Homotopy complex projective spaces with Pin(2)-action

Abstract

Let M be a manifold homotopy equivalent to the complex projective space CP m . Petrie conjectured that M has standard total Pontrjagin class if M admits a non-trivial action by S 1. We prove the conjecture for m<12 under the assumption that the action extends to a nice Pin(2)-action with fixed point. The proof involves equivariant index theory for Spin c -manifolds and Jacobi functions as well as classical results from the theory of transformation groups.