On the algebraic foundations of bounded cohomology

Bühler, Theo

doi:10.1090/S0065-9266-2011-00618-0

AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution

On the algebraic foundations of bounded cohomology

About this Title

Theo Bühler, Departement Mathematik der ETH Zürich, HG G 28.3, Rämistrasse 101, CH-8092 Zürich

Publication: Memoirs of the American Mathematical Society
Publication Year: 2011; Volume 214, Number 1006
ISBNs: 978-0-8218-5311-5 (print); 978-1-4704-0623-3 (online)
DOI: https://doi.org/10.1090/S0065-9266-2011-00618-0
Published electronically: March 11, 2011
Keywords: Bounded Cohomology, Derived Categories, Derived Functors
MSC: Primary 18E30; Secondary 18E10, 18G60, 46M18, 20J05, 57T

PDF View full volume as PDF

Abstract

Bounded cohomology for topological spaces was introduced by Gromov in the late seventies, mainly to describe the simplicial volume invariant. It is an exotic cohomology theory for spaces in that it fails excision and thus cannot be represented by spectra. Gromov’s basic vanishing result of bounded cohomology for simply connected spaces implies that bounded cohomology for spaces is an invariant of the fundamental group. To prove this, one is led to introduce a cohomology theory for groups and the present work is concerned with the latter, which has been studied by Gromov, Brooks, Ivanov and Noskov to name but the most important initial contributors. Generalizing these ideas from discrete groups to topological groups, Burger and Monod have developed continuous bounded cohomology in the late nineties.

It is a widespread opinion among experts that (continuous) bounded cohomology cannot be interpreted as a derived functor and that triangulated methods break down. We prove that this is wrong.

We use the formalism of exact categories and their derived categories in order to construct a classical derived functor on the category of Banach $G$-modules with values in Waelbroeck’s abelian category. This gives us an axiomatic characterization of this theory for free and it is a simple matter to reconstruct the classical semi-normed cohomology spaces out of Waelbroeck’s category.

We prove that the derived categories of right bounded and of left bounded complexes of Banach $G$-modules are equivalent to the derived category of two abelian categories (one for each boundedness condition), a consequence of the theory of abstract truncation and hearts of $t$-structures. Moreover, we prove that the derived categories of Banach $G$-modules can be constructed as the homotopy categories of model structures on the categories of chain complexes of Banach $G$-modules thus proving that the theory fits into yet another standard framework of homological and homotopical algebra.

