Coarse cohomology with twisted coefficients
-
Elisa Hartmann
Abstract
To a coarse structure we associate a Grothendieck topology which is determined by coarse covers. A coarse map between coarse spaces gives rise to a morphism of Grothendieck topologies. This way we define sheaves and sheaf cohomology on coarse spaces. We obtain that sheaf cohomology is a functor on the coarse category: if two coarse maps are close they induce the same map in cohomology. There is a coarse version of a Mayer-Vietoris sequence and for every inclusion of coarse spaces there is a coarse version of relative cohomology. Cohomology with constant coefficients can be computed using the number of ends of a coarse space.
(Communicated by Július Korbaš)
References
[Art62] Artin, M.: Grothendieck Topologies, Harvard University Press, Cambridge, Massachusetts, 1962.Search in Google Scholar
[BE17] Bunke, U.—Engel, A.: Coarse cohomology theories, http://arxiv.org/abs/1711.08599.Search in Google Scholar
[DKU98] Dranishnikov, A. N.—Keesling, J.—Uspenskij, V. V.: On the Higson corona of uniformly contractible spaces, Topology 37(4) (1998), 791–803.10.1016/S0040-9383(97)00048-7Search in Google Scholar
[dlH00] de la Harpe, P.: Topics in Geometric Group Theory. Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000.Search in Google Scholar
[Gra06] Grave, B.: Coarse Geometry and Asymptotic Dimension, PhD thesis, Georg-August Universitt Gttingen, 2006.Search in Google Scholar
[Har67] Hartshorne, R.: Local Cohomology, A seminar given by A. Grothendieck, Vol. 1961, Harvard University, Springer-Verlag, Berlin-New York, 1967.10.1007/BFb0073971Search in Google Scholar
[Har17] Hartmann, E.: A twisted version of controlled K-theory, http://arxiv.org/abs/1711.03746.Search in Google Scholar
[Har19a] Hartmann, E.: A totally bounded uniformity on coarse metric spaces, Topol. Appl. (2019), https://doi.org/10.1016/j.topol.2019.06.040.Search in Google Scholar
[Har19b] Hartmann, E.: Coarse homotopy on metric spaces and their corona, http://arxiv.org/abs/1907.03510.Search in Google Scholar
[Har19c] Hartmann, E.: Twisted coefficients on coarse spaces and their corona, http://arxiv.org/abs/1904.00380.Search in Google Scholar
[HR94] Higson, N.—Roe, J.: A homotopy invariance theorem in coarse cohomology andK-theory, Trans. Amer. Math. Soc. 345(1) (1994), 347–365.10.1090/S0002-9947-1994-1243611-8Search in Google Scholar
[HR00] Higson, N.—Roe, J.: AnalyticK-homology. Oxford Mathematical Monographs, Oxford University Press, Oxford, 2000.Search in Google Scholar
[HRY93] Higson, N.—Roe, J.—Yu, G.: A coarse Mayer-Vietoris principle, Math. Proc. Cambridge Philos. Soc. 114(1) (1993), 85–97.10.1017/S0305004100071425Search in Google Scholar
[Kee94] Keesling, J.: The one-dimensional Čech cohomology of the Higson compactification and its corona, Topology Proc. 19 (1994), 129–148.Search in Google Scholar
[McL07] McLarty, C.: The rising sea: Grothendieck on simplicity and generality. In: Episodes in the History of Modern Algebra (1800–1950), Hist. Math., vol. 32, Amer. Math. Soc., Providence, RI, 2007, pp. 301–325.Search in Google Scholar
[Mei08] Meier, J.: Groups, Graphs and Trees. London Mathematical Society Student Texts, vol. 73, Cambridge University Press, Cambridge, 2008.Search in Google Scholar
[Mit01] Mitchener, P. D.: Coarse homology theories, Algebr. Geom. Topol. 1 (2001), 271–297.10.2140/agt.2001.1.271Search in Google Scholar
[Roe03] Roe, J.: Lectures on Coarse Geometry. University Lecture Series, vol. 31, American Mathematical Society, Providence, RI, 2003.10.1090/ulect/031Search in Google Scholar
[Sch99] Schmidt, A.: Coarse geometry via Grothendieck topologies, Math. Nachr. 203 (1999), 159–173.10.1002/mana.1999.3212030111Search in Google Scholar
[Sha96] Shafarevich, I. R.: Algebraic Geometry 2, Springer-Verlag, Berlin Heidelberg, 1996.10.1007/978-3-642-60925-1Search in Google Scholar
[Sta68] Stallings, J. R.: On torsion-free groups with infinitely many ends, Ann. of Math. (2) 88 (1968), 312–334.10.2307/1970577Search in Google Scholar
[Tam94] Tamme, G.: Introduction to étale Cohomology. Universitext, Springer-Verlag, Berlin, 1994.10.1007/978-3-642-78421-7Search in Google Scholar
[Wei94] Weibel, C. A.: An Introduction to Homological Algebra. Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994.Search in Google Scholar
© 2020 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Regular papers
- Fuzzy deductive systems of RM algebras
- Congruence pairs of principal MS-algebras and perfect extensions
- The lattices of 𝔏-fuzzy state filters in state residuated lattices
- Central lifting property for orthomodular lattices
- EQ-Modules
- On the exponential Diophantine equation Pxn + Pxn+1 + ⋯ + Pxn+k-1 = Pm
- Remarks on some generalization of the notion of microscopic sets
- Disjointness of composition operators on Hv0 spaces
- A common fixed point theorem for non-self mappings in strictly convex menger PM-spaces
- The Poincaré-Cartan forms of one-dimensional variational integrals
- Coarse cohomology with twisted coefficients
- Divisible extension of probability
- Asymptotic behavior of the records of multivariate random sequences in a norm sense
- Strong convergence of the functional nonparametric relative error regression estimator under right censoring
- A new kumaraswamy generalized family of distributions: Properties and applications
- Efficient message transmission via twisted Edwards curves
- Computation of several Hessenberg determinants
Articles in the same Issue
- Regular papers
- Fuzzy deductive systems of RM algebras
- Congruence pairs of principal MS-algebras and perfect extensions
- The lattices of 𝔏-fuzzy state filters in state residuated lattices
- Central lifting property for orthomodular lattices
- EQ-Modules
- On the exponential Diophantine equation Pxn + Pxn+1 + ⋯ + Pxn+k-1 = Pm
- Remarks on some generalization of the notion of microscopic sets
- Disjointness of composition operators on Hv0 spaces
- A common fixed point theorem for non-self mappings in strictly convex menger PM-spaces
- The Poincaré-Cartan forms of one-dimensional variational integrals
- Coarse cohomology with twisted coefficients
- Divisible extension of probability
- Asymptotic behavior of the records of multivariate random sequences in a norm sense
- Strong convergence of the functional nonparametric relative error regression estimator under right censoring
- A new kumaraswamy generalized family of distributions: Properties and applications
- Efficient message transmission via twisted Edwards curves
- Computation of several Hessenberg determinants
