Bounded cohomology of certain groups of homeomorphisms.

*(English)*Zbl 0536.57023The authors consider relations among bounded cohomology, \(\ell^ 1\) homology and ordinary real cohomology of spaces or groups. They present in particular a necessary and sufficient condition for bounded cohomology to inject into ordinary cohomology and by using it they prove the vanishing of bounded cohomology and \(\ell^ 1\) homology of \(Homeo_ k({\mathbb{R}}^ n)\), the group of all the homeomorphisms of \({\mathbb{R}}^ n\) with compact supports. They also determine the second bounded cohomology of \(SL_ 2{\mathbb{R}}\).

##### MSC:

57T20 | Homotopy groups of topological groups and homogeneous spaces |

55N99 | Homology and cohomology theories in algebraic topology |

20J05 | Homological methods in group theory |

58D05 | Groups of diffeomorphisms and homeomorphisms as manifolds |

##### Keywords:

\(\ell^ 1\) homology; real cohomology of groups; bounded cohomology of the group of homeomorphisms of real n-space with compact support; bounded cohomology of SL(2,R)
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\textit{S. Matsumoto} and \textit{S. Morita}, Proc. Am. Math. Soc. 94, 539--544 (1985; Zbl 0536.57023)

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##### References:

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