Subject: Brayton's question
Date: Sat, 03 May 2003 23:27:53 +0900
From: Norio IWASE
To: dmd1@lehigh.edu
Dear Don,
The answer to Brayton's question depends on the choice of H(m)
: T*T -> ST the Hopf construction of the multiplication m. If
we replace H(m) : T*T -> ST by Stasheff's projection p_2, the
answer is 'yes'. I do not know the answer for the Sugawara's
projection H(m). (cf. [Morisugi, Contemp. Math. 239 (1999),
225--238] and [Stasheff, Tran. AMS. 108 (1963), 275--292]):
Assume m has a two-sided strict unit. Let E = Tu[0,1]xTxTuT/-
be the push out of two maps pr_1 : TxT -> T and m : TxT -> T.
By identifying the subspace [0,1]xTx* with T by the projection,
we obtain a space E_2 which is homotopy equivalent to T*T. We
regard ST as the reduced push out of two trivial maps T -> *.
Then pr_2 : TxT -> T induces Stasheff's p_2 : E_2 -> ST.
By [Stasheff], p_2 is a quasi-fibration if m has right and left
homotopy inversions. Let q : F -> T be a homotopy fibre
F = {v : [0,1] -> E_2 | v(0)=*, v(1)=i(x) for some x in T}
of the inclusion i : T -> E_2. There is a canonical projection
p : F -> LST by p(v) = p_2v, which is a homotopy equivalence,
since p_2 is a quasi-fibration. Let j be the inverse of p so
that r=qj : LST -> T is the map in question.
For any element x in T, let l(x) be the loop on ST defined by
l(x)(t) = [t,x] in ST. This gives a map l : T -> LST. We have
a lift L : T -> F of l as follows
L(x)(t) = [t,*,x] in E_2, and hence pL = l.
By the definition of the map q : F -> T, we have
qL(x) = L(x)(1) = [1,*,x] = m(*,x) = x, and hence qL=1.
Thus rl = qjpL is homotopic to qL = 1 : T -> T (see [Morisugi]).
I am afraid that I am not fully following the notations given
in [Morisugi]. I did also alter the definition of p_2 : E_2 ->
ST in [Morisugi] by the one in [Stasheff] to make our arguments
simpler. The following argument gives the answer:
Similarly, we have another map M : TxT -> F given by
M(x,y) = L(x)+L_x(y), L_x(y)(t) = [t,x,y] in E_2,
where u+v is the addition of paths u and v.
Then M : TxT -> F gives a lift of A(lxl) : TxT -> LST as
pM(x,y) = l(x)+l(y) = A(lxl)(x,y), and hence pM=A(lxl),
where A denotes the loop addition. On the other hand, we have
qM(x,y) = M(x,y)(1) = [1,x,y] = m(x,y), and hence qM=m.
Thus rA(lxl) = qjpM is homotopic to qM = m : TxT -> T.
This observation shall also imply that
r(l(x_1)+l(x_2)+...+l(x_{n-1})+l(x_n))
= (...(x_1.x_2)...).x_{n-1}).x_n.
Best Regards, Norio Iwase
--
> Subject: Re: 2 questions
> Date: Tue, 29 Apr 2003 15:23:43 -0500
> From: Brayton Gray
>
> for the list:
>
> A long time ago Sugawara proved that if T is a connected H space of the
> homotopy type of a CW complex, there is a fibering of the form:
> T-----> T*T ------>ST
> where the projection is the Hopf construction on the multiplication.
> Continuing the fiber sequence to the left, we have a map:
>
> LST ----> T
>
> where LST is the loop space of the suspension of T. Presumably this is
> an extension of the multiplication:
>
> T x T ----> LST ----> T
>
> Does anyone know of a reference for this or a proof? Since there are
> other maps T x T ----> T
> for connected CW H spaces involving inverse maps, it is conceivable
> that this presumption is false.
> ________________________________________________
--
_/_/ Norio IWASE _/_/
--
_/_/ Norio IWASE @ Fac Math Kyushu U _/_/