2024/07/14 by Osamu Saeki, Saeki, Osamu · 3 citations
Computer Science · Engineering · #57R45 (Primary) 58K30 #57R70 #58K05 (Secondary) #Advanced Numerical Analysis Techniques #Digital Image Processing Techniques #FOS: Mathematics #Geometric Topology (math.GT) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2407.10145
openalex publication_date 2024/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study how to construct explicit deformations of generic smooth maps from closed n--dimensional manifolds M with n ≥ 2 to the 2--sphere S2 and show that every smooth map M → S2 is homotopic to a C^∞ stable map with at most one cusp point and with only folds of the middle absolute index. Furthermore, if n is even, such a C^∞ stable map can be so constructed that the restriction to the singular point set is a topological embedding. As a corollary, we show that for n ≥ 2 even, there always exists a C^∞ stable map M → R2 with at most one cusp point such that the restriction to the singular point set is a topological embedding. As another corollary, we give a new proof to the existence of an open book structure on odd dimensional manifolds which extends a given one on the boundary, originally due to Quinn. Finally, using the open book structure thus constructed, we show that k--connected n--dimensional manifolds always admit a fold map into R2 without folds of absolute indices i with 1 ≤ i ≤ k, for n ≥ 7 odd and 1 ≤ k ≤ (n-5)/2.