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The simplicial volume of closed manifolds covered by \(\mathbb H^2\times \mathbb H^2\). (English) Zbl 1156.53018

The main purpose of the paper under review is the computation of the Gromov norm of the Riemannian volume form on the product \(\mathbb H^2\times \mathbb H^2\), where \(\mathbb H^2\) denotes the hyperbolic plane of constant curvature \(-1\).
The following main theorem is shown: Let \(\omega_{\mathbb H^2 \times \mathbb H^2} \in H_c^4(PSL_2\mathbb R \times PSL_2\mathbb R)\) be the image, under the Van Est isomorphism, of the Riemannian volume form on \(\mathbb H^2 \times \mathbb H^2\). Then \(| | \omega_{\mathbb H^2 \times \mathbb H^2} | | _{\infty}=(2/3) \pi^2\). Denote by \(| | M| | \) the simplicial volume introduced by M. Gromov in [Publ. Math., Inst. Hautes Étud. Sci. 56, 5–99 (1982; Zbl 0516.53046)]. Using the main theorem it is shown:
If \(M\) is a closed, oriented Riemannian manifold whose universal cover \(\tilde M\) is isometric to \(\mathbb H^2 \times \mathbb H^2\), then \(\| M\|=3/2\pi^2.Vol(M)\). Among some corollaries derived from the above result, we would like to mention: Let \(M\) and \(N\) be closed, oriented surfaces. Then \(\| M\times N\| =(3/2)\| M\| \| N\|\). As pointed out by the author this gives the first exact value of a nonvanishing simplicial volume for a manifold not admitting a constant curvature metric. A relevant tool used is the Van Est isomorphism \(\mathbb J :A^*(X,E)^G \to H_c^*(G, E)\) between the \(G-\)invariant \(E-\)valued differential forms on \(X\) and the continuous cohomology of \(G\) with coefficients in \(E\) where \(E\) is a \(G\)-module.

MSC:

53C20 Global Riemannian geometry, including pinching
55N10 Singular homology and cohomology theory
20J06 Cohomology of groups

Citations:

Zbl 0516.53046
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References:

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