S^{2}-BUNDLES OVER ASPHERICAL SURFACES

AND 4-DIMENSIONAL GEOMETRIES

Robin J. Cobb & Jonathan A. Hillman

ABSTRACT: Melvin [Me] has shown that closed 4-manifolds that arise as S^{2}-bundles over closed, connected aspherical surfaces are classified up to diffeomorphism by the Stiefel-Whitney classes of the associated bundles. We show that each such 4-manifold admits one of the geometries S^{2} ×E^{2} or S^{2} ×ℍ^{2} [depending on whether χ(M) = 0 or χ(M) < 0]. Conversely a geometric closed, connected 4-manifold M of type S^{2} × E^{2} or S^{2} × ℍ^{2} is the total space of an S^{2}-bundle over a closed, connected aspherical surface precisely when its fundamental group Π_{1}(M) is torsion free. Furthermore the total spaces of ℝℙ^{2}-bundles over closed, connected aspherical surfaces are all geometric. Conversely a geometric closed, connected 4-manifold M′ is the total space of an ℝℙ^{2}-bundle if and only if Π_{1}(M′) ≅ ℤ/2ℤ × K where K is torsion free.

Introduction.

Melvin [Me] has shown that closed 4-manifolds that arise as S^{2}-bundles over closed aspherical surfaces are classified up to diffeomorphism by the Stiefel-Whitney classes of the associated bundles. Moreover one may construct ξ from w(ξ), see [Me] Structure Lemma. In particular there are two orientable 4-manifolds that arise as total spaces of S^{2}-bundles over each closed surface; and are distinguished by whether or not w_{2}(ξ) = 0. Ue [Ue] has shown that the total spaces M of such orientable bundles over the surfaces of Euler characteristic χ = 0 have the structures of Seifert 4-manifolds and are geometric.

There are just two S^{2}-bundles over S^{2}, the product bundle and S^{2} ˜×S^{2}; the latter is not geometric. There are four S^{2}-bundles over ℝℙ^{2} each of which admits an S^{2}×S^{2} geometry and is distinguishable by its Stiefel-Whitney classes [H1]. There are four S^{2}-bundles over each closed orientable surface B with Euler characteristic χ(B) ≤ 0, six S^{2}-bundles over the Klein bottle Kb and eight S^{2}-bundles over each closed non-orientable surface B with χ(B) ≤ 0, [Me]. The aim of this paper is to prove the following:

Main Theorem.

(1) A closed S^{2} × E^{2} or S^{2} × ℍ^{2}-manifold M is the total space of an S^{2}-bundle over some aspherical surface precisely when its fundamental group Π_{1}(M) is torsion free. Conversely all total spaces of such bundles are geometric.

(2) A closed S^{2}×E^{2} or S^{2}×ℍ^{2}-manifold M′ is the total space of an ℝℙ^{2}-bundle over some aspherical surface precisely when its fundamental group splits as

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 57N50. Secondary 57N13, 55R25.

Key words and phrases. aspherical surface, S^{2}-bundle, 4-dimensional geometry, Stiefel-Whitney class.

Π_{1}(M′) = ℤ/2ℤ × K, where K is torsion free. Conversely all total spaces of such bundles are geometric.

Generalities.

Bundles ξ will be denoted by p: M → B and will have fibre the 2-sphere S^{2} throughout unless otherwise stated. Let E^{2} denote the 2-dimensional Euclidean geometry with underlying space ℝ^{2}. Similarly let ℍ^{2} denote the geometry of the hyperbolic plane and 𝕏^{2} denote either of the above two geometries with underlying space X^{2}, in arguments that encompass both. Henceforth a manifold refered to as being closed is a compact, connected manifold without boundary.

From the long exact homotopy sequence for a fibration we have that Π_{1}(M) ≅ Π_{1}(B); in particular H_{1}(M) ≅ H_{1}(B) with any coefficients. Also as p is a fibration the Euler characteristic χ(M) = χ(S^{2})χ(B) = 2χ(B) so by Poincaré duality with coefficients in ℤ/2ℤ we obtain β_{2}(M) = 2 and therefore H_{2}(M; ℤ/2ℤ) = ℤ/2ℤ ⊕ ℤ/2ℤ. In fact H_{2}(M; ℤ/2ℤ) is generated by the images of a cross-section σ_{∗}(B) and the inclusion of the fibre i_{∗}(S^{2}); as the fibre has zero self intersection and the intersection number of a cross-section and a fibre is 1, they are linearly independent. We shall explicitly construct such cross-sections where needed.

