Date: Wed, 8 Mar 2000 14:07:09 -0600 (CST)
From: Bill Richter <richter@math.nwu.edu>
Subject: Re: Richter preprint

   someone showed me a manuscript by Bill Richter in which he derives
   various stable splittings using Dold's fixed-point transfer.  I
   don't think it was ever published - I can't find it on MathSciNet
   or the Hopf archive, anyway.

Hi Neil, can you tell me more about what your interest is?  I have
several such cryptic manuscripts :) Here's what I've done:

Recently I proved a Cartan formula for my splitting of Omega SU(n).
So the Mitchell filtration F_k of Omega SU(n) splits multiplicatively.
The proof was an estimate that took me 12 years to get right.  John
Franks, one of the Dynamical Systems people at NWU, checked the hard
part of my estimate, so it's a legit proof.  I'll have this written up
& on the Hopf archive soon.  I have to merge the new code with my old
38 page proof of a partial Cartan formula (w.r.t. only Mitchell's
filtration 1, which is CP^{n-1}, the higher ones are singular
algebraic varieties).

This is a transfer splitting, analogous to the Kahn-Priddy splitting of
Q_0(S^0) using covering space transfers.  I use as you say Dold's
fixed-point transfer.  There's an intermediate result, Priddy & I
redid Snaith's splitting of BU using Becker-Gottlieb transfers.

So my proof of the Cartan formula is a double coset type calculation,
and the estimate was a proof that a certain map

m: F_p x F_q ---> F_q

is homotopic to the projection on the 2nd factor pi_2.  That doesn't
sound very hard, but my map m is really a map into an epsilon
neighborhood of a Grassmannian containing F_q.  It's easy to write
down a homotopy between m & pi_2 in the Grassmannian, but the homotopy
must be "controlled", stay in the epsilon neighborhood.  The problem
is that we don't have a nice deformation retraction map

r: Epsilon-nbhd ---> F_q

We only know that r exists by Hironaka's theorem on triangulating
singular algebraic varieties, or maybe by Durfee's extension of
Hironaka's result.

Well, this probably isn't the preprint you were referring to :)  I
also have a preprint where I showed that lots of splitting could be
called Hopf invariant splittings in that they all could be described
by Dold fixed point indices (not transfers).  But that wasn't a very
successful paper.  The point was to split the loops on a complex
Stiefel manifold this way, and I never succeeded.  The idea is that we
have an analogy

Q(S^0) = BU x Z = Omega U(n)

Q(S^p) = some BU-ish space which I can split = Omega V_{n+p,n}(C)

The starting point of the whole investigation was Miller's splitting
of U(n), which appeared to me to be a continuous geometry extension of
the discrete Hopf invariant splittings, (n choose k) is replaced by a
Grassmannian G_{n,k}.  But this preprint only has one new splitting in
it, the loops on the infinite Stiefel manifold.  I don't have a Hopf
invariant picture of Omega V_{n+p,n}, so the paper was mostly a
failure.

Greg Arone split Omega V_{n+p,n}(C) where I failed, using Weiss's
orthogonal Calculus.  But Greg cannot (according to Randy McCarthy)
show his splitting is multiplicative without doing a lotta work.  That
is, Greg split the Crabb model of Omega V_{n+p,n} which consists of
the polynomial based Stiefel loops

sum_{i=0}^k A_i z^i

where A_i is a complex (n+p,n) matrix.  Mitchell's work shows that the
Crabb model has a nice filtration whose associated graded is the right
kinda Thom complexes, so (by magic!) Greg has a stable splitting.  But
the Crabb model isn't multiplicative: if it were, we'd have a
multiplicative splitting of Omega V_{n+p,n}.  That the way that
Goodwillie recovered the multiplicative splitting of Q(S^p).  BTW

Does anyone know if Goodwillie's splitting is the same as the Hopf
invariant splitting (which goes back to Barratt)?  What about even the
James splitting of Omega Sigma X?

Back to my busted preprint: here's a way to head instead in a Calculus
direction :)

I'll certainly take another hack at coming up with a Dold fixed point
index splitting of Omega V_{n+p,n}.  Be nice to have 2 proofs, then we
could compare them, ask if the splitting maps are the same.  But
here's another idea.  My estimate is pretty close to a proof that the
Mitchell filtration (really the Mitchell-Segal group completion model
of Omega U(n)) is homotopy commutative.  My map

m: F_p x F_q ---> F_q

is actually the 1st component of a map

F_p x F_q ---> F_q x F_p

which I always thought was the flip map

(x,y) |---> (y,x).

So I've now proved "half" of this statement, and presumably the other
half follows, although I haven't succeeded yet.  It's a good exercise
for my "algebraic variety homotopy theory via estimates" idea.

But let's suppose that true, that the Mitchell-Segal group completion
model coprod_k F_k of Omega U(n) is homotopy commutative.  Seems like
you ought to be able to improve the Crabb model of Omega V_{n+p,n} to
a multiplicative model.  Maybe what's really needed is a C_2 structure
on the model coprod_k F_k, and I don't think I can do that, but you'd
guess that it exists.  Seems like there ought to be a slick Loop
Groups way of getting such a model, actually it must be a fattened
version of F_k, I'm thinking of my Epsilon-nbhd in the Grassmannian,
but maybe that's not the right way to do it.  Anyway, the Crabb model
of Omega V_{n+p,n} is a quotient of of Mitchell-Segal group completion
model of Omega U(n+p), so that's the place to start.

Bill