retrospective.tex

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I've been asked to describe what I remember about the genesis of the $\sigma$--orientation and related matters.
I'll do my best, with the caveat that my collaborators may have different memories, and if they do, theirs are just as right as mine.

The story starts around the fall of 1989.
I had spent a good deal of time in the late 1980s thinking about elliptic cohomology and not really getting anywhere, and I wanted to just think about homotopy theory for a while.
At the time Haynes Miller and I, inspired by work of Andy Baker and Jim McClure, had shown that the space of $A_{\infty}$ automorphisms of the Morava $E$--theory spectrum associated with the universal defomormation of a height $d$ formal group law over $\overline{\F}_{p}$ was homotopy discrete and equivalent to the semidirect product of the Morava stabilizer group and the Galois group of $\overline{\F}_{p}/\F_{p}$.
In fact Haynes and I worked out this proof at a reception for new faculty, in the very first conversation we had when I arrived at MIT in September 1989.
The reason for wanting this action was to construct the spectrum $EO_{d}=E_{d}^{hH}$ associated to a finite subgroup $H$ of the Morava stabilizer.
At the time that just seemed like a cool thing to do, but in retrospect the possibility of doing so must have been inspired by Ravenel's paper~\cite{RavenelNonexistenceArfInvariantElts}.
This led to the project of computing as many of the homotopy groups of $EO_{d}$ as one could.
The first interesting examples were for $d=(p-1)$, which is the smallest value of $d$ for which there is an element of order $p$ in the stabilizer group.
This was the case Ravenel pointed to in~\cite{RavenelNonexistenceArfInvariantElts} and it had immediate applications to the non-existence of Smith-Toda complexes.
The next case after that was the binary tetrahedral group in the height two stabilizer group at $p=2$.

Back in those days I drove an Alfa Romeo Spider, and at some point it needed a new head gasket.
I had dropped the car off at a small local shop, and when I went to pick it up the next day the mechanic told me he hadn't been able to get to it.
I asked how long it would be, and he told me four hours.
Conveniently, I was in that wonderful state of being obsessed with a math problem, so I just shrugged my shoulders and said that was cool, I'd wait.
(Also, conveniently, this was before cell phones and the distractions of the internet.)
I pulled out a pencil and paper, and I managed in that time to get a formula for the action of the group on the ring of functions and compute the cohomology.
I also had a method for getting the first differential, and after writing it down I realized it formally implied all the other differentials.
When I got home I wrote---by hand---a postscript file for a picture of the whole spectral sequence.

It happened that Mahowald made a visit to MIT shortly after that and I showed him the computation.
He immediately recognized it and told me he had published a paper with Don Davis proving that such a spectrum could not exist.
He also said he had never fully believed the proof but never could find anything wrong with it.
I still don't know how he recognized the computation.
What I had drawn was what we now think of as the Adams--Novikov spectral sequence for the $K(2)$--localization of $\tmf$, and what Mahowald had related it to was a spectrum whose cohomology is $\mathcal A^* \mmod \mathcal A(2)^*$.
Technically there wasn't quite a contradiction.
However we soon convinced ourselves this spectrum $EO_{2}$ probably did imply the existence of a spectrum whose cohomology is $\mathcal A^* \mmod \mathcal A(2)^*$, and that the Davis--Mahowald argument probably applied to $EO_{2}$ as well.
We both worked pretty hard trying to find the resolution.
I was worried about something foundational in the theory of $A_{\infty}$ ring spectra and devoted a lot of time to that, and Mahowald perused his argument with Davis.
Mark and I went through the Davis--Mahowald argument very, very carefully.
It involved an long series of incredibly dexterous moves, and I think I learned more about how homotopy groups work in checking that argument than from any other experience---but we
couldn't find a mistake.
On April Fool's day Mark found the error: his paper with Davis was completely fine, and the error was in the accepted computations of the homotopy groups of spheres.
Though I don't recall if this was 1990 or 1991, the day has stayed with me.
Adams hadn't been gone long and it was his tradition to give a lecture every April Fool's day proving two contradictory statements, and challenge the audience to find a mistake.
Mark and I felt he had given us one last private April Fool's lecture.

