Conformally flat metrics on 4-manifolds

Michael Kapovich

August 1, 2003

Abstract

We prove that for each closed smooth spin 4-manifold M there exists a closed smooth 4-manifold N such that M#N admits a conformally flat Riemannian metric.

Contents

1 Introduction 1

2 Definitions and notation 3

3 Reflection groups in S^{4} with prescribed combinatorics of the fundamental domains 4

4 Proof of Theorem 1.1 7

Bibliography 11

1 Introduction

The goal of this paper is to prove

Theorem 1.1. Let M^{4} be an closed connected smooth spin 4-manifold. Then there exists a closed orientable 4-manifold N such that M#N admits a conformally flat Riemannian metric. The manifold N is (in principle) computable in terms of triangulation of M.

Recall that there are many closed 4-dimensional spin manifold which admit no flat conformal structure; for instance, simply-connected manifolds (not diffeomorphic to S^{4}), manifolds with simple infinite fundamental group. (First examples of first 3manifolds not admitting flat conformal structure were constructed by W. Goldman in [1].) The above theorem shows that if M admits a flat conformal structure, it does not imply that all components of its connected sum decomposition are also conformally flat.

Our motivation comes from the following theorem of C. Taubes [12]:

Theorem 1.2. Let M be a smooth closed oriented 4-manifold. Then there exists a number k so that the connected sum of M with k copies of \overline{ℂℙ^{2}} admits a halfconformally flat structure.

Here \overline{ℂℙ^{2}} is the complex-projective plane with the reversed orientation. Recall that a Riemannian metric g on M is anti self-dual (or half-conformally flat) if the self-dual part W_{+} of the Weyl tensor vanishes. Vanishing of both self dual and anti self-dual parts of the Weyl tensor (i.e., vanishing of the entire Weyl tensor) is equivalent to local conformal flatness of the metric g.

Note that the assumption that M^{4} is spin is equivalent to vanishing of all Stiefel-Whitney classes, which is equivalent to triviality of the tangent bundle of M′ = M \ {p}. According to the Hirsch-Smale theory (see for instance [7, Theorem 4.7] or [11]), M′ := M \ {p} is parallelizable iff M′ admits an immersion into ℝ^{4}. Thus, by taking M to be simply-connected with nontrivial 2-nd Stiefel-Whitney class, one sees that M#N does not admit a flat conformal structure for any N: Otherwise the developing map would immerse M′ into S^{4}. Therefore the vanishing condition is, to some extent, necessary. Note also that (unlike in Taubes’ theorem) one cannot expect N to be simply-connected since the only closed conformally flat simply-connected Riemannian manifold is the sphere with the standard conformal structure.

Sonjong Hwang in his thesis [5], has verified that for 3-manifolds an analogue of Theorem 1.1 holds, moreover, one can use a connected sum of Haken manifolds as the manifold N. Similar arguments can be used to prove an analogous theorem in the context of locally spherical CR structures on 3-manifolds.

The arguments in both 3-dimensional and 4-dimensional cases, in spirit (although, not in the technique), are parallel to Taubes’: We start with a singular conformallyflat metric on M, where the singularity is localized in a ball B ⊂ M. The singular metric is obtained by pull-back of the standard metric on the 4-sphere under a branched covering M → S^{4}. Then we would like to “resolve the singularity”. To do so we remove an open tobular neighborhood U of the singular locus. Afterwards, use the “orbifold trick” (cf. for instance [2]) to eliminate the boundary of M \ U: Introduce a Moebius reflection orbifold structure on M \ U to get a closed Moebius orbifold O which is a connected sum of M with an orbifold. After passing to an appropriate finite manifold cover over O we get a conformally-flat manifold which has M as a connected summand.

Lack of reflection groups in higher-dimensional hyperbolic spaces limits this strategy to low dimensions. Using arguments somewaht similar to the strict hyperbolization of Charney and Davis (see [3]) one can generalize Theorem 1.1 to higher dimensional almost parallelizable manifolds. We will discuss this issue elsewhere.

Acknowledgments. During the work on this paper the author was supported by the NSF grants DMS-99-71404 and DMS-02-03045. Part of this work was done during the author’s stay at Max-Plank-Instutit für Mathematik in Bonn, whose hospitality is gratefully acknowledged. The author is also grateful to Tadeusz Januszkiewicz who had pointed out at a flaw in the original version of the paper.