References

Claire Anantharaman, Jean-Philippe Anker, Martine Babillot, Aline Bonami, Bruno Demange, Sandrine Grellier, François Havard, Philippe Jaming, Emmanuel Lesigne, Patrick Maheux, Jean-Pierre Otal, Barbara Schapira, and Jean-Pierre Schreiber, Théorèmes ergodiques pour les actions de groupes, Monographies de L’Enseignement Mathématique [Monographs of L’Enseignement Mathématique], vol. 41, L’Enseignement Mathématique, Geneva, 2010 (French). With a foreword in English by Amos Nevo. MR 2643350
William Arveson, An invitation to $C^*$-algebras, Graduate Texts in Mathematics, No. 39, Springer-Verlag, New York-Heidelberg, 1976. MR 0512360
Paul Balmer, Triangular Witt groups. I. The 12-term localization exact sequence, $K$-Theory 19 (2000), no. 4, 311–363. MR 1763933, DOI 10.1023/A:1007844609552
A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171 (French). MR 751966
A. Borel, P.-P. Grivel, B. Kaup, A. Haefliger, B. Malgrange, and F. Ehlers, Algebraic $D$-modules, Perspectives in Mathematics, vol. 2, Academic Press, Inc., Boston, MA, 1987. MR 882000
Yoav Benyamini and Joram Lindenstrauss, Geometric nonlinear functional analysis. Vol. 1, American Mathematical Society Colloquium Publications, vol. 48, American Mathematical Society, Providence, RI, 2000. MR 1727673, DOI 10.1090/coll/048
M. Burger and N. Monod, Continuous bounded cohomology and applications to rigidity theory, Geom. Funct. Anal. 12 (2002), no. 2, 219–280. MR 1911660, DOI 10.1007/s00039-002-8245-9
Abdesselam Bouarich, Théorèmes de Zilber-Eilenberg et de Brown en homologie $l_1$, Proyecciones 23 (2004), no. 2, 151–186 (French, with English summary). MR 2142264, DOI 10.4067/S0716-09172004000200007
Robert Brooks, Some remarks on bounded cohomology, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 53–63. MR 624804
Paul Balmer and Marco Schlichting, Idempotent completion of triangulated categories, J. Algebra 236 (2001), no. 2, 819–834. MR 1813503, DOI 10.1006/jabr.2000.8529
Theo Bühler, Exact categories, Expo. Math. 28 (2010), no. 1, 1–69. MR 2606234, DOI 10.1016/j.exmath.2009.04.004
Johann Cigler, Viktor Losert, and Peter Michor, Banach modules and functors on categories of Banach spaces, Lecture Notes in Pure and Applied Mathematics, vol. 46, Marcel Dekker, Inc., New York, 1979. MR 533819
Mahlon M. Day, Fixed-point theorems for compact convex sets, Illinois J. Math. 5 (1961), 585–590. MR 138100
Mahlon M. Day, Normed linear spaces, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 21, Springer-Verlag, New York-Heidelberg, 1973. MR 0344849
P. Deligne, Cohomologie étale, Lecture Notes in Mathematics, vol. 569, Springer-Verlag, Berlin, 1977 (French). Séminaire de géométrie algébrique du Bois-Marie SGA $4\frac {1}{2}$. MR 463174, DOI 10.1007/BFb0091526
Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
W. G. Dwyer and J. Spaliński, Homotopy theories and model categories, Handbook of algebraic topology, North-Holland, Amsterdam, 1995, pp. 73–126. MR 1361887, DOI 10.1016/B978-044481779-2/50003-1
Per Enflo, A counterexample to the approximation problem in Banach spaces, Acta Math. 130 (1973), 309–317. MR 402468, DOI 10.1007/BF02392270
David Fisher, Dave Witte Morris, and Kevin Whyte, Nonergodic actions, cocycles and superrigidity, New York J. Math. 10 (2004), 249–269. MR 2114789
Peter Freyd, Representations in abelian categories, Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965) Springer, New York, 1966, pp. 95–120. MR 0209333
Sergei I. Gelfand and Yuri I. Manin, Methods of homological algebra, 2nd ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. MR 1950475, DOI 10.1007/978-3-662-12492-5
Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955), Chapter 1: 196 pp.; Chapter 2: 140 (French). MR 75539
Michael Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. 56 (1982), 5–99 (1983). MR 686042
P. Gabriel and M. Zisman, Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer-Verlag New York, Inc., New York, 1967. MR 0210125
N. V. Ivanov, Foundations of the theory of bounded cohomology, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 143 (1985), 69–109, 177–178 (Russian, with English summary). Studies in topology, V. MR 806562
Birger Iversen, Cohomology of sheaves, Universitext, Springer-Verlag, Berlin, 1986. MR 842190, DOI 10.1007/978-3-642-82783-9
Barry Edward Johnson, Cohomology in Banach algebras, Memoirs of the American Mathematical Society, No. 127, American Mathematical Society, Providence, R.I., 1972. MR 0374934
Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
Bernhard Keller, Chain complexes and stable categories, Manuscripta Math. 67 (1990), no. 4, 379–417. MR 1052551, DOI 10.1007/BF02568439
Bernhard Keller, Derived categories and universal problems, Comm. Algebra 19 (1991), no. 3, 699–747. MR 1102982, DOI 10.1080/00927879108824166
Bernhard Keller, Derived categories and their uses, Handbook of algebra, Vol. 1, Handb. Algebr., vol. 1, Elsevier/North-Holland, Amsterdam, 1996, pp. 671–701. MR 1421815, DOI 10.1016/S1570-7954(96)80023-4
Gottfried Köthe, Hebbare lokalkonvexe Räume, Math. Ann. 165 (1966), 181–195 (German). MR 196464, DOI 10.1007/BF01343797
Bernhard Keller and Dieter Vossieck, Sous les catégories dérivées, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 6, 225–228 (French, with English summary). MR 907948
T. Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics, vol. 189, Springer-Verlag, New York, 1999. MR 1653294, DOI 10.1007/978-1-4612-0525-8
