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 _/_/