Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.02793 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915477040660480 |
|---|---|
| author | Granath, Elliot |
| author_facet | Granath, Elliot |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_02793 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Scalar Curvature And Transfer Maps In Spin And Spin^c Bordism Granath, Elliot Algebraic Topology 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. |
| title | Scalar Curvature And Transfer Maps In Spin And Spin^c Bordism |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2509.02793 |