2012/10/31 by Stefan Behrens, Kenta Hayano
Mathematics · #Advanced Combinatorial Mathematics #Class (philosophy) #Computer science #Cusp (singularity) #Dimension (graph theory) #Focus (optics) #Geometric and Algebraic Topology #Geometry #Homotopy #Homotopy and Cohomology in Algebraic Topology #Mathematical analysis #Mathematics #Pure mathematics #Set (abstract data type) #Simple (philosophy) #Surface (topology) #Uniqueness #math.GT
paper · pdf · doi:10.1112/plms/pdw042
This submission replaces the previous version entitled "Vanishing Cycles and Homotopies of Wrinkled Fibrations". The article has been completely rewritten to incorporate new and improved techniques, leading to a strengthening of the main results and a clearer exposition. (49 pages, 15 figures)
arxiv created 2014/12/08 · openalex publication_date 2016/10/04 · arxiv updated 2018/05/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We study a class of homotopies between maps from 4-manifolds to surfaces which we call cusp merges. These homotopies naturally appear in the uniqueness problems for certain pictorial descriptions of 4-manifolds derived from maps to the 2-sphere (for example, broken Lefschetz fibrations, wrinkled fibrations, or Morse 2-functions). Our main results provide a classification of cusp merge homotopies in terms of suitably framed curves in the source manifold, as well as a fairly explicit description of a parallel transport diffeomorphism associated to a cusp merge homotopy. The latter is the key ingredient in understanding how the aforementioned pictorial descriptions change under homotopies involving cusp merges. We apply our methods to the uniqueness problem of surface diagrams of 4-manifolds and describe algorithms to obtain surface diagrams for total spaces of (achiral) Lefschetz fibrations and 4-manifolds of the form M × S 1 , where M is a 3-manifold. Along the way we provide extensive background material about maps to surfaces and homotopies thereof and develop a theory of parallel transport that generalizes the use of gradient flows in Morse theory.