2011/02/25 by David Gay, David T. Gay, Robion Kirby · 1 citation
Computer Science · Mathematics · #Algebra over a field #Biology #Circle-valued Morse theory #Combinatorics #Computer science #Discrete Morse theory #Evolutionary biology #Fiber #Function (biology) #Geometric and Algebraic Topology #Homology (biology) #Homotopy and Cohomology in Algebraic Topology #Materials science #Mathematical analysis #Mathematics #Maxima and minima #Morse code #Morse homology #Morse theory #Pure mathematics #Range (aeronautics) #Topological and Geometric Data Analysis #Topology (electrical circuits) #math.GT #msc:57M50 #msc:57R17
paper · pdf · doi:10.1073/pnas.1018465108
12 pages, 8 figures, to appear in Proc. Nat. Acad. Sci. U.S.A. This contains a condensed presentation of most of the results in arXiv:1102.0750 as well as some alternative perspectives. Revised to add a reference
arxiv created 2011/02/25 · openalex publication_date 2011/04/25 · openalex created_date 2016/06/24 · arxiv updated 2016/07/13 · openalex updated_date 2026/08/05
We discuss generic smooth maps from smooth manifolds to smooth surfaces, which we call "Morse 2-functions," and homotopies between such maps. The two central issues are to keep the fibers connected, in which case the Morse 2-function is "fiber-connected," and to avoid local extrema over one-dimensional submanifolds of the range, in which case the Morse 2-function is "indefinite." This is foundational work for the long-range goal of defining smooth invariants from Morse 2-functions using tools analogous to classical Morse homology and Cerf theory.