On the Thom conjecture in CP³

2021/09/10 by Daniel Ruberman, Marko Slapar, Ruberman, Daniel +3
Mathematics · #14J70 (Secondary) #57K40 (Primary) 57R40 #57R95 #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2109.05089

openalex publication_date 2021/09/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

What is the simplest smooth simply connected 4-manifold embedded in CP³ homologous to a degree d hypersurface V_d? A version of this question associated with Thom asks if V_d has the smallest b₂ among all such manifolds. While this is true for degree at most 4, we show that for all d ≥ 5, there is a manifold M_d in this homology class with b₂(M_d) < b₂(V_d). This contrasts with the Kronheimer-Mrowka solution of the Thom conjecture about surfaces in CP², and is similar to results of Freedman for 2n-manifolds in CPⁿ⁺¹ with n odd and greater than 1.

Related

• Relative Thom Conjectures, symplectic and beyond
• The symplectic Thom conjecture
• Geometric proof of Thom conjecture
• Embeddings of 4--manifolds in \CP³
• Realizing a homology class of a compact manifold by a homology class of an explicit closed submanifold--a new approach to Thom's works on homology classes of submanifolds-