2 Definitions and notation

We let Mob(S^{4}) denote the full group of Moebius transformations of S^{4}, i.e. the group generated by inversions in round spheres. Equivalently, Mob(S^{4}) is the restriction of the full group of isometries Isom(ℍ^{5}) to the 4-sphere S^{4} which is the ideal boundary of ℍ^{5}. We will regard S^{4} as 1-point compactification ℝ^{4} ∪ {∞} of then Euclidean 4-space.

Definition 2.1. Let Q be a unit cube in ℝ^{4}. We define the PL inversion J in the boundary of Q as follows. Let h : S^{4} → S^{4} be a PL homeomorphism which sends Σ = ∂Q onto the round sphere S^{3} ⊂ ℝ^{4} and h(∞) = ∞. Let j : S^{4} → S^{4} be the ordinary inversion in S^{3}. Then J := h^{−1} ∘ j ∘ h.

Definition 2.2. A Moebius or a flat conformal structure on a smooth 4-manifold M is an atlas {(V_{α}, φ_{α}), α ∈ A} which consist of diffeomorphisms φ_{α} : V_{α} → U_{α} ⊂ S^{4} so that the transition mappings φ_{α} ∘ φ_{β}^{−1} are restrictions of Moebius transformations.

Equivalently, one can describe Moebius structures on M are conformal classes of conformally-Euclidean Riemannian metrics on M. Each conformal structure on M gives rise to a local conformal diffeomorphism, called a developing map, d : M̃ → S^{4}, where M̃ is the universal cover of M. If M is connected, the mapping d is equivariant with respect to a holonomy representation ρ : π_{1}(M) → Mob(S^{4}), where π_{1}(M) acts on M̃ as the group of deck-transformations. Given a pair (d, ρ), where ρ is a representation of π_{1}(M) into Mob(S^{4}) and d is a ρ-equivariant local diffeomorphism from M̃ to S^{4}, one constructs the corresponding Moebius structure on M by taking a pull-back of the standard flat conformal structure on S^{4} to M̃ via d and then projecting the structure to M.

Analogously, one defines a complex-projective structures on complex 3-manifold Z, as a ℂℙ^{3}-valued holomorphic atlas on Z so that the transition mappings belong to P GL(3, ℂ).

The concept of Moebius structure generalizes naturally to the category of orbifolds: A 4-dimensional Moebius orbifold O is a pair (X, A), where X is a Hausdorff topological space, the underlying space of the orbifold, A is a family of local parameterizations ψ_{α} : U_{α} → U_{α}/Γ_{α} = V_{α}, where {V_{α}, α ∈ A} is an open covering of X, U_{α} are open subsets in S^{4}, Γ_{α} are finite groups of Moebius automorphisms of U_{α} and the mappings ψ_{α} satisfy the usual compatibility conditions:

If V_{α} → V_{β} is the inclusion map then we have a Moebius embedding U_{α} → U_{β} which is equivariant with respect to a monomorphism Γ_{α} → Γ_{β}, so that the diagram

\begin{aligned}U_{α} & → & U_{β} \\ ↓ & & ↓ \\ V_{α} & → & V_{β} \end{aligned}

is commutative. The groups Γ_{α} are the local fundamental groups of the orbifold O.

For a Moebius orbifold one defines a developing mapping d : Õ → S^{4} (which is a local homeomorphism from the universal cover Õ of O) and, if O is connected, a holonomy homeomorphism ρ : π_{1}(O) → Mob(S^{4}), which satisfy the same equivariance condition as in the manifold case. Again, given a pair (d, ρ), where d is a ρ-equivariant homeomorphism, one defines the corresponding Moebius structure via pull-back.

Example 2.3. Let G ⊂ Mob(S^{4}) be a subgroup acting properly discontinuously on an open subset Ω ⊂ S^{4}. Then quotient space Ω/G has a natural Moebius orbifold structure. The local charts ϕ_{α} appear in this case as restrictions of the projection p : Ω → Ω/G to open subsets with finite stabilizers.

In particular, suppose that G is a finite subgroup of Mob(S^{4}) generated by reflections, the quotient Q := Ω/G can be identified with the intersection of a fundamental domain of G with Ω. The Moebius structures on 4-dimensional manifolds and orbifolds constructed in this paper definitely do not arise this way. A more interesting example is obtained by taking a manifold M and a local homeomorphism h : M → S^{4}, so that Q ⊂ h(M). Then we can pull-back the Moebius orbifold structure on Q to an appropriate subset X of M, to get a 4-dimensional Moebius orbifold. As another example of a pull-back construction, let O be a Moebius orbifold and M → O be an orbifold cover such that M is a manifold. Then one can pull-back the Moebius orbifold structure from O to an ordinary Moebius structure on M.

