The space of sections of a smooth function
This work addresses evasion path problems in mobile sensor networks, offering theoretical tools for analysis, but it appears incremental as it builds on existing topological methods.
The paper tackles the problem of computing connected components of the space of sections for a smooth function on a compact manifold, using homotopy groups of fibers, and applies this to derive new results on evasion paths in mobile sensor networks, providing necessary and sufficient conditions and lower bounds on homotopy classes without connectivity assumptions.
Given a compact manifold $X$ with boundary and a submersion $f : X \rightarrow Y$ 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.