Subject: Arkowitz's Barratt-Puppe sequence orbit question
From: Bill Richter <richter@math.northwestern.edu>
Date: Mon, 17 Oct 2005 23:27:39 -0500

   Let Z be the mapping cone of a map X-->Y. Then there is the
   Barratt-Puppe sequence X -f-> Y --> Z --> Sigma(X) -->...  and an
   operation of [Sigma(X),V] on [Z,V] for any space V.  Apart from
   certain obvious situations (eg, V an H-space or f the inclusion of
   X in the cone on X), does anyone know of concrete calculations of
   the operation in specific cases?  For instance, what is the orbit
   or stability group of a non-trivial element in [Z,V]?

Martin, the Barcus-Barratt theorem calculates this if X & Z are
suspensions.  The orbit of alpha in [Z,V] is the "cokernel" of a map

[Sigma(Y), V] ---> [Sigma(X), V]

which is a sum of Sigma(f)^* plus terms involving Whitehead products
with f & alpha composed with Hopf invariants of f.  I'm sure you knew
it's just Sigma(f)^* in case f desuspends.  My favorite proof of the
Barcus-Barratt theorem is an unpublished one by Chris Stover.

I published an embarrassing concrete calculation with John Baez:

[S1 x S2, S2]

You'd guess this is just Z + Z, but it's smaller.  I was proud of my
Barcus-Barratt calculation, since nobody I asked knew how to do it,
but Mark Mahowald tells me this is a special case of Pontryagin's
calculation of maps from any 3-manifold into S2, and furthermore,
Pontryagin's calculation was important in Steenrod discovering Sq^i.
I hope the physicists don't know about Pontryagin, and were interested
in John's interpretation of the 2 Zs as the soliton/instanton numbers.

I had an interesting example of the dual of the Barcus-Barratt theorem
(maps into a fiber F when E & B are loop spaces) in the non-published
version of my Duke paper on Poincare embeddings.  Poincare duality
showed that the orbits were much smaller than you'd expect.  But under
the referee's relentless pressure, I realized uniqueness followed
formally from existence, and took the argument out.