Let ξ be an S^{2}-bundle. The homomorphism p^{∗}: H^{∗}(B; ℤ/2ℤ) → H^{∗}(M; ℤ/2ℤ), induced by p is injective and the Whitney sum formula gives:

(a) w_{1}(M) = p^{∗}(w_{1}(B) + w_{1}(ξ)), and

(b) w_{2}(M) = p^{∗}(w_{2}(B) + w_{1}(B) ⌣ w_{1}(ξ) + w_{2}(ξ)). See Melvin [Me] Lemma 1.

The Wu class v_{2}(M) is characterised by:

v_{2}(M) ⌣ x = x ⌣ x, ∀x ∈ H^{2}(M; ℤ/2ℤ).

Let D: H^{2}(M; ℤ/2ℤ)→^{∼}H_{2}(M; ℤ/2ℤ) be the Poincaré duality isomorphism. Then we may dualize the above equation in terms of intersections of surfaces by:

D(x) · D(v_{2}(M)) ≡ D(x) · D(x) mod 2, ∀D(x) ∈ H_{2}(M);

so that D(v_{2}(M)) is the characteristic element for the intersection form. Together with the Wu relation v_{2}(M) = w_{2}(M) + w_{1}(M)^{2} we have that w_{2}(M) = w_{1}(M)^{2} precisely when all self intersections in M are even. Moreover

v_{2}(M) = p^{∗}(w_{2}(B) + w_{1}(B) ⌣ w_{1}(ξ) + w_{2}(ξ)) + p^{∗}(w_{1}(B) + w_{1}(ξ))^{2}
= p^{∗}(w_{2}(B) + w_{1}(B) ⌣ w_{1}(ξ) + w_{2}(ξ) + w_{1}(B)^{2} + w_{1}(ξ)^{2})
= p^{∗}(w_{2}(ξ)) from the Wu relations for B.

and as p^{∗} is injective we obtain w_{2}(ξ) = 0 ⟺ v_{2}(M) = 0.

If ξ is a geometric S^{2}-bundle with fundamental group Π < O(3)×I som(𝕏^{2}) then w_{1}(ξ) is detected by the determinant: det(g_{S}) = (−1)^{w_{1}(ξ)(g_{S},g_{X})} for all (g_{S}, g_{X}) ∈ Π.

Geometic 4-Manifolds That Arise as Total Spaces of S^{2}-Bundles Over Closed Surfaces

Before proving that all S^{2}-bundles over closed aspherical surfaces are geometric we shall determine which geometric 4-manifolds are total spaces of S^{2}-bundles over closed aspherical surfaces.

Lemma. Let M be a closed S^{2} × 𝕏^{2}-manifold with fundamental group Π_{1}(M) < I som(S^{2} × 𝕏^{2}) and Π_{𝕏^{2}} denote the image of Π_{1}(M) in I som(𝕏^{2}).

(1) If Π_{1}(M) is torsion free then it acts freely on 𝕏^{2} and M = (S^{2} ×𝕏^{2})/Π_{1}(M) is the total space of an S^{2}-bundle over E^{2}/Π_{𝕏^{2}}.

(2) Π_{1}(M) has a torsion free subgroup of index two.

Proof : (1) Let g ∈ Π_{1}(M) fix e ∈ 𝕏^{2}. Then ∀s ∈ S^{2} the orbit of (s, e) ∈ S^{2} ×𝕏^{2} under ⟨g⟩ is homeomorphic to the the orbit of s ∈ S^{2} by ⟨g⟩; moreover the stabilizer of the orbit is trivial as Π_{1}(M) acts fixed point free. Now as S^{2} is compact these orbits must be finite so that |⟨g⟩| < ∞ and Π_{1}(M) has torsion. Similarly, the kernel of the projection of Π_{1}(M) onto Π_{𝕏^{2}} is a finite group which acts freely on S^{2}.

