2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/60912Let $M$ 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.16 pages, no figuresGeometric TopologyAlgebraic Topology19J35; 19L47; 57R20; 57S25text