Homotopy pullback of AnAn-spaces and its applications to AnAn-types of gauge groups

We construct the homotopy pullback of AnAn-spaces and show some universal property of it. As the first application, we review Zabrodsky's result which states that for each prime p  , there is a finite CW complex which admits an Ap−1Ap−1-form but no ApAp-form. As the second application, we investigate AnAn-types of gauge groups. In particular, we give a new result on AnAn-types of the gauge groups of principal SU(2)SU(2)-bundles over S4S4, which is a complete classification when they are localized away from 2.