H-spaces as direct product factors of loop spaces

Any H-space X can be imbedded in the loop space of its suspension ΩΣX as a direct product factor: X×Y≃ΩΣX for some Y. In this paper, we first study a condition on which the image of the cohomology suspension σ∗(x)∈H∗(ΩΣX;F) for any cohomology class x∈H∗(X;F) is identified with x⊗1∈H∗(X;F)⊗H∗(Y;F), where F is a field. We also give an explanation of any class in H∗(ΩΣX;F) as a class in H∗(X;F)⊗H∗(Y;F). We next generalize this argument to An-spaces. We show that any An-space X can be imbedded in the loop space of its projective n−1-space Pn−1X as a direct product factor: X×Yn−1≃ΩPn−1X for some Yn−1. Then we study a condition on which the class σ∗(x̂)∈H∗(ΩPn−1X;F) for any An−1-class x∈H∗(X;F) with εn−1(x̂)=x is identified with x⊗1∈H∗(X;F)⊗H∗(Yn−1;F), where εn−1 :ΣX→Pn−1X is the inclusion.