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The purpouse of this work is twofold. Firstly we give necessary and suficient conditions in order to have a vector bundle ζ^mk → Mⁿ to be cobordant to one of the form n^k ⊕ n^k ⊕ ... ⊕ n^k &rarr. Wⁿ in the sense of Conner and Floyd as defined in their book "Periodic Differentiable Maps". Secondly, among other results concerning involutions in low dimensional manifolds, we have complete results on non existence of involutions fixing complex projective spaces, and involutions fixing the disjoint union of product of spheres and one point.