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Continuity properties of measurable group cohomology

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Abstract

A version of group cohomology for locally compact groups and Polish modules has previously been developed using a bar resolution restricted to measurable cochains. That theory was shown to enjoy analogs of most of the standard algebraic properties of group cohomology, but various analytic features of those cohomology groups were only partially understood. This paper re-examines some of those issues. At its heart is a simple dimension-shifting argument which enables one to ‘regularize’ measurable cocycles, leading to some simplifications in the description of the cohomology groups. A range of consequences are then derived from this argument. First, we prove that for target modules that are Fréchet spaces, the cohomology groups agree with those defined using continuous cocycles, and hence they vanish in positive degrees when the acting group is compact. Using this, we then show that for Fréchet, discrete or toral modules the cohomology groups are continuous under forming inverse limits of compact base groups, and also under forming direct limits of discrete target modules. Lastly, these results together enable us to establish various circumstances under which the measurable-cochains cohomology groups coincide with others defined using sheaves on a semi-simplicial space associated to the underlying group, or sheaves on a classifying space for that group. We also prove in some cases that the natural quotient topologies on the measurable-cochains cohomology groups are Hausdorff.

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Notes

  1. In fact, earlier works such as Hochschild and Mostow’s [26] simply narrowed the requirement for long exact sequences to only those short exact sequences of modules that admit continuous left-inverses as sequences of topological spaces. With this convention, the theory \({\text{ H}}_\mathrm{cts}^*\) always has long exact sequences, but on the other hand the theory is not effaceable in \(\mathsf{P }(G)\) if one allows only inclusions of modules that give rise to such distinguished short exact sequences.

References

  1. Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos. Lecture Notes in Mathematics, vol. 269. Springer, Berlin (1972). Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Dirigé par M. Artin, A. Grothendieck, et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat

  2. Théorie des topos et cohomologie étale des schémas. Tome 2. Lecture Notes in Mathematics, vol. 270. Springer, Berlin (1972). Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat

  3. Théorie des topos et cohomologie étale des schémas. Tome 3. Lecture Notes in Mathematics, vol. 305. Springer, Berlin (1973). Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de P. Deligne et B. Saint-Donat

  4. Artin, E., Tate, J.: Class Field Theory. Princeton University Seminar Notes (1951)

  5. Artin, E., Tate, J.: Class Field Theory. American Mathematical Society, Providence (2009). Reprinted with corrections from the 1967 original

  6. Austin, T.: Pleasant extensions retaining algebraic structure, I. Preprint. arXiv.org:0905.0518

  7. Austin, T.: Pleasant extensions retaining algebraic structure, II. Preprint. arXiv.org:0910.0907

  8. Banach, S.: Théorie des opérations linéaires. Chelsea, New York (1955) (reprint of 1932 edition)

  9. Benyamini, Y., Lindenstrauss, J.: Geometric Nonlinear Functional Analysis. American Mathematical Society, Providence (2000)

    MATH  Google Scholar 

  10. Bochnak, J., Coste, M., Roy, M.-F.: Real algebraic geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 36. Springer, Berlin (1998). Translated from the 1987 French original, Revised by the authors

  11. Borel, A., Wallach, N.: Continuous cohomology, discrete subgroups, and representations of reductive groups. In: Mathematical Surveys and Monographs, vol. 67, 2nd edn. American Mathematical Society, Providence (2000)

  12. Bott, R., Tu, L.W.: Differential forms in algebraic topology. In: Graduate Texts in Mathematics, vol. 82. Springer, New York (1982)

  13. Bourbaki, N.: General Topology, vols. 1 and 2. Elements of Mathematics. Springer, Berlin (1989)

  14. Bredon, G.E.: Equivariant cohomology theories. In: Lecture Notes in Mathematics, No. 34. Springer, Berlin (1967)

  15. Brown, K.S.: Cohomology of groups. In: Graduate Texts in Mathematics, vol. 87. Springer, New York (1982)

  16. Buchsbaum, D.: Satellites and universal functors. Ann. Math. (2) 71, 199–209 (1960)

    Google Scholar 

  17. Davis, J.F., Kirk, P.: Lecture notes in algebraic topology. In: Graduate Studies in Mathematics, vol. 35. American Mathematical Society, Providence (2001)

  18. de Cornulier, Y., Tessera, R., Valette, A.: Isometric group actions on Hilbert spaces: growth of cocycles. Geom. Funct. Anal. 17(3), 770–792 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Eilenberg, S., MacLane, S.: Cohomology theory in abstract groups I. Ann. Math. (2) 48, 51–78 (1947)

    Google Scholar 

  20. Eilenberg, S., MacLane, S.: Cohomology theory in abstract groups II. Group extensions with a non-Abelian kernel. Ann. Math. (2) 48, 326–341 (1947)

    Google Scholar 

  21. Flach, M.: Cohomology of topological groups with applications to the Weil group. Compos. Math. 144(3), 633–656 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Grothendieck, A.: Produits tensoriels topologiques et espaces nucléaires, chap. 1: Produits tensoriels topologiques. Number 16 in Mem. Amer. Math. Soc. American Mathematical Society, Providence (1955)

  23. Guillemin, V.W., Sternberg, S.: Supersymmetry and equivariant de Rham Theory. In: Mathematics Past and Present, vol. 2. Springer, Berlin (1999)

  24. Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  25. Hochschild, G.: Cohomology of algebraic linear groups. Illinois J. Math. 5, 492–519 (1961)