We refer the reader to [9] for a more detailed discussion of geometric structures on orbifolds.

3 Reflection groups in S^{4} with prescribed combinatorics of the fundamental domains

Figure 1: Barycentric subdivision of a square.

Consider the standard cubulation Q of ℝ^{4} by the Euclidean cubes with the edges of length 2 and let X denote the 2-skeleton of this cubulation. Given a collection of round balls {B_{i}, i ∈ I} in ℝ^{4}, with the nerve N, we define the canonical simplicial mapping f : N → ℝ^{4} by sending each vertex of N to the center of the corresponding ball and extending f linearly to the simplices of N. For a subcomplex K ⊂ Q define its barycentric subdivision β(K) to be the following simplicial complex. Subdivide each edge of K by its midpoint. Then inductively subdivide each k-cube Q in K by coning off the barycentric subdivision of ∂Q from the center of Q.

Figure 2:

Proposition 3.1. Suppose that K ⊂ X is a 2-dimensional compact subcomplex such that each vertex belongs to a 2-cell. Then there exists a collection of open round 4-balls B_{i}, i = 1, ..., k, centered at the vertices of β(K), so that:

(1) The Moebius inversions R_{i} in the round spheres S_{i} = ∂B_{i} generate a discrete reflection group G ⊂ Mob(S^{4}).

(2) The complement S^{4} \ ∪_{i=1}^{k}B_{i} is a fundamental domain Φ of G.

(3) The canonical mapping from the nerve of {B_{i}, i = 1, ..., k} to ℝ^{4} is a simplicial isomorphism onto β(K).

Proof: We begin by constructing the family of spheres S_{i}, i ∈ ℕ centered at certain points of X. For each square Q in X we pick 9 points x_{1}, ...., x_{9}: x_{5}, ..., x_{8} are the vertices of Q, x_{1}, ..., x_{4} are midpoints of the edges of Q and x_{9} is the center of Q. Pick radii r, R, ρ so that:

(a) The spheres S_{1} = S(x_{1}, r), S_{5} = S(x_{5}, R) are orthogonal.

(b) The (exterior) angle of intersection between the spheres S(x_{1}, r), S_{9} = S(x_{9}, ρ) equals π/3.

(c) The (exterior) angle of intersection between the spheres S(x_{5}, R), S(x_{9}, ρ) equals π/5. (Actually the latter angle can be taken π/4 as well, but π/3 would not suffice.)

Then (by doing MAPLE calculations) for the radii r, R, ρ we get:

R ≈ 0.8534646790, r ≈ 0.5211506901, ρ ≈ 0.6317819089

In particular, r and ρ are both less than 1/\sqrt{2} ≈ 0.7071067810.

Remark 3.2. Here is a proof that ρ < 1/\sqrt{2} without help of MAPLE. We have system of equations:

\{ \begin{matrix}R^{2} + r^{2} = 1, \\ R^{2} + ρ^{2} + Rρ = 2, \\ r^{2} + ρ^{2} + \sqrt{2}ρr = 1 \end{matrix}

Observe that R = R(ρ) and r = r(ρ) are both decreasing functions of ρ, for ρ > 0. Let’s estimate R^{2}(ρ) + r^{2}(ρ) with ρ = 1/\sqrt{2}. We get:

R = \frac{\sqrt{13} − 1}{2\sqrt{2}}, r = \frac{\sqrt{3} − 1}{2}.

Then R^{2} + r^{2} < 1. Thus, by monotonicity, R(ρ)^{2} + r(ρ)^{2} < 1 for all ρ ≥ 1/\sqrt{2}. Hence each triple of solutions of our system satisfies: ρ < 1/\sqrt{2}. From the equation R^{2} + r^{2} = 1 we also see that R < 1 < \sqrt{2}.

Similar calculation shows that r < 1/\sqrt{2}.

We then consider the collection of round balls B(x_{9}, ρ), B(x_{i}, r), i = 1, ..., 4 and B(x_{i}, R), i = 5, ..., 8; see Figure 2. The condition r < \sqrt{2}/2 and our choice of the angles of intersection between the spheres imply that the nerve of the above collection of balls is the barycentric subdivision of Q. See Figure 1 for the Coxeter graph of the Coxeter group generated by inversions in the spheres S_{1}, ..., S_{9}.

