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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 $\eta$?

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    $\begingroup$ Depending on your intended application, you might look for a strange model and ordinary action rather than a strange action and ordinary model: one might imagine there to be a different manifold-level model $\Omega^{gl}$ for $\pi_* gl_1 \mathbb S$ which more naturally forms a "module" over the usual framed bordism model $\Omega^{fr}$ for $\pi_* \mathbb S$, in the sense that there's some (nice) operation on manifolds $\Omega^{fr} \times \Omega^{gl} \to \Omega^{gl}$ which descends to bordism classes and recovers the action you indicate. Beats me, though. $\endgroup$
    – Eric Peterson
    Commented Oct 14, 2020 at 23:59
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    $\begingroup$ I think the Freudenthal suspension theorem tells you that an element of $\pi_kS$ will act on $\pi_*gl_1S$ in the same way as it acts on $\pi_*S$, for $*>k$. So on the one hand the construction won't be symmetric in the two manifolds but on the other hand will "almost always" agree with direct product (up to framed bordism of course.) And in the specific case of $\eta$, the action only differs from the direct product at $\pi_1$ where it acts by zero I think...so as @EricPeterson suggested, it will be a strange operation indeed $\endgroup$
    – kiran
    Commented Oct 16, 2020 at 5:02
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    $\begingroup$ $\eta^3 = 0$ in $\pi_*BGl_1(S)$ because this is true in $\pi_* BO$. The Freudenthal suspension theorem and the 'sphere of origin' come into play because we're comparing the composition and the smash product, and the usual argument that they are equal stably can only desuspend as far as the sphere of origin. Much more can be said, but I am not prepared to do so right now. Possibly relevant: Bökstedt, Marcel ; Waldhausen, Friedhelm . The map BSG→A(∗)→QS0. $\endgroup$
    – Robert Bruner
    Commented Oct 21, 2020 at 1:58
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    $\begingroup$ This is somewhat related to this question about how to describe the ring structure on the sphere spectrum in terms of the framed bordism category. $\endgroup$
    – Noah Snyder
    Commented Oct 21, 2020 at 18:16

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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.

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