Čech cohomology groups are the simplest topological invariants of such spaces.
Juan García Escudero 2011, 'Randomness and Topological Invariants in Pentagonal Tiling Spaces', Discrete Dynamics in Nature and Societyhttp://dx.doi.org/10.1155/2011/946913. Retrieved from DOAJ CC BY 4.0 (https://creativecommons.org/licenses/by-sa/4.0/legalcode)
We propose a concrete description of that spectrum and of the corresponding cohomology theory.
Davide Gaiotto, Theo Johnson-Freyd 2019, 'Symmetry protected topological phases and generalized cohomology', Journal of High Energy Physicshttp://link.springer.com/article/10.1007/JHEP05(2019)007. Retrieved from DOAJ CC BY 4.0 (https://creativecommons.org/licenses/by-sa/4.0/legalcode)
Its cohomology contains torsion classes in various degrees.
Volker Braun, Mirjam Cvetič, Ron Donagi, Maximilian Poretschkin 2017, 'Type II string theory on Calabi-Yau manifolds with torsion and non-Abelian discretegauge symmetries', Journal of High Energy Physicshttp://link.springer.com/article/10.1007/JHEP07(2017)129. Retrieved from DOAJ CC BY 4.0 (https://creativecommons.org/licenses/by-sa/4.0/legalcode)
In this sense, cohomology underlies band topology.
A. Alexandradinata, Zhijun Wang, B. Andrei Bernevig 2016, 'Topological Insulators from Group Cohomology', Physical Review Xhttp://doi.org/10.1103/PhysRevX.6.021008. Retrieved from DOAJ CC BY 4.0 (https://creativecommons.org/licenses/by-sa/4.0/legalcode)
In particular, we determine its cohomology with trivial coefficients both for the basic and reduced complexes.
Yuan Lamei, Wu Henan 2016, 'Structures of W(2.2) Lie conformal algebra', Open Mathematicshttp://www.degruyter.com/view/j/math.2016.14.issue-1/math-2016-0054/math-2016-0054.xml?format=INT. Retrieved from DOAJ CC BY 4.0 (https://creativecommons.org/licenses/by-sa/4.0/legalcode)
We propose that group cohomology provides a symmetry-based classification of quasimomentum manifolds, which in turn determines the band topology.
A. Alexandradinata, Zhijun Wang, B. Andrei Bernevig 2016, 'Topological Insulators from Group Cohomology', Physical Review Xhttp://doi.org/10.1103/PhysRevX.6.021008. Retrieved from DOAJ CC BY 4.0 (https://creativecommons.org/licenses/by-sa/4.0/legalcode)
Moreover, we study extension theory and introduce a cohomology theory for ternary, and more generally n-ary, f-quandles.
Churchill Indu R. U., Elhamdadi M., Green M., Makhlouf A. 2018, 'Ternary and n-ary f-distributive structures', Open Mathematicshttp://www.degruyter.com/view/j/math.2018.16.issue-1/math-2018-0006/math-2018-0006.xml?format=INT. Retrieved from DOAJ CC BY 4.0 (https://creativecommons.org/licenses/by-sa/4.0/legalcode)
While a formal cohomology-based classification of such states was recently discovered, their physical properties remain mysterious.
Ashvin Vishwanath, T. Senthil 2013, 'Physics of Three-Dimensional Bosonic Topological Insulators: Surface-Deconfined Criticalityand Quantized Magnetoelectric Effect', Physical Review Xhttp://doi.org/10.1103/PhysRevX.3.011016. Retrieved from DOAJ CC BY 4.0 (https://creativecommons.org/licenses/by-sa/4.0/legalcode)
Hence, the uniqueness theorem connects all cohomology theories with partially compact supports with the singular cohomology theory.
L. Mdzinarishvili 2018, 'The uniqueness theorem for cohomologies on the category of polyhedral pairs', Transactions of A. Razmadze Mathematical Institutehttp://www.sciencedirect.com/science/article/pii/S2346809217301137. Retrieved from DOAJ CC BY 4.0 (https://creativecommons.org/licenses/by-sa/4.0/legalcode)
In particular, we argue that the structures can be understood in the language of extended equivariant cohomology.
Felix M. Haehl, R. Loganayagam, Mukund Rangamani 2017, 'Schwinger-Keldysh formalism. Part II: thermal equivariant cohomology', Journal of High Energy Physicshttp://link.springer.com/article/10.1007/JHEP06(2017)070. Retrieved from DOAJ CC BY 4.0 (https://creativecommons.org/licenses/by-sa/4.0/legalcode)