On manifolds homeomorphic to spheres

2025/09/16 by Somnath Basu, Basu, Somnath, Sachchidanand Prasad +1
Computer Science · Mathematics · #55M99 #55Q40 #Algebraic Topology (math.AT) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Primary: 57R70 #Secondary: 55R10 #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2509.13008

openalex publication_date 2025/09/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove a result analogous to Reeb's theorem in the context of Morse-Bott functions: if a closed, smooth manifold M admits a Morse-Bott function having two critical submanifolds S^k and S^l (k ≠ l), then M has dimension k+l+1 and is homeomorphic to the standard sphere S^k+l+1 but not necessarily diffeomorphic to it. We also prove similar results for projective spaces over the real numbers, complex numbers and quaternions.

Citations

• On the generalized brace product: relation to H-splitting of loop space fibrations & the J-homomorphism
• On smooth functions with two critical values
• A minimal Brieskorn 5-sphere in the Gromoll-Meyer sphere and its applications
• The entropy formula for the Ricci flow and its geometric applications

Related

• On smooth functions with two critical values
• Characterizing 3-dimensional manifolds represented as connected sums of Lens spaces, S² × S¹, and torus bundles over the circle by certain Morse-Bott functions
• Approximations of Lipschitz maps via Ehresmann fibrations and Reeb's sphere theorem for Lipschitz functions
• Extending functions from a neighborhood of the sphere to the ball
• Perfect Morse functions and exotic S² x S²'s