In particular if Π_{1}(M) is torsion free then we have the following commutative diagram:

[figure: S^{2} × 𝕏^{2}; ρ; f; 𝕏^{2}; \overline{f}; M = (S^{2} × 𝕏^{2})/Π_{1}(M); p; 𝕏^{2}/Π_{𝕏^{2}}]

where ρ is the canonical projection and f and \overline{f} are universal covering projections. We show that the fibres of p are S^{2}. Let e′ = \overline{f}(e) ∈ 𝕏^{2}/Π_{𝕏^{2}}. Then p^{−1}(e′) = fρ^{−1}\overline{f}^{−1}(e′) = fρ^{−1}(Orb_{Π_{𝕏^{2}}} (e)) = f(S^{2} × Orb_{Π_{𝕏^{2}}} (e)). Since Π_{𝕏^{2}} acts freely on 𝕏^{2}, the map S^{2} × {e} ↦ f(S^{2} × Orb_{Π_{𝕏^{2}}} (e)) is a bijection so that p^{−1}(e′) ≃ S^{2}. Moreover the projection [figure: Π_{1}(M); Π_{𝕏^{2}}] is an isomorphism.

(2)Now suppose that g = (g_{S}, g_{X}) is a nontrivial element of Π_{1}(M) of finite order. As g_{X} is a self homeomorphism of the plane of finite order it fixes a point. Therfore g_{S} must act freely and so g_{S} = −I since {I, −I} is the only nontrivial subgroup of O(3) that acts freely and properly discontinuously on S^{2}. Hence if we denote the projection of Π_{1}(M) onto its geometric factor in O(3) by α; the composite map:

[figure: I som(S^{2} × 𝕏^{2}) > Π_{1}(M); α; O(3); Det; {1, −1} = ℤ/2ℤ]

will have kernel K = {I} × Π_{𝕏^{2}} ≅ Π_{𝕏^{2}} < Π_{1}(M) of index two and must be torsion free. □

Since the fundamental groups of total spaces of S^{2}-bundles over closed aspherical surfaces are torsion free we have proved the following:

Theorem 1. A closed S^{2} × E^{2} or S^{2} × ℍ^{2}-manifold M is the total space of an S^{2}-bundle over a closed aspherical surface if and only if Π_{1}(M) is torsion free.

Geometric 4-Manifolds That Arise as Total Spaces of ℝℙ^{2}-Bundles Over Closed Surfaces

Let M be either a closed S^{2} × E^{2} or S^{2} × ℍ^{2}-manifold. Suppose Π_{1}(M) ≅ ℤ/2ℤ × K. By the preceeding lemma we know K is torsion free and acts freely and properly discontinuously on X^{2}, with compact quotient.

Let g ∈ Π_{1}(M) generate the torsion subgroup. Viewing Π_{1}(M) as a subgroup of O(3)×I som(𝕏^{2}), we may write g = (g_{S}, g_{X}) where g_{S} ∈ O(3) and g_{X} ∈ I som(𝕏^{2}). Since g_{X}^{2} = id_{𝕏^{2}} the fixed point set F = F ix(g_{X}) = {x ∈ X^{2}|g_{X}(x) = x} is nonempty, and is either a point, a line, or the whole of X^{2}. Since g_{X} commutes with the action of K on X^{2} we have KF = F, and so K acts freely and properly discontinuously on F. But K is neither trivial nor infinite cyclic, and so we must have F = X^{2}. Hence g_{X} = id_{𝕏^{2}}. It now follows as in part (2) of the above lemma that g_{S} = −I and so Π_{1}(M) = ⟨−I⟩ × K < O(3) × I som(𝕏^{2}); the product decomposition corresponding to the geometric factors. Furthermore we have the following commutative diagram:

[figure: S^{2} × 𝕏; ρ; f; 𝕏; \overline{f}; M; p; 𝕏^{2}/K]

and M is the total space of an ℝℙ^{2}-bundle over 𝕏^{2}/K, by an argument such as part (1) of the lemma above.

Conversely if M is the total space of an ℝℙ^{2}-bundle over B then from the long exact homotopy sequence for a fibration we have:

[figure: 1; ℤ/2ℤ; i; Π_{1}(M); p_{⋆}; Π_{1}(B); 1 .]

