# Framed bordism

## 1 Introduction

The framed bordism groups $\Omega_n^{fr}$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}\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$.

• this table from the Wikipedia article on homotopy groups of spheres for more values.

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

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.