Harmonic maps, hyperbolic cohomology and higher Milnor inequalities.

*(English)*Zbl 0804.57013This 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 \(| c_ 1 (E_ -) | \leq {3 \over 2} (g - 1).\}\).

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 \(| c_ 1 (E_ -) | \leq {3 \over 2} (g - 1).\}\).

Reviewer: J.Korbaš (Bratislava)

##### 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 |