Is there a framed bordism model for the units of the sphere spectrum, $gl_1(S)$?
At the level of individual homotopy groups, the Pontrjagin-Thom construction identifies the group of bordism classes of stably framed $k$-manifolds with the stable homotopy group $\pi_k(S)$, and even captures the product. But for $k > 0$ this group agrees with $\pi_k(gl_1(S))$. Is there an operation on framed manifolds corresponding to the action of $\pi_*(S)$ on $\pi_*(gl_1(S))$? What if we just asked for multiplication by $\eta$?