrjagin-Thom isomorphism|Pontrjagin-Thom isomorphism]], discovered by Pontryagin. The Thom spectrum $MPBO$ corresponding to the path fibration over $BO$ is homotopy equivalent to the sphere spectrum $S$ since the path space is contractible. Thus we get $$\Omega_n^{fr} \cong \pi_n(MPBO) \cong \pi_n^s.$$ Consequently most of the classification results use homotopy theory. Adams spectral sequence and Novikov's generalization {{cite|Ravenel1986}}. Toda brackets. Nishida {{cite|Nishida1973}} proved that in the ring $\Omega_*^{fr}$ all elements of positive degree are nilpotent. </wikitex> == Further topics == <wikitex>; Kervaire invariant 1, Hopf invariant 1 problems, J-homomorphism, first $p$-torsion in degree. </wikitex> == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]] [[Category:Bordism]]\Omega_n^{fr}$ of manifolds with a framing of the stable normal bundle (or equivalently the stable tangent bundle) are isomorphic to the stable homotopy groups of spheres $\pi_n^{s}$$\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$$p$-torsion in $\Omega_*^{fr}$$\Omega_*^{fr}$ for all primes $p$$p$.)

2 Generators

• $\Omega_0^{fr}=\Zz$$\Omega_0^{fr}=\Zz$, generated by a point.

• $\Omega_1^{fr}=\Zz_2$$\Omega_1^{fr}=\Zz_2$, generated by $S^1$$S^1$ with the Lie group framing.

• $\Omega_2^{fr}=\Zz_2$$\Omega_2^{fr}=\Zz_2$,

• $\Omega_3^{fr}=\Zz_{24}$$\Omega_3^{fr}=\Zz_{24}$, generated by $S^3=SU(2)$$S^3=SU(2)$ with the Lie group framing of

• $\Omega_4^{fr}=\Omega_5^{fr}=0$$\Omega_4^{fr}=\Omega_5^{fr}=0$.

• $\Omega_6^{fr} = \Zz_2$$\Omega_6^{fr} = \Zz_2$, generated $S^3 \times S^3$$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$$\Omega_7^{fr} \cong \Zz_{240} \cong \Zz_{16} \oplus \Zz_3 \oplus \Zz_5$, generated by $S^7$$S^7$ with twisted framing defined by the generator of $\pi_7(O) \cong \Zz$$\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}$$\Omega_n^{fr}$ is a finite abelian group for $n>0$$n>0$.

3 Invariants

Degree of a map $S^n\to S^n$$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$$MPBO$ corresponding to the path fibration over $BO$$BO$ is homotopy equivalent to the sphere spectrum $S$$S$ since the path space is contractible. Thus we get

$\displaystyle \Omega_n^{fr} \cong \pi_n(MPBO) \cong \pi_n^s.$ $$\displaystyle \Omega_n^{fr} \cong \pi_n(MPBO) \cong \pi_n^s.$$

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

5 Further topics

Kervaire invariant 1, Hopf invariant 1 problems, J-homomorphism, first $p$$p$-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