On manifolds with the homotopy type of complex projective space

Abstract:

It is known that in every even dimension greater than four there are infinitely many nonhomeomorphic smooth manifolds with the homotopy type of complex projective space. In this paper we provide an explicit construction of homotopy complex projective spaces. Our initial data will be a manifold X with the homotopy type of ${\mathbf {C}}{{\mathbf {P}}^3}$ and an embedding ${\gamma _3}:{S^5} \to {S^7}$ . A homotopy 7-sphere ${\Sigma ^7}$ is constructed and an embedding ${\gamma _4}:{\Sigma ^7} \to {S^9}$ may be chosen. The procedure continues inductively until either an obstruction or the desired dimension is reached; in the latter case the final obstruction is the class of ${\Sigma ^{2n - 1}}$ in ${\Theta _{2n - 1}}$. Should this obstruction vanish, the final choice is of a diffeomorphism ${\gamma _n}:{\Sigma ^{2n - 1}} \to {S^{2n - 1}}$. There results a manifold, denoted $(X,{\gamma _3}, \cdots ,{\gamma _{n - 1}},{\gamma _n})$, with the homotopy type of ${\mathbf {C}}{{\mathbf {P}}^n}$. We describe the obstructions encountered, but are able to evaluate only the primary ones. It is shown that every homotopy complex projective space may be so constructed, and in terms of this construction, necessary and sufficient conditions for two homotopy complex projective spaces to be diffeomorphic are stated.