Primitive obstructions in the cohomology of loopspaces

Abstract

Let X X and X ′ X’ be H H -spaces. If f : Ω X → Ω X ′ f:\Omega X \to \Omega X’ is an H H -map then the obstruction to f f being a homotopy-commutative map is a subset { c 2 ( f ) } ⊂ [ Ω X Λ Ω X ; Ω 2 X ′ ] \left \{ {{c_2}(f)} \right \} \subset \left [ {\Omega X\Lambda \Omega X;{\Omega ^2}X’} \right ] . In this paper we prove: I f [ f ] If[f] is in the image of the composition \[ [ P k + m Ω X ; X ′ ] → [ Σ Ω X ; X ′ ] → ≈ [ Ω X ; Ω X ′ ] , \left [ P_{k + m} \Omega X;X’ \right ] \rightarrow \left [ {\Sigma \Omega X;X’} \right ] \stackrel {\approx }{\rightarrow } \left [ {\Omega X;\Omega X’} \right ], \] then { c 2 ( f ) } \left \{ {{c_2}(f)} \right \} is in the image of the composition \[ [ P k Ω X Λ P m Ω X ; X ′ ] → [ Σ Ω X Λ Σ Ω X ; X ′ ] → ≈ [ Ω X Λ Ω X ; Ω 2 X ′ ] . \left [ P_k \Omega X\Lambda {P_m}\Omega X;X’ \right ] \rightarrow \left [ \Sigma \Omega X\Lambda \Sigma \Omega X;X’ \right ] \stackrel {\approx }{\rightarrow } \left [ \Omega X\Lambda \Omega X;{\Omega ^2}X’ \right ]. \] Consequently if α ∈ H n ( Ω X ; Z p ) \alpha \in {H^n}(\Omega X;{Z_p}) is an A 3 {A_3} -class in the sense of Stasheff then each element of { c 2 ( f ) } \left \{ {{c_2}(f)} \right \} is of the form ∑ c ′ i ⊗ c i \sum {{{c’}_i}} \otimes {c_i} where the c i {c_i} are primitive.