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 designed to serve a dual purpose: the first part acts as an invitation for advanced graduate students and beyond, in mathematics and theoretical physics, leading them from basic topological charges to modern research applications. The subsequent parts provide a comprehensive resource for academic researchers in al- gebraic topology, differential geometry, and string theory interested in a cutting edge formulation of geometric cohomology and its applications to quantum systems.