Pontrjagin-Thom model for units of the sphere spectrum? 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 $\eta$?
 A: You should definitely have a look at Bokstedt and Waldhausen's "The map $BG \to A(*) \to QS^0$":
MR0921487  Bökstedt, Marcel ;  Waldhausen, Friedhelm . The map BSG→A(∗)→QS0.
Algebraic topology and algebraic K-theory (Princeton, N.J., 1983),
418--431, Ann. of Math. Stud., 113, Princeton Univ. Press, Princeton, NJ,  1987.
They study a geometrically defined transfer map  $BG \to G$, and show it is multiplication by $\eta$, and that this agrees with the usual action of $\eta$ under the iso to $\pi_*(S)$ in dimensions 3 and up, but not on $\pi_2$ as I mentioned above.  This appearance of $\eta$ reminds me of the theorem of Blumberg, Cohen and Schlichtkrull on THH of Thom  spectra, which might also be worth looking at in this connection.
MR2651551
Blumberg, Andrew J.; Cohen, Ralph L.; Schlichtkrull, Christian.
Topological Hochschild homology of Thom spectra and the free loop space.
Geom. Topol.  14  (2010),  no. 2, 1165--1242.
Sorry if this turns out to be irrelevant, but these are fun papers to read in any case.
