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Harmonic maps, hyperbolic cohomology and higher Milnor inequalities. (English) Zbl 0804.57013
This paper contains various applications of existence results on harmonic maps and of versions of Thurston’s straightening technique. For example, denote by $$\Sigma^ g$$ a closed orientable surface of genus $$g>1$$, with a metric of constant negative curvature. Let $$N$$ be a complete simply connected Riemannian manifold with curvature $$K(N)$$ satisfying $$-K \leq K (N) \leq - k < 0$$. Consider a flat bundle $$E \to \Sigma^ g$$, with fiber $$N$$, whose holonomy group lies in $$\text{ISO} (N)$$ (the isometry group of $$N)$$, and suppose that the natural action of the holonomy subgroup of $$\text{ISO} (N)$$ on the sphere at infinity $$S_ \infty (N)$$ is fixed- point-free. Then the Donaldson theorem on the existence of harmonic cross-sections of flat bundles enables the author to prove that there exists a section $$s : \Sigma^ g \to E$$ whose area does not exceed $$4 \pi (g - 1)/k$$.
The latter result is then the main ingredient of author’s proof of the following theorem due to W. Goldman: if $$\xi$$ is the oriented circle bundle associated with a representation $$\pi : \pi_ 1 (\Sigma^ g) \to \text{PSL} (2, \mathbb{R})$$, then the absolute value of the Euler number $$\chi (\xi)$$ does not exceed $$2g - 2$$, and if $$\chi(\xi)=2-2g$$, then the image of $$\pi$$ acts discontinuously and co-compactly in the hyperbolic plane.
Using Thurston’s straightening technique, the author derives an estimate for the value of a volume class $$\text{Vol} (\pi) \in H^ n (M; \mathbb{R})$$ on the fundamental class $$[M]$$ of a compact manifold $$M$$, where $$\pi : \pi_ 1 (M) \to \text{SO} (1,n)$$ is a representation of the fundamental group of $$M$$. Then he proves similar results for representations $$\pi : \pi_ 1 (M) \to \text{Sp} (1,n)$$.
The author also presents some $$\mathbb{Z}$$-cohomology restrictions for 4- dimensional compact manifolds with negative curvature.
{Reviewer’s remark: Several misprints occur in the paper; for example, the inequality in Theorem D.2 (p. 902) should be $$| c_ 1 (E_ -) | \leq {3 \over 2} (g - 1).\}$$.

##### MSC:
 57R22 Topology of vector bundles and fiber bundles 57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.) 53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 57R99 Differential topology
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