**Background**

For compact Lie groups, Atiyah and Segal proved a strong relationship between Borel-equivariant K-theory, defined in terms of the K-theory of $X \times_G EG$, and the equivariant K-theory of X defined in terms of equivariant vector bundles. Roughly, for "nice" spaces X the K-theory of $X \times_G EG$ is a completion of the equivariant K-theory of X, and in particular the K-theory of BG is a completion of the complex representation ring of G.

The Segal conjecture is an analogous result proven in subsequent years (by many authors, with Carlsson completing the proof). It's less well-known outside the subject, and obtained by roughly replacing "vector bundles" with "covering spaces" - the original conjecture is that for a finite group $G$, the abelian group of stable classes of maps $\varinjlim[S^n \wedge BG, S^n]$ has as limit to the Burnside ring of finite $G$-sets. There are further statements describing $\varinjlim [S^{n+k} \wedge BG, S^n]$ in terms of a completion of certain equivariant stable homotopy groups. It's notable for the fact that it's not really a computational result - we describe two objects as being isomorphic, without any knowledge of what the resulting groups on either side really are.

There are a number of steps in this proof, and over the years most of them have been recast and reinterpreted in a number of ways. However, the initial steps in the proof are computational. Lin proved this conjecture for the case $G = \mathbb{Z}/2$, and Gunawardena proved it for the case $G = \mathbb{Z}/p$ for odd primes $p$. Lin's original proof involved some very difficult computations in the Lambda algebra and a simplified proof was ultimately written up by Lin-Davis-Mahowald-Adams. It amounts to a computation of certain Ext or Tor groups over the Steenrod algebra - namely, if $H^* \mathbb{RP}^\infty = \mathbb{Z}/2[x]$ has its standard module structure over the Steenrod algebra, then $Ext^{**}(\mathbb{Z}/2[x^{\pm 1}],\mathbb{Z}/2)$ degenerates down to a single nonzero group.

**Bordism theory**

A lot of the contemporary work in stable homotopy theory uses the relationship between stable homotopy theory and the moduli of formal groups, rather than the Adams-spectral-sequence calculations that are used in the above proofs. The analogous calculation would be the following.

Let L be the Lazard ring carrying the universal formal group law, with 2-series $[2](t)$ whose zeros are the "2-torsion" of the formal group law. Then there is an L-algebra $$ Q = t^{-1} L[[t]]/[2](t) $$ whose functor of points would be described (up to completion issues) as taking a ring R to the set of formal group laws on R equipped with a nowhere-zero 2-torsion point. This comes equipped with natural descent data for change-of-coordinates on the formal group law, and so it describes a sheaf on the moduli stack of formal group laws $\mathcal{M}$.

A student of Doug Ravenel's (Binhua Mao) proved in his thesis that the analogous Ext-computation is valid in the formal-group setting: namely, if one computes the Ext-groups $$Ext_{\mathcal M}(\mathcal{O}, Q \otimes \omega^{\otimes t})$$ where $\omega$ is the sheaf of invariant 1-forms on $\mathcal{M}$, it converges to a completion of $$Ext_{\mathcal M}({\mathcal O}, \omega^{\otimes t}).$$ (The result was stated in different language, and I am still ignoring completion issues.)

However, as I understand the proof (and I don't claim that I really do!) it essentially uses a reduction to the Adams spectral sequence case by using a filtration that reduces to the group scheme of automorphisms of the additive formal group law, and this is very closely connected to the Steenrod algebra. I would regard it as still being computationally focused, and I don't really have a grip on why one might *expect* it to be true without carrying the motivation from homotopy theory all the way through.

**Question (finally)**

Is there is a more conceptual interpretation of this computation in terms of the geometry of the moduli of formal groups?