“…Since the image of p 2 plays a key role, we define nonnegative integers called the reactivity of M , R(M ), and the (co)homologically fixed reactivity of M , R H (M ), by the following equations For other problems, for example counting the number of deformation equivalence classes of G 2 -structures on M as in [11], it is important to know the value of p 2 for diffeomorphisms. Hence, we defined the smooth reactivity of M , R Diff (M ), and the smooth (co)homologically fixed reactivity of M , R Diff H (M ) by the equations…”