Moreover w_{1}(M) restricts nontrivially to the fibre as shown below:

[figure: Π_{1}(M); w_{1}(M); ℤ/2ℤ; Π_{1}(ℝℙ^{2}); i]

so that the above sequence splits and we have proved the following:

Theorem 2. A closed S^{2} × E^{2} or S^{2} × ℍ^{2}, manifold M is the total space of an ℝℙ^{2}-bundle over a closed aspherical surface if and only if Π_{1}(M) ≅ ℤ/2ℤ × K, where K is torsion free.

Now for every total space M of an ℝℙ^{2}-bundle over B, w_{1}(M) ≠ 0 therefore M is non-orientable and hence has a unique orientation double cover M̃. Moreover this orientation cover is the total space of of an S^{2}-bundle over B; the ℝℙ^{2}-bundle being obtained from the S^{2}-bundle by factoring out the fibre by the antipodal map. That is we have a bijective correspondence:

[figure: \begin{cases}orientable S^{2}-bundles \\ over B \end{cases}\}; \begin{cases}ℝℙ^{2}-bundles \\ over B \end{cases}\} ;]

and there are two ℝℙ^{2}-bundles over each closed aspherical surface.

S^{2}-Bundles Over T

In the spirit of Ue we show that the non-orientable S^{2}-bundles over the torus T = S^{1} × S^{1} and the Klein bottle Kb admit geometries of type S^{2} × E^{2} before considering the case where χ(B) < 0.

Now we may take the fundamental domain of T = ℝ^{2}/ℤ^{2} to be 𝔗 = {(s, t) ∈ ℝ^{2}|0 ≤ s, t ≤ 1}, and use this to calculate w_{2}(ξ) by constructing a section σ: 𝔗 → S^{2} × E^{2} of 𝔗 so that σ induces a section σ̃: T → M of ξ as illustrated below:

[figure: S^{2} × E^{2}; f; E^{2}; d; σ; 𝔗; g; i; M; p; T; σ̃]

That is σ̃(g(t)) = fσ(i(t)), ∀t ∈ 𝔗 where f and d are universal covering projections, and g is the standard gluing map. In fact the cross-sections we shall describe are maps σ: 𝔗 → S^{2} with the action of the gluing map g predetermined by construction.

Let R_{i} ∈ O(3) be the reflection in ℝ^{3} that changes the sign of the i^{th} coordinate, for i = 1, 2, 3. Then from [H2] we have that the four 4-manifolds that are total spaces of S^{2}-bundles over T are given by the quotient of S^{2} × E^{2} by the subgroups of I som(S^{2} × E^{2}) = I som(S^{2}) × I som(E^{2}) = O(3) × (ℝ^{2} ⋊ O(2)) generated by (A_{j}, (1, 0), I) and (B_{j}, (0, 1), I) for j = 1, 2, each of which acts freely and cocompactly on S^{2} × E^{2} and is isomorphic to ℤ^{2} = Π_{1}(T) = Π_{1}(M) ; where:

(I)ξ_{1} : A_{1} = I, B_{1} = −I;
(II)ξ_{2} : A_{2} = R_{1}, B_{2} = R_{1}R_{2};
(III)ξ_{3} : A_{3} = I, B_{3} = I; and
(IV)ξ_{4} : A_{4} = R_{1}R_{2}, B_{4} = R_{1}R_{3}.

Cases (III) and (IV) yield orientable 4-manifolds that are total spaces of S^{2}- bundles over T and have been shown to be distinguished by w_{2}(ξ_{3}) = 0 and w_{2}(ξ_{4}) ≠ 0, [Ue].

Using the standard stereographic projection of S^{2} ⊂ ℝ^{3} = ℂ × ℝ onto ℂ̂ = ℂ ∪ {∞}, we may identify the reflections R_{i}: S^{2} → S^{2} with the antiholomorphic involutions:

z⟼^{R_{1}} \frac{1}{\overline{z}}, z⟼^{R_{2}}\overline{z}, z⟼^{R_{3}} − \overline{z}.

