[ALGTOP-L] Michael Barratt

Bill Richter richter at math.northwestern.edu
Fri Oct 23 01:31:07 EDT 2015

Michael Barratt was a big influence on each of my 11 papers, which mostly used Michael's type of unstable homotopy theory to explain Mark Mahowald's ideas.  There's a lot more work left to be done in this area, and it all starts with Mike Hopkins's Oxford thesis which explains how to modernize 50s homotopy theory, e.g. why a Hopf invariant is the natural map from the inverse limit to the homotopy inverse limit.  Let me explain how I met Michael and also his last paper, a 1995 PAMS joint paper with Mark, Fred Cohen, Brayton Gray and I, 
Two results on the 2-local EHP spectral sequence.

I was an undergraduate at Princeton, and thought of myself as a surgeon, working with Bill Browder and Andrew Ranicki, with help from Dan Chess, who gave me a paper by Bruce Williams stressing Boardman-Steer's Hopf Invariants paper.  Unfortunately I didn't graduate, but Mark was at Princeton my un-senior year, and he brought me back with him to Northwestern in 1980.  On the Amtrak train to Chicago I proved my 1st theorem, about high dimensional knots and Z-equivariant Hopf invs, using Andrew's ideas about knots and CW-pi complexes for pi = Z`.  When I got to NWU, Michael grabbed me and started explaining to me what I was actually doing with my Z-equivariant Hopf invs, and I was about as happy as I've ever been.  I thought, ``This is where I'm supposed to be.  I've found my teacher.''  The next conversation with Michael I remember was a year and a half later, the night before Michael left for his trip to Israel that Giora Dula mentioned.  We met at his office about 10PM and t!
 alked until after midnight.  I was getting a wrong answer with my Z-equivariant Hopf invs, and Michael straightened me out, using his ideas about symmetry formulas for un-suspended Hopf invs, which fixed a sign error in Boardman-Steer's paper.  So I thanked Michael profusely, but I knew things were going south for me at NWU, and I said something like I hope I see you again,  and Michael said, ``Of course I'll see you again.  You're a mathematician.''  That was really important confirmation for me.  Unfortunately I never got my Z-equivariant knot theory written up, due to model category theory problems I didn't solve for another decade, but the main theorem I did publish with a non-equivariant proof in 1997, 
A homotopy-theoretic proof of Williams's metastable Poincare embedding theorem.

One of Michael's people, Fred Cohen, took me under his wing in 1982 and taught me about the Snaith splitting of O^nS^n X, and that was very important in my stable splitting of Omega SU(n), which again I failed to publish, but thankfully Michael Crabb and Steve Mitchell explained my proof in their generalization 
The loops on U(n)/O(n) and U(2n)/Sp(n).
By this time I was mostly a student of Mark's, but if it weren't for Michael, I wouldn't have had anything to offer Mark. 
Here's a conversation from about that time with Michael, Jeff Smith and Mike, in the NWU student center, which served beer for a while.  I think this shows Michael's incredible wit:
Jeff) Writing papers is terrible.
Mike) Sure.  Wouldn't it be great if there were these people you could talk to and they'd write up your paper?
Michael) Ah, and the next step is not to have to talk to these people!

As a number of folks have posted, Michael and his wife Eileen were great hosts, and I'm sure made a healthy contribution to our math community.  When my wife Kathleen and I ended up in Chicago in 1988, Michael and Eileen put us up for a few weeks, and I remember one brilliant interchange:
Michael) John Moore and I were walking around talking about this conjecture and we walked into a bar...
Eileen) A bar! Fancy that!
Michael at this time explained to me Giora's thesis, which I think was a very nice theorem of Michael's from the 50s about the attaching map of a Thom complex using the relative Hopf invariant.   

In 1992 I was at NWU and worked with Brayton Gray, another one of Michael's people, and Michael on a conjecture of Mark's about the EHP sequence, which led to my cryptic 1995 PAMS paper
The H-space squaring map on O^3 S^{4n+1} factors through the double suspension,
about which Jim Boardman wrote 
``The main tools are a Barratt-style unstable symmetry formula in Theorem 2.3 for the James-Hopf invariant H_2 and a dual Barratt-Toda formula in Theorem 3.1 which uses work of Steer and the reviewer.''
I probably should have made both Michael and Brayton (who had a nice colifting idea) coauthors, but the argument was dramatically improved much later by Brayton in my later odd-primary paper PAMS paper
A conjecture of Gray and the p-th power map on Ω^2 S^{2np+1},
which again Brayton should have been a coauthor, but here I have an excuse: somebody had to referee the paper :) Continuing to explain how Michael's people were able to work together, I come to Michael's final paper in 1995, which was both a simplified version and a generalization of my paper above, a paper mostly due to Fred.  When I explained to Fred the title theorem of my 1995 PAMS paper, Fred told me over the phone how a simpler version of my argument would prove Mark's conjecture, that the E_2 term of the Z/2 EHP spectral sequence is a Z/2 vector space. But if Michael hadn't explained to me how to use his unstable Hopf invariant symmetry formulas here, none of this would have happened. 

I'll end with a quote of Mark's, when I told him that Michael had forgotten something he told me much earlier:
Mark) Sure!  Michael's already forgotten more than you'll ever know!
I really miss both of them, Mark and Michael. 


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