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Bounded and unbounded cohomology of homeomorphism and diffeomorphism groups

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We determine the bounded cohomology of the group of homeomorphisms of certain low-dimensional manifolds. In particular, for the group of orientation-preserving homeomorphisms of the circle and of the closed 2-disc, it is isomorphic to the polynomial ring generated by the bounded Euler class. These seem to be the first examples of groups for which the entire bounded cohomology can be described without being trivial. We further prove that the \(C^r\)-diffeomorphisms groups of the circle and of the closed 2-disc have the same bounded cohomology as their homeomorphism groups, so that both differ from the ordinary cohomology of \(C^r\)-diffeomorphisms when \(r>1\). Finally, we determine the low-dimensional bounded cohomology of homeo- and diffeomorphism of the spheres \(S^n\) and of certain 3-manifolds. In particular, we answer a question of Ghys by showing that the Euler class in \(H^4(\textrm{Homeo}_\circ (S^3))\) is unbounded.

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Upon becoming aware of our work, Fournier-Facio, Löh and Moraschini have kindly shown us a draft of their work [19]. We are grateful for their comments. Although the two lines of investigation have essentially no overlap, their work gives an alternative proof of the bounded acyclicity of some of the stabilisers appearing within some proofs below. We are grateful to Jonathan Bowden for his comments on the group of homeomorphisms and diffeomorphisms of 3-dimensional pair of pants. We thank Shigeyuki Morita for his comment about the powers of the Euler class for flat \(D^2\)-bundles and also we thank Mehdi Yazdi for his careful reading and critical comments on the earlier draft of this paper. We are grateful to the anonymous referee for many comments which improved our exposition. S.N. was partially supported by NSF CAREER Grant DMS-2239106, NSF DMS-2113828 and Simons Foundation (855209, SN).

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Monod, N., Nariman, S. Bounded and unbounded cohomology of homeomorphism and diffeomorphism groups. Invent. math. 232, 1439–1475 (2023).

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