The first Pontrjagin classes of homotopy complex projective spaces

Abstract

Let M2n be an oriented closed smooth manifold homotopy equivalent to the complex projective space CP(n). The main purpose of this paper is to show that when n is even, the difference between the first Pontrjagin class of M2n and that of CP(n) is divisible by 16. As a geometric application of this result, we prove that the Kervaire sphere of dimension 4k+1 does not admit any free circle group action if k+1 is not a power of 2.