Topological materials can exhibit robust properties that are protected against disorder even in the absence of quantum effects. In mechanics and optics, topological protection has been primarily applied to linear waves and non-interacting systems at zero temperature. In this talk, we demonstrate how to construct topologically protected states that arise from the combination of strong interactions and thermal fluctuations inherent to soft matter. Specifically, we consider fluctuating lines under tension, subject to a class of spatially modulated substrate potentials. At equilibrium, the lines acquire a collective tilt proportional to an integer topological invariant called the Chern number. This quantized tilt is robust against substrate disorder, as verified by classical Langevin dynamics simulations. We establish the topological underpinning of this pattern via a mapping that we develop between the line fluid and Thouless pumping of an imaginary-time Mott insulator in which excitations are gapped by interactions. Our work points to a new class of classical topological phenomena in which the topological signature manifests itself in a structural property rather than a transport measurement.