Homotopy groups of spheres correspond to framed submanifolds of Euclidean space through the Pontrjagin-Thom construction.  For example, the Hopf map corresponds to a circle in $\mathbb{R}^3$ framed “with a twist”.  The homotopy groups of $S^1$ thus correspond to framed codimension one submanifolds.  But such are canonically framed and all bound, so there are no interesting/ non-trivial examples.