On generalized Whitehead products

Abstract

We define a symmetric monodical pairing G ◦ H among simply connected co-H spaces G and H with the property that S(G◦H) is equivalent to the smash product G∧H as co-H spaces. We further generalize the Whitehead product map to a map G ◦ H → G ∨ H whose mapping cone is the cartesian product. Whitehead products have played an important role in unstable homotopy. They were originally introduced [Whi41] as a bilinear pairing of homotopy groups: πm(X)⊗ πn(X) → πm+n−1(X), m, n > 1. This was generalized ([Ark62], [Coh57], [Hil59]) by constructing a map W: S(A ∧B) → SA ∨ SB. Precomposition with W defines a function on based homotopy classes: [SA,X]× [SB,X] → [S(A ∧B), X], which is bilinear in case A and B are suspensions. The case where A and B are Moore spaces was central to the work of Cohen, Moore and Neisendorfer ([CMN79]). In [Ani93] and in particular [AG95], this work was generalized. Much of this has since been simplified in [GT10], but further understanding will require a generalization from suspensions to co-H spaces. The purpose of this work is to carry out and study such a generalization. Let CO be the category of simply connected co-H spaces and co-H maps. We define a functor CO × CO → CO, (G,H) → G ◦H, and a natural transformation (1) W: G ◦H → G ∨H generalizing the Whitehead product map. The existence of G◦H generalizes a result of Theriault [The03] who showed that the smash product of two simply connected co-associative co-H spaces is the suspension of a co-H space. We do not need the coH spaces to be co-associative and require only one of them to be simply connected. We call G ◦H the Theriault product of G and H. Received by the editors November 28, 2009 and, in revised form, June 8, 2010. 2010 Mathematics Subject Classification. Primary 55P99, 55Q15, 55Q20, 55Q25. 1In fact we can define G ◦ H for any two co-H spaces but require at least one of them to be either simply connected or a suspension in order to obtain the co-H space structure map on G◦H. Recently Grbic, Theriault and Wu have shown that the smash product of any two co-H spaces is the suspension of a co-H space, but their construction cannot satisfy Theorem 1(a) below [GTW]. c ©2011 American Mathematical Society Reverts to public domain 28 years from publication 6143