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 $\pi_0$ and $\pi_1$ 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.

中文翻译：

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光滑函数的截面空间

给定一个带有边界的紧凑流形 $X$ 和一个淹没 $f ： X \rightarrow Y$ 其对 $X$ 边界的限制已经隔离了具有不同临界值的临界点，其中 $Y$ 是 $[0,1]$或$S^1$，$f$截面空间的连通分量由$f$纤维的$\pi_0$和$\pi_1$计算。然后利用该计算为移动传感器网络的回避路径问题的平滑版本提供新结果：从覆盖区域的时变同调和边界上同调的时变杯积，一个必要的和计算了回避路径存在的充分条件和回避路径同伦类数量的下界。不需要连接假设。