Let ri > 0(for each i) be an integer, and be the disjoint union of many real, complex, or quaterulonic projective (2ri + 1)-spaces according as or = R, C, H respectively.In this paper we show that smooth involutions(M, T)with fixed point set are not cobordat zero if and only if the total Stiefel Whitney class of all spheres whose dimensions larger than ac (m.α=1, 2, 4 respectively, when α= R, C, H) is nontrivial.