2024/10/28 by Katherine Williams Booth, Booth, Katherine Williams
Engineering · Mathematics · #20F65 #57K20 #57S05 #Advanced Numerical Analysis Techniques #Advanced Theoretical and Applied Studies in Material Sciences and Geometry #FOS: Mathematics #Geometric Topology (math.GT) #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2410.21460
openalex publication_date 2024/10/28 · openalex created_date 2024/11/14 · openalex updated_date 2026/07/28
In this paper, we study Homeo1(S), the group of homeomorphisms of a surface that preserve the set of one-dimensional C1 submanifolds of that surface. The group Homeo1(S) belongs to a family of similarly defined groups Homeok(S) that were recently introduced by the author. In a separate paper, we have shown that for most closed surfaces, Homeok(S) is naturally isomorphic to the automorphisms of a smooth fine curve graph. By contrast, the work in this paper gives local conditions that characterize Homeo1(S). We show that there exists a collection of conditions that are both necessary and sufficient for a homeomorphism of the surface to be an element of this group. These conditions primarily depend upon the structure of the induced map on the projective tangent bundle. Additionally, we provide examples of several types of elements of Homeo1(S) that are not diffeomorphisms. These include inducing discontinuous maps on the projective tangent bundle and having infinitely many non-differentiable points.