Suppose now that Q^{4} is a 4-cube in the cubulation Q, apply the above construction to each 2-face of Q^{4}. Then the condition ρ, r < \sqrt{2}/2 implies that the the nerve N_{Q^{4}} of the resulting collection of balls {B_{i}} is such that the canonical mapping N_{Q^{4}} → β((Q^{4})^{(2)}) is a simplicial isomorphism.

Now we are ready to construct the covering {B_{i} : i = 1, ..., k} of the 2-complex K. For each 2-face Q of K introduce the family of nine round spheres S_{i} constructed above, consider the inversions R_{i} is these spheres; the spheres S_{i} bound balls {B_{i} : i = 1, ..., k}. The fact that for each 4-cube Q^{4} the mapping N_{Q^{4}} → β((Q^{4})^{(2)}) is a simplicial isomorphism, implies that the mapping from the nerve of the covering {B_{i} : i = 1, ..., k} to K is a simplicial isomorphism as well. Thus the exterior angles of intersections between the spheres equal \frac{π}{2} and \frac{π}{3}, thus we can apply Poincare’s fundamental polyhedron theorem [10] to ensure that the intersection of the complements to the balls B_{i} is a fundamental domain for the Moebius group G generated by the above reflections.

Remark 3.3. Instead of collections of round balls based on a cubulation of ℝ^{4} one could use a periodic triangulation of ℝ^{4}, however in this case the construction of a collection of balls covering the 2-skeleton of a 4-simplex would be more complicated.

4 Proof of Theorem 1.1

Recall that the manifold M is almost parallelizable, i.e M^{∘} = M \{p} is parallelizable; hence, by [7], there exists an immersion f : M^{∘} → ℝ^{4}. Let B denote a small open round ball centered at p and let M′ denote the complement M \ B. We retain the notation f for the restriction f|M′. We next convert to the piecewise-cubical setting: Consider the pull-back of the standard cubulation Q of ℝ^{4} to M′ via f. The image f(M′) is compact; hence, after replacing if necessary Q by its sufficiently fine cubical subdivision, we may assume that for each point x ∈ M′ there exists a cube Q_{x} ∈ Q and a component Q̃_{x} ⊂ M^{∘} of f^{−1}(Q) so that f|Q̃_{x} : Q̃_{x} → Q_{x} is a homeomorphism. In particular, if C ⊂ M′ denotes the union of the cubes Q̃_{x}, x ∈ M′, then C is a compact cubical complex. After replacing B by a slightly larger open ball B′ we get: ∂B′ ⊂ int(C). Let M^{•} denote the complement M \ B′. Now consider the 2-nd cubical subdivision C′′ of C and the regular neighorhood N := N(F r_{M}(C)) in C′′ of the frontier F r_{M}(C): The frontier of N in C is a 3-dimensional submanifold Y ⊂ B′ which is contained in the 3-skeleton of C′′. Thus f(Y) is also contained in the 3skeleton of Q′′. Let M′′ denote the closure of the component of M \Y which is disjoint from p. Clearly, M′′ is a compact cubulated manifold with the boundary Y. We retain the notation f for the restriction f|M′′. We now double M′′ across its boundary Y, the result is a closed cubulated manifold DM, let τ : DM → DM denote the involution fixing Y pointwise; the mapping f extends to DM \ M′′ by f ∘ τ. Thus we get a globally defined pieciewise-linear map F : DM → ℝ^{4}, which is a homeomorphism on each 4-cube in DM and is a local diffeomorphism on M^{•}. By cutting DM along the sphere ∂M^{•} we get a connected sum decomposition DM = M#W. We orient DM so that F preserves the orientation on M^{•}.

We now borrow the standard arguments from the proof of Alexander’s theorem which states that each closed n-dimensional PL manifold is a branched cover over the n-sphere, see e.g. [4]. For each cube Q′ ⊂ DM such that F |Q′ is orientationreversing we replace F |Q′ with the composition J ∘ F |Q′, where J is the PL inversion in the boundary of the unit cube F (∂Q′) (see Definition 2.1). The resulting mapping h : DM → S^{4} has the property that it is a local PL homeomorphism away from a 2-dimensional subcomplex L ⊂ DM \ M′′, which is therefore disjoint from M^{•}. (Note that L has dimension 2 near every point: Each vertex in L belongs to a 2cube in L.) Thus the mapping h is a branched covering over S^{4} with the singular locus L contained in DM \ M^{•}, the branch-locus of h is the compact subcomplex K = h(L) ⊂ ℝ^{4}. The branched covering h has the property that for each point x ∈ K there exists a neighborhood U(x) ⊂ ℝ^{4} such that h^{−1}(U(x)) is a disjoint union of balls V (y), y ∈ h^{−1}(x) ∈ DM \ M′′, (whose interiors contain y), so that for each y ∈ h^{−1}(x), the restriction h|V (y) is a branched covering onto U(x). Moreover, each branched covering h|V (y) is obtained by coning off a branched covering from the 3-sphere ∂V (y) to the 3-sphere U(x).

