Subject: RE: four postings
Date: Wed, 18 Feb 2004 01:01:35 +0800
From: "Wu Jie" <matwuj@nus.edu.sg>

Just a little note on Bill's arguments about the relations between H_2\circ H_2 and H_4:

For James-Hopf invariants, the map H_2\circ H_2: J(\Sigma X) ----> J((\Sigma X)^{\smash 4})

is H_4 composed with the loop suspension of the sum of the three permutations:

id +(32)+(231): (\Sigma X)^{\smash 4} ---> (\Sigma X)^{\smash 4}

In the case that \Sigma X=S^1,

H_2\circ H_2 =H_4

integrally.

In general, H_a\circ H_b is H_{ab} composed with the loop suspension of the summation of certain permutations, which can be done by computations in Fred's groups. "On combinatorial calculations of the James-Hopf maps, Topology, 37(1998), No. 5, 1011-1023."

I am not sure, but it looks possible to work out the composite H_2\circ \Omega\eta by combinatorial computations  because \Omega\eta: \Omega S^3=J(S^2)--> \Omega S^2=J(S^1)  restricted to J_n(S^2)can be written down as certain product element in the group [S^1\times ....\times S^1, J(S^1)].

On the other hand, it seems that the (left) inverse of \Omega\eta can be given by the Ganea-Hopf invariants:

Observe the Hopf fibration

\Sigma \Omega X \smash \Omega X --\eta-->\Sigma \Omega X ----> X

for X=CP^\infty in our case. The (left) inverse of \eta can be given as the composite:

\Omega \Sigma \Omega X ---\phi--->\Omega (X\vee X) --H--> \Omega\Sigma (\Omega X\smash \Omega X),

where \phi is the unique (up to homotopy) H-map induced from \Omega X ---> \Omega (X\vee X), the product of two canonical inclusions, and H is the Ganea-Hopf invariant. I made some combinatorial computations on this type of Hopf invariants in "On co-$H$-maps to the suspension of the projective plane, Topology and its Applications, 123(2002), 547-571."

Note: From the combinatorial view, \Omega (X \vee Y) is the free product of the (simplicial) groups G_1=\Omega X and G_2=\Omega Y. The kernel of

G_1* G_2 ---> G_1 \times G_2

is the free group which is exactly the Milnor's construction on G_1\smash G_2, where g_1\smash g_2 is identified with the commutator [g_1,g_2] in the group G_1*G_2. The (simplicial) map

H: G_1*G_2---> F(G_1\smash G_2)

can be written down explicitly. By composing the explicit map F(G)---> G*G, one gets an explicit (combinatorial) construction for the inverse of \Omega\eta. In our case, we can make computation in the self free product of K(\Z,2)=CP^\infty. It seems possible to set up relations between Ganea-Hopf invariants and the James-Hopf invariants in combinatorial way, which might give the (right) inverse of
H_2:\Omega S^2 --->\Omega S^3.

Best Regards,

Jie
______________________________________________________________\\   

Subject: Re: four postings
Date: Tue, 17 Feb 2004 20:27:28 -0600
From: Bill Richter <richter@math.northwestern.edu>

Neil Strickland wrote:

   Corollary III.6.3 in Baues's "Commutator calculus" book says that
   in \pi_*(S^*) we have

     H(b \circ a) = H(b) \circ a + (b # b) \circ H(a),

   provided that H is defined by the James-Hopf procedure with the
   left lexicographic ordering.

I don't believe this formula.  Possibly it's correct because we're on
spheres, and I don't have a counterexample ready, but Baues makes a
related error in his book.  Baues falsely claims a Cartan formula for
the left lex James-Hopf inv, & not just for spheres:

H_2(b + a) = H_2(b) + (b U a) + H_2(a)

See if his composition result uses his false Cartan formula, & find
the error in his (short) Cartan proof.  Baues's book is helpful
otherwise, it's a reasonable exposition of Barratt's work.  It's not
perfect, but it's better than Michael's version :D In general, I think
Baues is a good promoter of subjects.  For instance, his book
"Algebraic Homotopy" is a nice intro to model category theory.
Baues's axioms aren't as powerful as Quillen's, so it's not efficient,
but Baues's gives a great answer to, "Why model categories?"

The obvious reason to doubt these formulas is that there are 2
definitions of b # b and b U a.  After a suspension, the 2 definitions
are equal, and that's evidence for Boardman & Steer's formulas.
Boardman & Steer's composition formula is a straightforward
application of the Cartan formula (with sign errors for H_{k>2}).