2025/02/24 by Lawande, Shital, Kuldeep Saha, Saha, Kuldeep · 5 citations
Mathematics · #57K20 (Primary) 57R15 (Secondary) #57R52 #Advanced Operator Algebra Research #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2502.17640
openalex publication_date 2025/02/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study smooth proper embeddings of compact orientable surfaces in compact orientable 4-manifolds and elements in the mapping class group of that surface which are induced by diffeomorphisms of the ambient 4-manifolds. We call such mapping classes extendible. An embedding for which all mapping classes are extendible is called flexible. We show that for most of the surfaces there exists no flexible embedding in a 4-manifold with homology type of a 4-ball or of a 4-sphere. As an application of our method, we address a question of Etnyre and Lekili and show that there exists no simple open book decomposition of S5 with a spin page where all 3-dimensional open books admit open book embeddings. We also provide many constructions and criteria for extendible and non-extendible mapping classes, and discuss a connection between extendibility and sliceness of links in a homology 4-ball with S3 boundary. Finally, we give a new generating set of the group of extendible mapping classes for the trivial embedding of a closed genus g surface in S4, consisting of 3g generators. This improves a previous result of Hirose giving a generating set of size 6g-1.