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
