Manifolds Carrying Large Scalar Curvature

Let W = S E be a complex spinor bundle with vanishing first Chern class over a simply connected spin manifold M of dimension 5. Up to connected sums we prove that W admits a twisted Dirac operator with positive order-0-term in the Weitzenb¨ock decomposition if and only if the characteristic numbers Aˆ(TM)[M] and ch (E)Aˆ(TM)[M] vanish. This is achieved by generalizing [2] to twisted Dirac operators