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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.02793 |
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Table of Contents:
- In 1992, Stolz proved that, among simply connected Spin-manifolds of dimension 5 or greater, the vanishing of a particular invariant $α$ is necessary and sufficient for the existence of a metric of positive scalar curvature. More precisely, there is a map $α\colonΩ_*^{\rm Spin}\to {\rm ko}$ (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 $\kerα$ is the image of a transfer map $Ω_{*-8}^{\rm Spin}{\rm BPSp}(3)\toΩ_*^{\rm Spin}$. In this paper we prove an analogous result for Spin$^c$-manifolds and a related invariant $α^c: Ω_*^{{\rm Spin}^c} \to {\rm ku}$. We show that $\kerα^x$ is the sum of the image of Stolz's transfer $Ω_{*-8}^{\rm Spin}{\rm BPSp}(3) \to Ω_*^{{\rm Spin}^c}$ and an analogous map $Ω_{*-4}^{{\rm Spin}^c}{\rm BSU}(3) \to Ω_*^{{\rm Spin}^c}$. Finally, we expand on some details in Stolz's original paper and provide alternate proofs for some parts.