From Metamaterials to Micromorphic Continua: A Spectral and K-Theoretic Approach to Topological Elasticity

Topological phases are usually discussed in the context of discrete lattice systems, where band gaps and Bloch theory allow for K-theoretic classification. In elasticity, however, classical homogeneous continua are gapless and therefore topologically trivial. Enriched models such as Cosserat or relaxed-micromorphic elasticity introduce internal degrees of freedom that generate optical gaps and reopen the possibility of strong topological phases. In this talk I present a rigorous operator-theoretic framework for comparing a band-gap metamaterial with its homogeneous micromorphic counterpart. Because the enriched model acts on a larger Hilbert space, the relevant subspaces must be selected spectrally via Riesz projections rather than by fixed coordinate decompositions. This leads to a cluster-dependent identification of Hilbert spaces and a stable notion of bulk topological equivalence. I will explain how groupoid C^*-algebras provide the natural compactification of continuous momentum space and yield bulk–boundary and bulk–defect correspondences. In chiral micromorphic media, integrating out optical modes generates effective Chern–Simons terms, linking bulk invariants to protected surface waves. The goal is to clarify when homogenised enriched continua genuinely retain the strong topological content of the underlying metamaterial.