Serge Lang, Rapport sur la cohomologie des groupes, W. A. Benjamin, Inc., New York-Amsterdam, 1967 (French). MR 0212073
G. Laumon, Sur la catégorie dérivée des ${\cal D}$-modules filtrés, Algebraic geometry (Tokyo/Kyoto, 1982) Lecture Notes in Math., vol. 1016, Springer, Berlin, 1983, pp. 151–237 (French). MR 726427, DOI 10.1007/BFb0099964
Clara Löh, $\ell ^{1}$-Homology and Simplicial Volume, Ph.D. thesis, Westfälische Wilhelms-Universität Münster, 2007.
J. Lindenstrauss and A. Pełczyński, Absolutely summing operators in $L_{p}$-spaces and their applications, Studia Math. 29 (1968), 275–326. MR 231188, DOI 10.4064/sm-29-3-275-326
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056
Georges Maltsiniotis, Le théorème de Quillen, d’adjonction des foncteurs dérivés, revisité, C. R. Math. Acad. Sci. Paris 344 (2007), no. 9, 549–552 (French, with English and French summaries). MR 2323740, DOI 10.1016/j.crma.2007.03.011
J. P. May, The additivity of traces in triangulated categories, Adv. Math. 163 (2001), no. 1, 34–73. MR 1867203, DOI 10.1006/aima.2001.1995
Ernest Michael, Continuous selections. I, Ann. of Math. (2) 63 (1956), 361–382. MR 77107, DOI 10.2307/1969615
Saunders Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998. MR 1712872
Shigenori Matsumoto and Shigeyuki Morita, Bounded cohomology of certain groups of homeomorphisms, Proc. Amer. Math. Soc. 94 (1985), no. 3, 539–544. MR 787909, DOI 10.1090/S0002-9939-1985-0787909-6
Nicolas Monod, Continuous bounded cohomology of locally compact groups, Lecture Notes in Mathematics, vol. 1758, Springer-Verlag, Berlin, 2001. MR 1840942, DOI 10.1007/b80626
Calvin C. Moore, Group extensions and cohomology for locally compact groups. III, Trans. Amer. Math. Soc. 221 (1976), no. 1, 1–33. MR 414775, DOI 10.1090/S0002-9947-1976-0414775-X
Amnon Neeman, The derived category of an exact category, J. Algebra 135 (1990), no. 2, 388–394. MR 1080854, DOI 10.1016/0021-8693(90)90296-Z
Amnon Neeman, Triangulated categories, Annals of Mathematics Studies, vol. 148, Princeton University Press, Princeton, NJ, 2001. MR 1812507, DOI 10.1515/9781400837212
HeeSook Park, Foundations of the theory of $l_1$ homology, J. Korean Math. Soc. 41 (2004), no. 4, 591–615. MR 2068142, DOI 10.4134/JKMS.2004.41.4.591
A. Pełczyński, Projections in certain Banach spaces, Studia Math. 19 (1960), 209–228. MR 126145, DOI 10.4064/sm-19-2-209-228
R. S. Phillips, On linear transformations, Trans. Amer. Math. Soc. 48 (1940), 516–541. MR 4094, DOI 10.1090/S0002-9947-1940-0004094-3
Brian Parshall and Leonard L. Scott, Derived categories, quasi-hereditary algebras, and algebraic groups, Carlton U. Math. Notes 3 (1988), 1–105.
Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. MR 0223432
Daniel Quillen, Higher algebraic $K$-theory. I, Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 341, Springer, Berlin, 1973, pp. 85–147. MR 0338129
Jeremy Rickard, Derived categories and stable equivalence, J. Pure Appl. Algebra 61 (1989), no. 3, 303–317. MR 1027750, DOI 10.1016/0022-4049(89)90081-9
Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. MR 1157815
Robert Schatten, On the direct product of Banach spaces, Trans. Amer. Math. Soc. 53 (1943), 195–217. MR 7568, DOI 10.1090/S0002-9947-1943-0007568-7
Jean-Pierre Schneiders, Quasi-abelian categories and sheaves, Mém. Soc. Math. Fr. (N.S.) 76 (1999), vi+134 (English, with English and French summaries). MR 1779315
A. Szankowski, The space of all bounded operators on Hilbert space does not have the approximation property, Séminaire d’Analyse Fonctionnelle (1978–1979), École Polytech., Palaiseau, 1979, pp. Exp. No. 14-15, 28. MR 557369
M. Takesaki, Theory of operator algebras. III, Encyclopaedia of Mathematical Sciences, vol. 127, Springer-Verlag, Berlin, 2003. Operator Algebras and Non-commutative Geometry, 8. MR 1943007, DOI 10.1007/978-3-662-10453-8
R. W. Thomason and Thomas Trobaugh, Higher algebraic $K$-theory of schemes and of derived categories, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 247–435. MR 1106918, DOI 10.1007/978-0-8176-4576-2_{1}0
Jean-Louis Verdier, Des catégories dérivées des catégories abéliennes, Astérisque 239 (1996), xii+253 pp. (1997) (French, with French summary). With a preface by Luc Illusie; Edited and with a note by Georges Maltsiniotis. MR 1453167
Lucien Waelbroeck, Some theorems about bounded structures, J. Functional Analysis 1 (1967), 392–408. MR 0220040, DOI 10.1016/0022-1236(67)90009-2
Lucien Waelbroeck, Bornological quotients, Mémoire de la Classe des Sciences. Collection in-4$^\textrm {o}$. 3$^\textrm {e}$ Série [Memoir of the Science Section. Collection in-4$^\textrm {o}$. 3rd Series], VII, Académie Royale de Belgique. Classe des Sciences, Brussels, 2005. With the collaboration of Guy Noël. MR 2128718
Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324, DOI 10.1017/CBO9781139644136
Robert Whitley, Mathematical Notes: Projecting $m$ onto $c_0$, Amer. Math. Monthly 73 (1966), no. 3, 285–286. MR 1533692, DOI 10.2307/2315346

AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution

On the algebraic foundations of bounded cohomology

About this Title

Table of Contents

Abstract

References [Enhancements On Off] (What's this?)

memo_has_moved_text();On the algebraic foundations of bounded cohomology

About this Title

Table of Contents

Abstract

References [Enhancements On Off] (What's this?)

On the algebraic foundations of bounded cohomology