On framed cobordism classes of classical Lie groups

Abstract

It is known that any compact connected Lie group with its left invariant framing is framed null-cobordant in the $p$-component for any prime $p\ne 2,3$. In this paper we will prove that the 3-components of $\mathrm{SO}\left(2n+1\right)$ and $Sp\left(n\right)$ are zero for $n\ge 3$, $n\ne 5,7,11$. Combining this with the previously known results on $\mathrm{SO}\left(2n\right)$ and $SU\left(n\right)$ consequently we see that any classical group has at most only the 2-component with some exceptions.