Question 4.1. Given a local diffeomorphism f : M^{∘} → ℝ^{4}, is it possible to modify f within the ball B to make it a branched cover M → S^{4} which is ramified over a smooth surface in S^{4}? This can be easily arranged in the case of 3-manifolds. In dimension 4 this would be is a relative version of a recent theorem of Iori and Piergallini [8].

Let T denote a regular neighborhood of K in ℝ^{4}, so that U(x) ⊂ T for each x ∈ K. Next, subdivide the cubulation of ℝ^{4} and scale the subdivision up to the standard cubulation Q by regular cubes with edges of length 2, so that the discrete group G and the collection of balls {B_{j}, j = 1, ..., k} associated with the subcomplex K in section 3 have the properties:

1. T ⊂ ∪_{i=1}^{k}B_{k}.

2. Each ball B_{j}, j = 1, ..., k, (centered at x_{j} ∈ K) is contained in the neighborhood U(x_{j}).

We now use the branched covering h to introduce a Moebius orbifold structure O on the complement X_{O} to an open tubular neighborhood N^{0}(L) of L in M as follows:

For each ball B_{j} ⊂ U(x_{j}) centered at x_{j} ∈ K and for each y_{j} ∈ h^{−1}(x_{j}) ∩ L, such that the restriction h|V (y_{j}) is not a homeomorphism onto its image, we let B̃(y_{j}) denote the inverse image h^{−1}(B_{j}) ∩ V (y_{j}). It follows that each B̃(y_{j}) is a polyhedral 4-ball in M and the union of these balls is a tubular neighborhood N (L) of L. The boundary of N (L) has a natural partition into subcomplexes: “vertices”, “edges”, “2-faces” and “3-faces”:

• The “vertices” are the points of triple intersections of the 3-spheres ∂B̃(y_{j}), ∂B̃(y_{i}), ∂B̃(y_{l}).

• The “2-faces” are the connected components of the double intersections of the 3-spheres ∂B̃(y_{j}), ∂B̃(y_{i}).

• The “3-faces” are the connected components of the complements

∂B̃(y_{j}) \ ∪_{i≠j}B̃(y_{i}).

We declare each “3-face” a boundary reflector of the orbifold O. The dihedral angles between the balls B_{j} define the dihedral angles between the boundary reflectors in O. Since the restriction h|M \ L is a local homeomorphism, this construction defines a Moebius orbifold O. The mapping h|X_{O} is the projection of the developing mapping h̃ : Õ → S^{4} of this Moebius orbifold. Let O^{•} denote the orbifold with boundary O \ M^{•}; let O′ be the closed orbifold obtained by attaching 4-disk D^{4} to O^{•} along the boundary sphere S^{3}.

We next convert back to the smooth category. It is clear from the construction that the orbifold O is obtained by (smooth) gluing of the manifold with boundary M \B and the orbifold with boundary O^{•}. Hence O is diffeomorphic to the connected sum of the manifold M with the orbifold O′.

It remains to construct a finite manifold covering M̂ over the orbifold O, so that M \ B lifts homeomorphically to M̂; the construction is analogous to the one used by M. Davis in [2]. The universal cover Õ is a manifold since it admits a (locally homeomorphic) developing mapping to S^{4}. The fundamental group π_{1}(O) is the free product π_{1}(M) ∗ π_{1}(Q). We have holonomy homomorphism

ϕ : π_{1}(O) → G,

the subgroup π_{1}(M) is contained in the kernel of this homomorphism; by construction, the kernel of ϕ acts freely on Õ. The Coxeter group G is virtually torsion-free, let θ :

G → A be a homomorphism onto a finite group A, so that Ker(θ) is torsion-free and orientation-preserving. Then the kernel of the homomorphism ψ = θ ∘ ϕ : π_{1}(O) → A is a finite index subgroup of π_{1}(O), which contains π_{1}(M) and still acts freely on Õ. Let M̂ → O denote the finite orbifold cover corresponding to the subgroup Ker(ψ). Then M̂ is a smooth oriented conformally flat manifold, the submanifold M^{•} lifts diffeomorphically to M^{•} ⊂ M̂. Thus the connected sum decomposition O = M#O′ also lifts to M̂, so that the latter manifold is diffeomorphic to the connected sum of M and a 4-manifold N.

