The space of sections of a smooth function
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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 S^1, the connected components of the space of sections of f are computed from π_0 and π_1 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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