  Let $M$ be a manifold homotopy equivalent to the complex projective space $\C
P^m$. 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.
