In recent years, topology has profoundly deepened our understanding of collective phases of waves, from electrons to magnons. Here, I focus on mechanical metamaterials: can topology illuminate their behavior, and conversely, can mechanics reveal general properties of topological phases?

The topological characterization of mechanical systems dates back to Maxwell's seminal work, yet a recent reformulation enables a direct experimental measurement of topology in mechanical configurations [1]. Remarkably, this approach requires no a priori theoretical model of the metamaterial, yet predicts the spatial location of low-energy deformations and self-stresses.

I will then discuss a distinct topological property emerging in the mechanics of non-orientable structures, exemplified by the Möbius strip [2]. Its buckling modes vanish along the ribbon, effectively embedding an “edge state” within the bulk and challenging the conventional bulk–boundary correspondence that underlies most topological phases. Finally, I will show that this notion of non-orientable mechanics extends far beyond mechanical systems, shedding light on frustrated phases of matter across different physical contexts [3].

[1] M. Guzman et al., PNAS 121, e2305287121 (2024).
[2] D. Bartolo and D. Carpentier, Phys. Rev. X 9, 041058 (2019).
[3] X. Guo et al., Nature 618, 506 (2023).