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 $\alpha \text{\hspace{0.17em}⁣}:{\mathrm{\Omega }}_{\ast }^{\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}\to \mathrm{k}\mathrm{o}$ (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 ${\mathrm{\Omega }}_{\ast -8}^{\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}\mathrm{B}\mathrm{P}\mathrm{S}\mathrm{p}(3)\to {\mathrm{\Omega }}_{\ast }^{\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}$. In this paper we prove an analogous result for Spin${}^c$-manifolds and a related invariant ${\alpha }^c:{\mathrm{\Omega }}_{\ast }^{{\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^c}\to \mathrm{k}\mathrm{u}$. We show that $\ker {\alpha }^x$ is the sum of the image of Stolz’s transfer ${\mathrm{\Omega }}_{\ast -8}^{\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}\mathrm{B}\mathrm{P}\mathrm{S}\mathrm{p}(3)\to {\mathrm{\Omega }}_{\ast }^{{\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^c}$ and an analogous map ${\mathrm{\Omega }}_{\ast -4}^{{\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^c}\mathrm{B}\mathrm{S}\mathrm{U}(3)\to {\mathrm{\Omega }}_{\ast }^{{\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^c}$. Finally, we expand on some details in Stolz’s original paper and provide alternate proofs for some parts.