Introduction x In [9, 10, 11], Cohen introduced a combinatorial group S) with a representa- tion to the group of the homotopy classes of functorial self maps of loop suspensions. This group is important for studying the classical exponent problem in homotopy theory, for instance the classical results in [2] can be obtained by some simple com- binatorial computations in the group 5}. In [39, 40, 41, 45, 48], the group S) has been successfully applied to solve some problems in the classical homotopy the- ory including the Cohen conjecture. As a combinatorial group, 5} has connections with homotopy string links studied by Milnor and Habegger-Lin in low dimensional topology [27, 28, 19, 23], as well as braid groups and simplicial groups [3, 13, 46]. The purpose of this article is to study the maps from loop suspensions to loop spaces using group representations. In particular, we obtain a generalization of the Cohen group $). Our generalization provides a way to construct various Cohen type groups related to maps from loop suspensions to loop spaces. By investigating the reduced diagonals, we obtain the shuffle relations on the Cohen groups. The quotient of the Cohen group S) by the shuffle relations gives a universal ring 9\ for functorial self maps of double loop spaces of double suspensions. The ring 91 is related to the extension groups of the important symmetric group modules Lie(rc) by investigating functors to coalgebras. Moreover the obstructions to the exponent problem in homotopy theory are displayed in these extension groups. In addition to homotopy theory, the representation theory of the ring 91 is related to the functorial version of the Poincare-Birkhoff-Witt Theorem, with connections to the modular representation theory of the symmetric groups. The remainder of this introduction describes our results in more details. Let X be a pointed space. Recall that the James construction J(X) is the free monoid generated by points in X modulo the single relation that the basepoint * = 1, with the weak topology. The classical James theorem [21] states that J(X) is (weak) homotopy equivalent to QEX if X is path-connected. The James filtration {Jn(X)} is the word filtration, namely Jn(X) is the quotient space of Xn by the equivalence relation generated by \X\, . . . , X{ i , *, X^, . . . , Xn—i) ~ v^l ' %j 1' *' x ji x n l) for any 2 i, j n. Let X^ denote the n-fold self smash product of X. Then Jn(X)/Jn-i(X) is homeomorphic to X^n\ Let qn: Xn Jn(X) be the quotient map. By the suspension splitting theorem [21], the inclusions J n _i(X) C Jn(X) induce a tower of group epimorphisms [J(X), QY] » [Jn(X), QY] » [X, nY) 1 Received by the editor January 25, 2005. Research is supported in part by the Academic Research Fund of the National University of Singapore. 1
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