Florian Frick, Matt Superdock

Lazarev and Lieb showed that finitely many integrable functions from the unit interval to $\u2102$ 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.