Schreiber Geometric Orbifold Cohomology

A monograph that we are finalizing:

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• Hisham Sati$\;$ and$\;$ Urs Schreiber:

  
  

  Geometric Orbifold Cohomology

  
  

  CRC Press (2026, to appear)

  ISBN:9781041147510

  • download draft: pdf (355 pages)

  
  

  based on these preprints:

  • Parts I, II: arXiv:2511.12720

  • Parts IV, V: arXiv:2008.01101

  
  

Abstract. Topological phases of quantum materials and generally brane charges in quantum gravity are measured by extraordinary cohomology of orbifolded spaces, and fragilely/microscopically by nonabelian cohomology twisted by tangential and other geometric orbifold structure. These applications, with more or less direct significance for cutting-edge experimental physics, motivate the further development of the ancient algebro-topological concept of cohomology to a robust theory of generalized nonabelian twisted geometric orbifold cohomology.

After surveying the general outlook and key example applications, the book begins gently with a pedagogical development of a streamlined construction of topological twisted nonabelian orbifold cohomology, realized as connected components of topological slice mapping stacks. We showcase in this manner a neat novel construction of twisted orbi-orientifold K-theory.

To refine this notion of twisted nonabelian orbifold cohomology and capture geometric (tangential) orbifold structure, the second half of the book transitions to modern geometric homotopy theory (higher topos theory). We introduce a notion of singular-cohesive $\infty$-toposes faithfully embedding manifolds, orbifolds and higher etale stacks and orbispaces into a modal synthetic geometry. This allows for the elegant axiomatization of higher orbi-differential Cartan geometry. Again by forming connected components of suitable slice mapping stacks, but now in the singular-cohesive $\infty$-topos, this yields relevant notions of generalized twisted nonabelian geometric orbifold cohomology.

A key result established in this generality is that tangentially twisted cohomology of orbifolds reduces in the vicinity of $G$-orbifold singularities to $RO(G)$-graded equivariant cohomology. We close by detailing the construction of twisted nonabelian (unstable) orbifold Cohomotopy in this manner, which in the motivating applications is the fragile/microscopic classifier for phases of Chern insulator materials and M-branes.

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Related monograph:

• Hisham Sati, Urs Schreiber:

  Equivariant principal $\infty$-bundles

  Cambridge University Press (2025)

Related articles

• Domenico Fiorenza, Hisham Sati, Urs Schreiber:

  Twisted Cohomotopy implies M-theory anomaly cancellation on 8-manifolds

  arXiv:1904.10207

• Domenico Fiorenza, Hisham Sati, Urs Schreiber:

  Twisted Cohomotopy implies M5 WZ term level quantization

  arXiv:1906.07417

• Hisham Sati, Urs Schreiber:

  Equivariant Cohomotopy implies orientifold tadpole cancellation

  arXiv:1909.12277

• Hisham Sati, Urs Schreiber:

  Lift of fractional D-brane charge to equivariant Cohomotopy theory

  arXiv:1812.09679

• Hisham Sati, Urs Schreiber:

  Differential Cohomotopy implies intersecting brane observables

  arXiv:1912.10425

• Hisham Sati, Urs Schreiber:

  Twisted Cohomotopy implies M5-brane anomaly cancellation

  arXiv:2002.07737

• Domenico Fiorenza, Hisham Sati, Urs Schreiber:

  Twisted Cohomotopy implies twisted String structure on M5-branes

  arXiv:2002.11093

• Domenico Fiorenza, Hisham Sati, Urs Schreiber:

  Twistorial Cohomotopy implies Green-Schwarz anomaly cancellation

  arXiv:2008.08544

• Domenico Fiorenza, Hisham Sati, Urs Schreiber:

  The Character Map in Twisted Non-Abelian Cohomology

  arXiv:2009.11909

• Hisham Sati, Urs Schreiber:

  The Character Map in Equivariant Twistorial Cohomotopy

  arXiv:2011.06533

Related talks:

• Geometric Orbifold Cohomology in Singular-Cohesive ∞-Topoi

  talk at ItaCa Fest 2025, 17 June 2025

• Higher and Equivariant Bundles

  talk via Feza Gürsey Higher Structures Seminar,

  Feza Gürsey Institute Istanbul, 2022

• Equivariant Super Homotopy Theory

  talk at Geometry in Modal Homotopy Type Theory

  CMU Pittsburgh, 2019

• Equivariant Cohomotopy of toroidal orbifolds

  talk at Prof. Sadok Kallel‘s group seminar

  AUS Sharjah, 2019