Scalar Curvature And Transfer Maps In Spin And Spin^c Bordism

In 1992, Stolz proved that, among simply connected Spin-manifolds of dimension 5 or greater, the vanishing of a particular invariant $\alpha$ is necessary and sufficient for the existence of a metric of positive scalar curvature. More precisely, there is a map $\alpha\colon\Omega_*^{\rm Spin}\to {\rm ko}$ (which may be realized as the index of a Dirac operator) which Hitchin established vanishes on bordism classes containing a manifold with a metric of positive scalar curvature. Stolz showed $\ker\alpha$ is the image of a transfer map $\Omega_{*-8}^{\rm Spin}{\rm BPSp}(3)\to\Omega_*^{\rm Spin}$. In this paper we prove an analogous result for Spin$^c$-manifolds and a related invariant $\alpha^c: \Omega_*^{{\rm Spin}^c} \to {\rm ku}$. We show that $\ker\alpha^x$ is the sum of the image of Stolz's transfer $\Omega_{*-8}^{\rm Spin}{\rm BPSp}(3) \to \Omega_*^{{\rm Spin}^c}$ and an analogous map $\Omega_{*-4}^{{\rm Spin}^c}{\rm BSU}(3) \to \Omega_*^{{\rm Spin}^c}$. Finally, we expand on some details in Stolz's original paper and provide alternate proofs for some parts.