On the cobordism groups of $O⟨n⟩$-manifolds

Given a smooth manifold, a tangential $O⟨n⟩$-structure is a lift of the classifying map of its tangent bundle to $BO⟨n⟩$. An $O⟨n⟩$-manifold is a manifold equipped with an equivalence class of tangential $O⟨n⟩$-structures. The cases $n=2$ and $n=4$ recover the familiar notions of oriented and spin manifolds. Larger values of $n$ furnish higher analogs of these with additional geometric properties. In this thesis, we calculate the 2-primary component of the cobordism groups of compact $O⟨n⟩$-manifolds in a range of degrees, for $n$ between 9 and 17. We derive various consequences for the boundaries of almost-closed manifolds.