Lazarev and Lieb showed that finitely many integrable functions from the unit interval to $ℂ$ can be simultaneously annihilated in the $L^2$ inner product by a smooth function to the unit circle. Here, we answer a question of Lazarev and Lieb proving a generalization of their result by lower bounding the equivariant topology of the space of smooth circle-valued functions with a certain $W^{1,1}$-norm bound. Our proof uses a variety of motion planning algorithms that instead of contractibility yield a lower bound for the $\mathbb{Z}/2$-coindex of a space.

中文翻译：

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非线性 Lazarev-Lieb 定理：通过运动规划的 L2 正交性

Lazarev 和 Lieb 证明了从单位区间到$ℂ$可以同时被歼灭${\mathrm{大号}}^2$单位圆的平滑函数的内积。在这里，我们回答了 Lazarev 和 Lieb 的问题，通过将光滑圆值函数空间的等变拓扑下限与某个$W^{1,1}$-范数限制。我们的证明使用了各种运动规划算法，而不是可收缩性，而是为$\mathbb{Z}/2$-空间的coindex。