Brillouin Platycosms and Topological Phases

There exist ten distinct closed flat 3D manifolds, known as "platycosms," that hold significance in mathematics and have been postulated as potential geometric models for our Universe. In this Letter, we demonstrate their manifestation as universes of Bloch particles, namely as momentum-space units referred to as Brillouin platycosms, which are natural extensions of the Brillouin torus within a broader framework of projective crystallographic symmetries. Moreover, we provide exact K-theoretical classifications of topological insulators over these platycosms by the Atiyah-Hirzebruch spectral sequence, and formulate a complete set of topological invariants for their identification. Topological phase transitions are generically characterized by Weyl semimetals, adhering to the generalized Nielsen-Ninomiya theorem: the total chirality number over a Brillouin platycosm is even (zero) if the platycosm is nonorientable (orientable). Our Letter generalizes the notion of Brillouin torus to ten Brillouin platycosms and therefore fundamentally diversifies the stages on which Bloch wave functions can perform their topological dance.