Abstract.We consider the condition when bounded cohomology injects into ordinary cohomology and prove the vanishing of bounded cohomology of the group of all compactly supported homeomorphisms of R".Introduction. In this note we consider relations among bounded cohomology, ordinary real cohomology and Z1 homology of spaces or groups. In particular we present a necessary and sufficient condition under which bounded cohomology injects into ordinary cohomology and by using it prove the vanishing of bounded cohomology and lx homology of Horneo^R", the group of all homeomorphisms of R" with compact support. We also determine the second bounded cohomology of SL2R.

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ABSTRACT. We give numerical invariants for deciding when two actions of a given group on the circle are semiconjugate. We give conditions for vanishing of an invariant called bounded real Euler class. We also determine endomorphisms of the discrete group PSL(2, R).

This paper is concerned about the orbit equivalence types of C ∞ diffeomorphisms of S 1 seen as nonsingular automorphisms of (S 1 , m), where m is the Lebesgue measure. Given any Liouville number α, it is shown that each of the subspace formed by type II 1 , II∞, III λ (λ > 1), III∞ and III 0 diffeomorphisms are C ∞ -dense in the space of the orientation preserving C ∞ diffeomorphisms with rotation number α.1991 Mathematics Subject Classification. Primary 37E10, secondary 37E45.