We observe that the proof of Theorem 1.1 can be modified to prove the following:

Theorem 4.2. Suppose that M is a closed smooth 4-manifold whose orientable 2-fold cover is Spin. Then there exists a closed smooth 4-manifold N so that M̂ = M#N admits a conformally-Euclidean Riemannian metric.

Proof: The difference with Theorem 1.1 is that M can be nonorientable. Let M̃ → M be the orientable double cover with the deck-transformation group D ≅ ℤ/2. Then all Stiefel-Whitney classes of M̃ are trivial. As before, let p ∈ M, {p_{1}, p_{2}} be the preimage of {p} in M. Consider a Euclidean reflection τ in ℝ^{4} and an epimorphism θ : D → ⟨τ⟩. Then, arguing as in the proof of Hirsch’s theorem [7, Theorem 4.7], one gets a θ-equivariant immersion f̃: M̃ \ {p_{1}, p_{2}} → ℝ^{4}. This yields a D-invariant flat conformal structure on M̃ \ {p_{1}, p_{2}} via pull-back of the flat conformal structure from ℝ^{4}. Let B_{1} ⊔B_{2} be a D-invariant disjoint union of open balls around the points p_{1}, p_{2}. Then the rest of the proof of Theorem 1.1 goes through: Replace the ball B_{1} with a manifold with boundary N_{1} so that the flat conformal structure on M̃ \ (B_{1} ∪ B_{2}) extends over N_{1}. Then glue a copy of N_{1} along the boundary of B_{2} in D-invariant fashion. Note that the quotient of the manifold P := (M̃ \ (B_{1} ∪ B_{2})) ∪ (N_{1} ∪ N_{2}) by the group D is diffeomorphic to a closed manifold M#N, where N is obtained from N_{1} by attaching the 4-ball along the boundary. Finally, project the D-invariant Moebius structure on P to a Moebius structure on the manifold M#N.

As a corollary of Theorem 1.1 we get:

Corollary 4.3. Let Γ be a finitely-presented group. Then there exists a 3-dimensional complex manifold Z which admits a complex-projective structure, so that the fundamental group of Z splits as Γ ∗ Γ′.

Proof: Our argument is similar to the one used to construct (via Taubes’ theorem) 3-dimensional complex manifolds with the prescribed finitely-presented fundamental group. We first construct a smooth closed oriented 4-dimensional spin manifold M with the fundamental group Γ. This can be done for instance as follows. Let ⟨x_{1}, ..., x_{n}|R_{1}, ..., R_{ℓ}⟩ be a presentation of Γ. Consider a 4-manifold X which is the connected sum of n copies of S^{3}×S^{1}. This manifold is clearly spin. Pick a collection of disjoint embedded smooth loops γ_{1}, ..., γ_{ℓ} in X, which represent the conjugacy classes of the words R_{1}, ..., R_{ℓ} in the free group π_{1}(X). Consider the pair (S^{4}, γ), where γ is an embedded smooth loop in S^{4}. For each i pick a diffeomorphism f_{i} between a tubular neighborhood T (γ) of γ in S^{4} and a tubular neighborhood T (γ_{i}) of γ_{i} in X. We can choose f_{i} so that it matches the spin structures of T (γ) and T (γ_{i}). Now, attach n copies of S^{4} \ T (γ) to X \ ∪_{i}T (γ_{i}) via the diffeomorphisms f_{i}. The result is a smooth spin 4-manifold M with the fundamental group Γ.

Next, by Theorem 1.1 there exists a smooth 4-manifold N (with the fundamental group Γ′) such that M̂ = M#N admits a conformally-Euclidean Riemannian metric. Applying the twistor construction to the manifold M̂ we get a complex 3-manifold Z which is an S^{2}-bundle over M̂ and the flat conformal structure on M̂ lifts to a complexprojective structure on Z, see for instance [6]. Clearly, π_{1}(Z) ≅ π_{1}(M̂) = Γ ∗ Γ′.

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Department of Mathematics, University of Utah, Salt Lake City, UT 84112-0090, kapovich@math.utah.edu

Department of Mathematics, University of California, Davis, 1 Shields Ave., Davis, CA 95616-8633