    MathSciNet  MATH  Google Scholar 

  26. Hochschild, G., Mostow, G.: Cohomology of Lie groups. Illinois J. Math. 6, 367–401 (1962)

    MathSciNet  MATH  Google Scholar 

  27. Hofmann, K.: Category theoretical methods in topological algebra. In: Categorical Topology (Proc. Conf., Mannheim, 1975). Lecture Notes in Mathematics, vol. 540, pp. 345–403. Springer, Berlin (1976)

  28. Hofmann, K.H., Mostert, P.S.: Cohomology Theories for Compact Abelian Groups. With an appendix by Eric C. Nummela. Springer, New York (1973)

    Book  Google Scholar 

  29. Hurewicz, W.: Beiträge zur Topologie der Deformationen IV. Asphärische Räume. Nederl. Akad. Wetensch. Proc. (1936)

  30. Husemöller, D.: Fibre Bundles. Graduate Texts in Mathematics, vol. 20. Springer, New York (1995)

  31. Lichtenbaum, S.: The Weil-étale topology for number rings. Ann. Math. (2) 170(2), 657–683 (2009)

    Google Scholar 

  32. Mackey, G.W.: Borel structure in groups and their duals. Trans. Am. Math. Soc. 85, 134–165 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  33. Michael, E.: Continuous selections II. Ann. Math. 64(2), 562–580 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  34. Michael, E.: Continuous selections III. Ann. Math. 65(2), 375–390 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  35. Monod, N.: Continuous bounded cohomology of locally compact groups. Lecture Notes in Mathematics, vol. 1758. Springer, Berlin (2001)

  36. Montgomery, D., Zippin, L.: Topological Transformation Groups. Interscience Publishers, New York-London (1955)

    MATH  Google Scholar 

  37. Moore, C.C.: Extensions and low dimensional cohomology theory of locally compact groups I, II. Trans. Am. Math. Soc. 113, 40–63 (1964)

    MATH  Google Scholar 

  38. Moore, C.C.: Group extensions and cohomology for locally compact groups III. Trans. Am. Math. Soc. 221(1), 1–33 (1976)

    Article  MATH  Google Scholar 

  39. Moore, C.C.: Group extensions and cohomology for locally compact groups IV. Trans. Am. Math. Soc. 221(1), 35–58 (1976)

    Article  MATH  Google Scholar 

  40. Mostow, G.: Cohomology of topological groups and solvmanifolds. Ann. Math. 68(2), 20–48 (1961)

    Google Scholar 

  41. Onishchik, A., Vinberg, E.: Lie Groups and Algebraic Groups. Springer Series in Soviet Mathematics. Springer, Berlin (1990)

    Book  Google Scholar 

  42. Rudin, W.: Fourier Analysis on Groups. Wiley Classics Library. Wiley, New York (1962)

    Google Scholar 

  43. Schapira, P.: Abelian sheaves. Lecture Notes from Université Paris VI. http://people.math.jussieu.fr/schapira/lectnotes/

  44. Segal, G.: Categories and classifying spaces. Publ. Math. I.H.E.S. 34, 105–112 (1968)

    Google Scholar 

  45. Segal, G.: Cohomology of topological groups. In: Instituto Nazionale di alta Matematica, Symposia Mathematica, vol. IV, pp. 377–387. Academic Press, New York (1970)

  46. Thomas, C.B.: Characteristic classes and the cohomology of finite groups. Cambridge Studies in Advanced Mathematics, vol. 9. Cambridge University Press, Cambridge (1986)

  47. Thomas, G.E.F.: Integration of functions with values in locally convex Suslin spaces. Trans. Am. Math. Soc. 212, 61–81 (1975)

    Article  MATH  Google Scholar 

  48. Tu, J.-L.: Groupoid cohomology and extensions. Trans. Am. Math. Soc. 358(11), 4721–4747 (2006)

    Article  MATH  Google Scholar 

  49. van Est, W.: Group cohomology and Lie algebra cohomology in Lie groups I, II. Indag. Math. 15, 484–504 (1953)

    Google Scholar 

  50. Weibel, C.A.: An introduction to homological algebra. In: Cambridge Studies in Advanced Mathematics, vol. 38. Cambridge University Press, Cambridge (1994)

  51. Weil, A.: Sur certains groupes d’opérateurs unitaires. Acta. Math. 111, 143–211 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  52. Wigner, D.: Algebraic cohomology of topological groups. Trans. Am. Math. Soc. 178, 83–93 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  53. Willis, G.: The structure of totally disconnected, locally compact groups. Math. Ann. 300(2), 341–363 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  54. Yoneda, N.: On ext and exact sequences. J. Fac. Sci. Univ. Tokyo Sect. I 8, 507–576 (1960)

    MathSciNet  MATH  Google Scholar 

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Acknowledgments

Our thanks go to Matthias Flach, Karl Hofmann, Arati Khairnar, Stephen Lichtenbaum, Nicolas Monod, C.S. Rajan and Terence Tao for several helpful communications.

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Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, New York University, New York, NY, 10012, USA

    Tim Austin

  2. Department of Mathematics, University of California, Berkeley, CA, 94720, USA

    Calvin C. Moore

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  1. Tim Austin
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Correspondence to Tim Austin.

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T. Austin’s research supported by fellowships from Microsoft Corporation and from the Clay Mathematics Institute.

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Austin, T., Moore, C.C. Continuity properties of measurable group cohomology. Math. Ann. 356, 885–937 (2013). https://doi.org/10.1007/s00208-012-0868-z

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