Geometric Orbifold Cohomology

Topological phases of quantum materials and brane charges in M-theory are measured by extraordinary cohomology theories defined on orbifold spacetimes. Geometric Orbifold Cohomology presents a modernized and enhanced formulation of these theories, establishing a rigorous framework for nonabelian and differential cohomology in the setting of higher geometry. Motivated by cutting-edge problems in mathematical physics—specifically the analysis of M-brane charges and topological insulators—this monograph bridges the gap between classical equivariant topology and modern cohesive homotopy theory. It fills a critical gap in the literature by offering a unified perspective where orbifold geometry is treated synthetically via modal operators in higher topos theory, allowing for a precise treatment of geometric singularities and differential structures. • Accessible Foundations: Begins with a streamlined reconstruction of twisted nonabelian orbifold cohomology and orbifold K-theory using the language of topological groupoids and stacks, offering a pedagogical entry point for new- comers. • Higher Topos Theory: Provides a detailed exposition of the transition from classical perspectives to the modern context of cohesive higher topos theory and global equivariant unstable homotopy theory. • Synthetic Orbifold Geometry: Lays out a powerful synthetic differential theory of higher orbifold Cartan geometry, utilizing systems of modal operators to rigorously capture orbi-singularities and proper equivariant homotopy types. • Unification of Methods: Unifies orbifold geometry with classical differential geometry and cohomology, compatibly enhancing the picture with modalities that reflect proper equivariant structures. • Concrete Applications: Showcases the theory’s utility through the analysis of tangentially twisted nonabelian orbifold cohomology, specifically J-twisted orbifold Cohomotopy, applied to "Hypothesis H" and the classification of topo- logical phases. This volume is desi