On the Cohomology of Topological Semigroups
Abstract
In this short note, we give some new results on continuous bounded cohomology groups of topological semigroups with values in complex field. We show that the second continuous bounded cohomology group of a compact metrizable semigroup, is a Banach space. Also, we study cohomology groups of amenable topological semigroups, and we show that cohomology groups of rank greater than one of a compact left or right amenable semigroup, are trivial. Also, we give some examples and applications about topological lattices.
Keywords
References
- [1] B. E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc., 127 (1972).
- [2] M. Gromov, Volume and bounded cohomology, Publ. Math. IHES, 56 (1982), 5-100.
- [3] R. Frigerio, Bounded cohomology of discrete groups, Amer. Math. Soc., 227 (2017).
- [4] E. H. Brown, R. H. Szczarba, Continuous cohomology and real homotopy type, Trans. Amer. Math. Soc., 311(1) (1989), 57-106.
- [5] L. Mdzinarishvili, Continuous singular cohomology, Georgian Math. J., 16 (2009), 321–342.
- [6] M. A. Mostow, Continuous cohomology of spaces with two topologies, Mem. Amer. Math. Soc., 7 (1976), 1–142.
- [7] J. D. Stasheff, Continuous cohomology of groups and classifying spaces, Bull. Amer. Math. Soc., 84(4) (1978), 513-530
- [8] R. Frigerio, (Bounded) Continuous cohomology and Gromov’s proportionality principle, Manuscripta Math., 134(3-4) (2011), 435-474.
- [9] N. Monod, Continuous bounded cohomology of locally compact groups, Lecture notes in Mathematics, no. 1758, Springer- Verlag, Berlin, 2001.
- [10] R. Brooks, Some remarks on bounded cohomology, Ann. Math. Studies, 97 (1980), 53-63.
