Geometric Decompositions of 4-dimensional Bundle Spaces

We consider geometric decompositions of aspherical 4manifolds which fibre over 2-orbifolds. We show first that no such manifold admits infinitely many fibrations over hyperbolic base orbifolds. If E is Seifert fibred over a hyperbolic surface B and either B has at most one cone point of order 2 or the monodromy has image in SL(2,Z) then E it has a decomposition induced from a decomposition of B. An n-manifold M admits a geometric decomposition if it has a finite collection S of disjoint connected 2-sided hypersurfaces such that each component of M −∪S∈SS is geometric of finite volume, i.e., is homeomorphic to Γ\X, for some geometry X and lattice Γ. We shall call the hypersurfaces S cusps and the components of M −∪S∈SS pieces of M . The decomposition is proper if the set of cusps is nonempty. We shall consider the possible geometric decompositions of aspherical orbifold bundles in dimension 4. An orbifold bundle E is the total space of an orbifold fibration p : E → B over a 2-dimensional base orbifold, with regular fibre F an aspherical surface. (Here “surface” shall mean closed 2-orbifold without exceptional points.) Let π = π1(E), φ = π1(F ) and β = π orb 1 (B), and let θ : β → Out(φ) be the characteristic homomorphism (or monodromy). We show first that if χ(E) > 0 then E admits only finitely many orbifold fibrations. In §2 we extend and correct some results on geometries on bundle spaces from Chapter 13 of [4] to the case of orbifold bundles. In §3 we constrain the possible geometries of pieces of a given orbifold bundle. In §4 and §5 we introduce the notions of (algebraically) horizontal and vertical decompositions. In particular, we show that no decomposition can have both algebraically horizontal and algebraically vertical cusps, and that if E is Seifert fibred and Im(θ) ≤ SL(2,Z) or 1991 Mathematics Subject Classification. 57N13.