Davis and Mahowald had really wanted to have a spectrum whose cohomology is $\mathcal A^* \mmod \mathcal A(2)^*$, and without it they had made do with the Thom spectrum $MO[8, \infty)$ associated to the $7$--connected cover $BO[8, \infty)$.
Bahri and Mahowald~\cite{BahriMahowald} had shown that there was an isomorphism of $\mathcal A$--modules
\[
H\F_2^{\ast}(MO[8, \infty)) \approx \mathcal A^* \mmod \mathcal A(2)^* \oplus M
\]
in which $M$ is $15$--connected.
This situation was meant to be an analogue of the situation with $\Spin$ cobordism (with cohomology $\mathcal A^* \mmod \mathcal A(1)^* \oplus N$ for some $7$--connected $N$) and connected $K$--theory $kO$ (with cohomology $\mathcal A^* \mmod \mathcal A(1)^*$).
This made it natural to construct a non-periodic version $eo_{2}$ of $EO_{2}$ with cohomology $\mathcal A^* \mmod \mathcal A(2)^*$.
Mahowald and I succeeded in doing so at the Mittag-Lefler institute in the fall of 1993.
It also made it natural to look for an ``orientation'' \[MO[8, \infty) \to eo_{2}\] analogous to the Atiyah--Bott--Shapiro orientation $M\Spin \to kO$.
At the time there was no known method of construction of the Atiyah--Bott--Shapiro orientation that did not rely on the interpretation of $KO$--theory in terms of vector bundles, so this seemed to be a tricky problem.
I conceived of a two-stage program for doing this: the first step was to produce a map $MO[8, \infty) \to E_{2}$ invariant up to homotopy under the action of the binary tetrahedral group, and the second step was to rigidify everything in sight by requiring all of the maps to be $A_{\infty}$ or $E_{\infty}$.

When I got back to MIT in the winter of 1994, Matt Ando and Neil Strickland were around and we got to thinking pretty hard about trying to understand the $E_{\infty}$ or $A_{\infty}$ maps from $MO[8, \infty)$ to $E_{2}$.
We weren't getting anywhere when Mark Hovey told us that computations he and Ravenel had done seemed to indicate that there couldn't be a map of spectra $MO[8, \infty) \to E_{2}$ invariant up to homotopy under the action of the binary tetrahedral group.
This didn't look good for the first step of the program, so Matt and Neil and I started thinking about how one might understand the cohomology of $MO[8, \infty)$ in terms of formal groups.
We found an answer in terms of cubical structures on formal groups and realized that there was a canonical map from $MO[8, \infty)$ to any complex oriented cohomology theory $E$ whose formal group was the formal completion of an elliptic curve~\cite{AHSTheoremOfTheCube}.
This led to the picture conjectured in my 1994 ICM talk~\cite{HopkinsICMZurich} of the $\tmf$ sheaf and the relationship between the conjectured $MO[8, \infty)$ orientation and the Witten genus.
It took a while but eventually the $\tmf$ sheaf was constructed, we found the right way to think about $E_{\infty}$ orientations of Thom spectra, and with Charles Rezk were able to produce the $\sigma$--orientation as well as the Atiyah-Bott-Shapiro orientation using homotopy theory.
I announced those results at the 2002 ICM~\cite{HopkinsICMBeijing}.

In the end the homotopy fixed point spectra $E_{d}^{hH}$ turned out to have many applications, from the non-existence of Smith-Toda complexes, computations in chromatic homotopy theory at low primes, and even to the Kervaire invariant problem.
However, generalizing the whole package with the orientation (which after all, was the problem that led to $\tmf$) is still quite a mystery.
In the early 1990s Hovey found an ingenious argument showing that for height $d>2$ there can't be an orientation $MO[d, \infty) \to E^{hH}_{d}$ if $H$ contains a non-trivial element of order $p$.
What should play the role of an orientation in those cases is pretty much up for grabs.