## Geometric Seifert 4-manifolds with hyperbolic bases.(English)Zbl 1151.57023

Seifert fibred 3-manifolds were originally defined and classified by H. Seifert in [Acta Math. 60, 147–238 (1933; Zbl 0006.08304) and JFM 59.1241.02]. Scott gave a survey of results connected with these classical Seifert spaces, and showed that they correspond to 3-manifolds having one of six of the eight 3-dimensional geometries in the sense of Thurston [see P. Scott, The geometries of 3-manifolds, Bull. Lond. Math. Soc. 15, 401–487 (1983; Zbl 0561.57001)]. Thurston’s geometrization conjecture asserts that any 3-manifold can be decomposed into such geometric pieces and so Seifert 3-manifolds are important building blocks. The author generalises the concept of a Seifert manifold to dimension 4 and investigates when they have geometric structures. He defines a Seifert manifold to be the total space of a bundle over a 2-orbifold with flat fibres (for Seifert 4-manifolds the fibre can either be a 2-torus or a Klein bottle). Ue has studied the geometries of the orientable Seifert 4-manifolds which have general fibre a torus [see M. Ue, Geometric 4-manifolds in the sense of Thurston and Seifert 4-manifolds. I: J. Math. Soc. Japan 42, No. 3, 511–540 (1990; Zbl 0707.57010) and II: ibid. 43, No. 1, 149–183 (1991; Zbl 0724.57009)]. In those papers Ue proved that (with a finite number of exceptions) orientable manifolds of eight of the 4-dimensional geometries are Seifert fibred. However, Seifert manifolds with a hyperbolic base are not necessarily geometric.
In the paper under review, the author extends the Ue work to the non-orientable case. Ue proved that orientable Seifert 4-manifolds with hyperbolic bases are geometric if and only if the monodromies are periodic. The author proves that this is also true for nonorientable Seifert 4-manifolds. This result will be used to prove virtually geometric Seifert 4-manifolds (i.e., a Seifert manifold which is finitely covered by a geometric manifold) with hyperbolic bases are geometric and thus he gives a classification of such manifolds in terms of finite covers.

### MSC:

 57N16 Geometric structures on manifolds of high or arbitrary dimension

### Keywords:

4-manifolds; Seifert; geometric
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