Then σ_{λ}(s, t) = (2t−1)e^{iπ(2s+λ)} +λt(t−1) defines a 1-parameter family of crosssections on 𝔗 for ξ_{1} with σ_{0}(s, t) = σ_{1}(s, t) only when (s, t) ∈ \{(0, \frac{5−\sqrt{17}}{2}), (\frac{1}{2}, \frac{−3+\sqrt{17}}{2})\}. The tangent plane to σ_{0}(s, t) at (0, \frac{5−\sqrt{17}}{2}) is spanned by the 4-vectors \{(0, 2π(4 − \sqrt{17}), 1, 0), (2, 0, 0, 1)\} and the tangent plane to σ_{1}(s, t) at (0, \frac{5−\sqrt{17}}{2}) is spanned by the 4-vectors \{(0, 2π(\sqrt{17} − 4), 1, 0), (2 − \sqrt{17}, 0, 0, 1)\}. These two sets together constitute a linearly independent set and hence the intersection at this point is transverse. Similarly the intersection at (\frac{1}{2}, \frac{−3+\sqrt{17}}{2}) may be shown to be transverse so that the mod 2 intersection number is σ · σ = 0, [as σ_{0}(s, t) and σ_{1}(s, t) are clearly isotopic]. Hence the second Wu class v_{2}(M) = 0 and from above we get w_{2}(ξ_{1}) = 0.

Similarly σ_{λ}(s, t) = (λs(s − 1) + 1)^{(t^{2}+t−1)}e^{iπ(2t+λ)(2t−1)} defines a 1-parameter family of cross-sections on 𝔗 for ξ_{2} such that the sections σ_{0} and σ_{1} intersect transversely in a single point. Hence the intersection number σ · σ = 1, thus v_{2}(M) ≠ 0 and w_{2}(ξ_{2}) ≠ 0.

Thus these two bundles are distinguished by their second Stiefel-Whitney classes so that all S^{2}-bundles over T admit geometries of type S^{2} × E^{2}.

S^{2}-Bundles Over Kb

The following six pairs ((A_{j}, (1, 0), ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} )), (B_{j}, (0, 1), I)) for 1 ≤ j ≤ 6 of ele-

ments of I som(S^{2} ×E^{2}) each generate a subgroup isomorphic to ℤ˜×ℤ = Π_{1}(Kb) = Π_{1}(M) which acts freely and cocompactly on S^{2} × E^{2}; where:

(I)ξ_{1} : A_{1} = I, B_{1} = I;
(II)ξ_{2} : A_{2} = I, B_{2} = −I;
(III)ξ_{3} : A_{3} = I, B_{3} = R_{2}R_{3};
(IV)ξ_{4} : A_{4} = R_{1}R_{2}, B_{4} = R_{1};
(V)ξ_{5} : A_{5} = −I, B_{5} = I; and
(VI)ξ_{6} : A_{6} = R_{1}, B_{6} = R_{2}R_{3}.

[Remark: the generators given on page 30 of [H2] in the cases (iii) and (ix) are incorrect. The pair (A, B) should be (I, R_{2}R_{3}) or (R_{1}, R_{2}R_{3}) respectively.]

Cases (V) and (VI) yield orientable 4-manifolds that are total spaces of S^{2}- bundles over Kb and have been shown to be distinguished by w_{2}(ξ_{5}) = 0 and w_{2}(ξ_{6}) ≠ 0, [Ue].

Note that in cases (I)-(IV) the generator (A_{j}, (1, 0), ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} )) is an orientation

reversing homeomorphism of S^{2} × E^{2} so that the corresponding 4-manifolds are non-orientable. These 4-manifolds are all total spaces of S^{2}-bundles over Kb, and we shall show that they may be distinguished by the Stiefel-Whitney classes of these bundles.

Since ξ_{1} is the product bundle w_{2}(ξ_{1}) = 0 and w_{1}(ξ_{1}) = 0.

