Bundles with Non-multiplicative Â-Genus and Spaces of Metrics with Lower Curvature Bounds

2020/10/31 by Georg Frenck, Jens Reinhold
Mathematics · #Curvature #Diffeomorphism #Genus #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Homotopy #Homotopy and Cohomology in Algebraic Topology #Homotopy group #Manifold (fluid mechanics) #Mathematical analysis #Mathematics #Pure mathematics #Scalar curvature #Sectional curvature #math.AT #math.DG #math.GT #msc:53C21 #msc:57R20 #msc:57R22 #msc:58D17

paper · pdf · doi:10.1093/imrn/rnaa361

14 pages, 1 figure. v2: strengthened the main theorems and changed the title, reworked presentation following a referee report. Final version accepted for publication in Int. Math. Res. Not. (IMRN)

openalex publication_date 2020/11/19 · arxiv created 2020/11/23 · openalex created_date 2020/12/07 · arxiv updated 2021/03/01 · openalex updated_date 2026/08/05

Abstract

Abstract We construct smooth bundles with base and fiber products of two spheres whose total spaces have nonvanishing A-genus. We then use these bundles to locate nontrivial rational homotopy groups of spaces of Riemannian metrics with lower curvature bounds for all Spin manifolds of dimension 6 or at least 10, which admit such a metric and are a connected sum of some manifold and Sⁿ × Sⁿ or Sⁿ × Sⁿ⁺¹, respectively. We also construct manifolds M whose spaces of Riemannian metrics of positive scalar curvature have homotopy groups that contain elements of infinite order that lie in the image of the orbit map induced by the push-forward action of the diffeomorphism group of M.

Citations

• Ricci Flow and the Sphere Theorem
• The space of metrics of positive scalar curvature
• The index of elliptic operators on compact manifolds
• Differentiable Structures on Spheres
• The space of positive Ricci curvature metrics on spin manifolds
• Topological Methods in Algebraic Geometry