On generalized Whitehead products

Abstract

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 co-H 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. We summarize our results in the following theorems.