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Register a new userContinuous Bounded Cohomology of Topological Semigroups
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Authors
Maysam Maysami Sadr
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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.
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Measure homology was introduced by Thurston in his notes about the geometry and topology of 3-manifolds, where it was exploited in the computation of the simplicial volume of hyperbolic manifolds. Zastrow and Hansen independently proved that there exists a canonical isomorphism between measure homology and singular homology (on the category of CW-complexes), and it was then shown by Loeh that, in the absolute case, such isomorphism is in fact an isometry with respect to the L^1-seminorm on singular homology and the total variation seminorm on measure homology. Loehs result plays a fundamental role in the use of measure homology as a tool for computing the simplicial volume of Riemannian manifolds. This paper deals with an extension of Loehs result to the relative case. We prove that relative singular homology and relative measure homology are isometrically isomorphic for a wide class of topological pairs. Our results can be applied for instance in computing the simplicial volume of Riemannian manifolds with boundary. Our arguments are based on new results about continuous (bounded) cohomology of topological pairs, which are probably of independent interest.
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This work deals with relations between a bounded cohomological invariant and the geometry of Hermitian symmetric spaces of noncompact type. The invariant, obtained from the Kahler class, is used to define and characterize a special class of totally geodesic embeddings, called tight embeddings. In addition, special isometric actions of finitely generated groups on Hermitian symmetric spaces are studied. Results of a joint work with M. Burger and A. Iozzi about surface group representations are also discussed.
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