SYMMETRIES OF HOMOTOPY COMPLEX PROJECTIVE 3 SPACES

We study symmetry properties of six-dimensional, smooth, closed manifolds which are homotopy equivalent to CP3. There are infinitely differentiably distinct such manifolds. It is known that if m is an odd prime, infinitely many homotopy Cp3's admit Z(m)-actions whereas only the standard Cp3 admits an action of the group Z(m) x Z(m) x Z(m). We study the intermediate case of Z(m) x Z(m)-actions and show that infinitely many homotopy Cp3's do admit Z(m) x Z(m)-actions for a fixed prime m. The major tool involved is equivariant surgery theory. Using a transversality argument, we construct normal maps for which the relevant surgery obstructions vanish allowing the construction of Z(m) x Z(m)-actions on homotopy Cp3's which are Z(m) x Z(m)-homotopy equivalent to a specially chosen linear action on Cp3. A key idea is to exploit an extra bit of symmetry which is built into our set-up in a way that forces the signature obstruction to vanish. By varying the parameters of our construction and calculating Pontryagin classes, we may construct actions on infinitely many differentiably distinct homotopy Cp3's as claimed.