The subgroup $b{P}_{n+1}$

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. The subgroup $b{P}_{n+1}\subseteq {\mathrm{\Theta }}_n$, for $n\ge 5$, consists of the oriented homotopy-sphere classes represented by boundaries of compact oriented parallelizable smooth $(n+1)$-manifolds. Here parallelizable means that the tangent bundle $TV$ is trivial, while stably parallelizable means that $TV\oplus {\epsilon }^r$ is trivial for some $r\ge 0$; stable parallelizability is the weaker condition and is not substituted silently.

Representative independence and closure under addition, inverse and the zero class are proved in Parallelizable boundaries form a subgroup, so $b{P}_{n+1}$ is a subgroup of ${\mathrm{\Theta }}_n$ as displayed. The definition is conditional on the group structure of The homotopy-sphere group ${\mathrm{\Theta }}_n$ and uses no choice principle beyond ${\mathrm{A}\mathrm{C}}_{\omega }$.