The space of sections of a smooth function

2020/06/22 by Gunnar Carlsson, Benjamin Filippenko, Carlsson, Gunnar +1
Computer Science · Mathematics · #55Q05 #57R19 #57R35 #62R40 #Algebraic Topology (math.AT) #Computational Geometry (cs.CG) #FOS: Computer and information sciences #FOS: Mathematics #Geometric Topology (math.GT) #Robotics (cs.RO) #cs.CG #cs.RO #math.AT #math.GT #msc:55Q05 #msc:57R19 #msc:57R35 #msc:62R40

paper · pdf · doi:10.48550/arxiv.2006.12023

38 pages, 1 figure

arxiv created 2020/06/22 · arxiv updated 2020/06/23

Abstract

Given a compact manifold X with boundary and a submersion f : X → Y whose restriction to the boundary of X has isolated critical points with distinct critical values and where Y is [0,1] or S¹, the connected components of the space of sections of f are computed from π₀ and π₁ of the fibers of f. This computation is then leveraged to provide new results on a smoothed version of the evasion path problem for mobile sensor networks: From the time-varying homology of the covered region and the time-varying cup-product on cohomology of the boundary, a necessary and sufficient condition for existence of an evasion path and a lower bound on the number of homotopy classes of evasion paths are computed. No connectivity assumptions are required.

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