Let the fundamental domain of Kb = ℝ^{2}/(ℤ˜×ℤ) be 𝔎 = {(s, t) ∈ ℝ^{2}|0 ≤ s ≤ 1, −\frac{1}{2} ≤ t ≤ \frac{1}{2}\}. The functions σ_{λ}(s, t) = (t^{2} + \frac{3}{4})e^{iπ(2s+λ)} + λ(t^{2} − \frac{1}{4}) define a 1-parameter family of cross-sections on 𝔎 for ξ_{2} that induces a cross-section σ̃: Kb → M of Kb as above with the sections σ_{0} and σ_{1} disjoint, so that σ · σ = 0.

That is v_{2}(M) = 0 and therefore w_{2}(ξ_{2}) = 0. Also the orientation character w_{1}(ξ_{2}) is non-zero as det(−I) = −1.

The functions σ_{λ}(s, t) = λte^{iπs} +(1−λ)(t^{2} − \frac{1}{4})e^{iπ2s} define a 1-parameter family of cross-sections for ξ_{3} such that the sections σ_{0} and σ_{1} intersect transversely in one point, so that σ · σ = 1. Hence v_{2}(M) ≠ 0 and therefore w_{2}(ξ_{3}) ≠ 0. Furthermore w_{1}(ξ_{3}) = 0 as det(I) = det(R_{2}R_{3}) = 1.

For ξ_{4} the functions σ_{λ}(s, t) = (λ(t^{2} − \frac{1}{4}) + 1)^{(s^{2}+s−1)}e^{iπ(2s+λ)(2s−1)} define 1parameter family of cross-sections with the sections σ_{0} and σ_{1}(s, t) intersecting transversely in a point, so that σ·σ = 1. Hence v_{2}(M) ≠ 0 and therefore w_{2}(ξ_{3}) ≠ 0. Also w_{1}(ξ_{4}) ≠ 0 as det(R_{1}) = −1.

Thus all S^{2}-bundles over Kb are geometric of type S^{2} × E^{2}.

S^{2}-Bundles Over Closed Orientable Surfaces With χ < 0

Let T^{g} be the closed orientable surface of genus g. Then

Π_{1}(T^{g}) = ⟨X_{1}, Y_{1}, …, X_{g}, Y_{g}| ∏_{i=1}^{g}[X_{i}, Y_{i}] = 1⟩

and we have a degree 1 map [figure: Ω: Π_{1}(T^{g}); ℤ^{2}] that kills the generators X_{j}, Y_{j} for j > 1. If 𝔗^{g} ⊂ ℍ^{2} is a 2g-gon representing the fundamental domain of T^{g} then Ω may be induced by a ‘collapsing function’ [figure: Ω: 𝔗^{g}; 𝔗] that collapses 2g−4 sides of 𝔗^{g} to a single vertex in the rectangle 𝔗. In turn this induces a degree 1 collapsing function [figure: Ω̂: T^{g}; T] that collapses g − 1 handles on T^{g} to a single point on T. Hence given an S^{2}-bundle ξ, p: M_{α} → T over T where M_{α} = (S^{2} × E^{2})/Γ_{α} and Γ_{α} = {(α(h), h)|h ∈ Π_{1}(T)} ≤ I som(S^{2} × E^{2}) where α: ℤ^{2} → O(3) is as before [there are four choices for α]; we may pull back along Ω̂ to get the commutative diagram:

[figure: M_{αΩ}; p_{αΩ}; M_{α}; p; T^{g}; Ω̂; T]

and an S^{2}-bundle Ω̂^{∗}(ξ), p_{αΩ}: M_{αΩ} → T^{g} over T^{g}, where M_{αΩ} = (S^{2} × ℍ^{2})/Γ_{αΩ} and Γ_{αΩ} = {(αΩ(h), h)|h ∈ Π_{1}(T^{g})} ≤ I som(S^{2} × ℍ^{2}). As Ω̂ is of degree 1 it induces monomorphisms in cohomology so that all the Stiefel-Whitney classes are carried from ξ to Ω̂^{∗}(ξ) that is w(Ω̂^{∗}(ξ)) = Ω̂^{∗}w(ξ) = w(ξ). Hence all S^{2}-bundles over T^{g} for g ≥ 2 are geometric of type S^{2} × ℍ^{2}.

