A nonlinear Lazarev–Lieb theorem: L2-orthogonality via motion planning

Abstract

Lazarev and Lieb showed that finitely many integrable functions from the unit interval to [Formula: see text] can be simultaneously annihilated in the [Formula: see text] 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 [Formula: see text]-norm bound. Our proof uses a variety of motion planning algorithms that instead of contractibility yield a lower bound for the [Formula: see text]-coindex of a space.