S^{2}-Bundles Over Closed Non-Orientable Surfaces With χ < 0

Let B be the closed surface #^{3}ℝℙ^{2} = T #ℝℙ^{2} = Kb#ℝℙ^{2}. Then we have a collapsing function map [figure: Ω̂: T #ℝℙ^{2}; ℝℙ_{2}] ; that collapses the torus sum-

mand to a single point. This map Ω̂ is of degree 1 and so induces monomorphisms in cohomology. In particular Ω̂^{∗} preserves the orientation character, that is w_{1}(Ω̂^{∗}(ξ)) = Ω̂^{∗}w_{1}(ℝℙ^{2}) = w_{1}(B); and is an isomorphism on H^{2}. We may pull back the four S^{2}-bundles ξ over ℝℙ^{2} along Ω̂ to obtain the four bundles over B with first Stiefel-Whitney class w_{1}(Ω̂^{∗}ξ) either 0 or w_{1}(B).

Similarly we have a collasping function [figure: Υ̂: Kb#ℝℙ^{2}; ℝℙ_{2}] ; that collapses

the Klein bottle summand to a single point. This map Υ̂ is of degree 1 mod 2 so that Υ̂^{∗}w_{1}(ℝℙ^{2}) has nonzero square since w_{1}(ℝℙ^{2})^{2} ≠ 0. Note that in this case Υ̂^{∗}w_{1}(ℝℙ^{2}) ≠ w_{1}(B). Hence we may pull back the two S^{2}-bundles ξ over ℝℙ^{2} with w_{1}(ξ) = w_{1}(ℝℙ^{2}) to obtain a further two bundles over B with w_{1}(Υ̂^{∗}(ξ))^{2} = Υ̂^{∗}w_{1}(ξ)^{2} ≠ 0, as Υ̂ is a ring monomorphism.

Again there is a collapsing function [figure: Δ̂: Kb#ℝℙ^{2}; Kb] ; that collapses the

Klein bottle summand to a single point. Once again Δ̂ is of degree 1 mod 2 so that we may pull back the two S^{2}-bundles ξ over Kb with w_{1}(ξ) = w_{1}(Kb) along Δ̂ to obtain the remaining two S^{2}-bundles over B. These two bundles Δ̂^{∗}(ξ) have w_{1}(Δ̂^{∗}(ξ)) ≠ 0 but w_{1}(Δ̂^{∗}(ξ))^{2} = 0; as w_{1}(Kb) ≠ 0 but w_{1}(Kb)^{2} = 0 and Δ̂^{∗} is a monomorphism.

Similar arguments apply to bundles over #^{n}ℝℙ^{2} where n > 3.

Thus all S^{2}-bundles over all closed aspherical surfaces are geometric. Furthermore since the antipodal involution of a geometric S^{2}-bundle is induced by an isometry (−I, id_{𝕏^{2}}) ∈ O(3) × I som(𝕏^{2}) we have that all ℝℙ^{2}-bundles over closed aspherical surfaces are geometric. These final two results together with theorems 1 and 2 prove the main theorem.

An alternative route to the main theorem would be to first show that orientable 4-manifolds which are total spaces of S^{2}-bundles are geometric, then deduce that ℝℙ^{2}-bundles are geometric [as above]; and finally observe that every S^{2}-bundle space double covers an ℝℙ^{2}-bundle space.

References

[H1]Jonathan A. Hillman, On 4-manifolds with universal covering space S^{2} × ℝ^{2} or S^{3} × ℝ, Top.

Appl. 52 (1993), 23-42. [H2]Jonathan A. Hillman, On 4-manifolds with universal covering space a compact geometric

manifold, J. Austral. Math. Soc. (Series A). 55 (1993), 137-148. [Hu]Dale Husemoller, Fibre bundles, Springer-Verlag, New York, 1994. [Me]Paul Melvin, 2-sphere bundles over compact surfaces, Proc. Amer. Math. Soc. 92 (1984),

567-572. [Ue]Masaaki Ue, Geometric 4-manifolds in the sense of Thurston and Seifert 4-manifolds II, J.

Math. Soc. Japan. 43 (1991), 149-183.

School of Mathematics and Statistics, The University of Sydney, Sydney, NSW 2006, Australia.

E-mail address: cobb r@ maths.su.oz.au ; hillman j@ maths.su.oz.au