Finitely Dominated Subnormal Covers of 4-manifolds

Jonathan A. Hillman

School of Mathematics and Statistics, The University of Sydney, Sydney, NSW 2006, AUSTRALIA

e-mail: jonh@ maths.usyd.edu.au

ABSTRACT

Let M be a closed 4-manifold which has a finitely dominated covering space associated to a subnormal subgroup G of infinite index in π = π_{1}(M). If G is F P_{3}, has finitely many ends and π is virtually torsion free then either M is aspherical or its universal covering space is homotopy equivalent to S^{2} or S^{3}. In the aspherical case such a subnormal subgroup is usually Z, a surface group or a P D_{3}-group. (This is a revision of Research Report 1994-23).

AMS Subject Classification (1991): Primary 57N13. Secondary 20J05 Key words and phrases: finitely dominated, 4-manifold, Poincaré duality group, subnormal subgroup.

In this note we shall extend earlier work on 4-manifolds with a finitely dominated infinite covering space from the regular to the subnormal case. (See §2 of Chapter 3 of [9]). Finitely dominated covering spaces of aspherical manifolds correspond to F P subgroups of the fundamental group. In §1 we show that an F P_{3} subnormal subgroup of a P D_{4}-group is usually a P D-group. However we have not been able to eliminate other possibilities completely. For instance, it is not known whether a Baumslag-Solitar group may be a subnormal subgroup of a P D_{4}-group. In §2 we assume that M is a closed 4-manifold and that π = π_{1}(M) has an F P_{3} subnormal subgroup G of infinite index such that the associated covering space is finitely dominated, and give homological conditions on π and G under which either M is aspherical or its universal covering space is homotopy equivalent to S^{2} or S^{3}.

§1. Poincaré duality groups

The Hirsch-Plotkin radical \sqrt{π} of a group π is the maximal locally nilpotent, normal subgroup of π. The Hirsch length h(ν) of a finitely generated nilpotent group ν is the number of infinite cyclic factors of a composition series for the group; h(\sqrt{π}) is the least upper bound of h(ν) as ν varies over finitely generated subgroups of \sqrt{π}. If G is a subgroup of π then C_{π}(G) and N_{π}(G) are the centralizer and normalizer of G in π, respectively. The centre of G is ζG = G ∩ C_{π}(G).

Theorem 1. Let G be a nontrivial F P_{3} normal subgroup of infinite index in a P D_{4}-group π. Then either

(i) G is a P D_{3}-group and π/G has two ends; (ii) G is a P D_{2}-group and π/G is virtually a P D_{2}-group; or (iii) G ≅ Z, H^{s}(π/G; Z[π/G]) = 0 for s ≤ 2 and H^{3}(π/G; Z[π/G]) ≅ Z.

Proof. The subgroup G is F P, since c.d.G < 4 [13], and hence so is π/G. The E_{2} terms of the LHS spectral sequence with coefficients Q[π] can then be expressed as tensor products H^{p}(π/G; Q[π/G]) ⊗ H^{q}(G; Q[G]). If H^{j}(π/G; Q[π/G]) and H^{k}(G; Q[G]) are the first nonzero such cohomology groups then H^{j}(π/G; Q[π/G])⊗H^{k}(G; Q[G]) persists to E_{∞} and hence j+k = 4 and this tensor product is Q. Hence H^{j}(π/G; Q[π/G]) ≅ H^{n−j}(G; Q[G]) ≅ Q. In particular, π/G has one or two ends and G is a P D_{4−j}-group over Q [6]. If π/G has two ends then it is virtually Z, and then G is a P D_{3}-group (over Z) by Theorem 9.11 of [1]. If H^{2}(G; Q[G]) ≅ H^{2}(π/G; Q[π/G]) ≅ Q then G and π/G are virtually P D_{2}-groups [3]. Since G is torsion free it is then in fact a P D_{2}-group. The only remaining possibility is (iii). □

Is it sufficient that G be F P_{2}? Must the quotient π/G be virtually a P D-group in case (iii) also?

Corollary. If K is F P_{2} and is subnormal in N where N is an F P_{3} normal subgroup of infinite index in the P D_{4}-group π then K is a P D_{k}-group for some k < 4. Proof. This follows immediately from Theorem 1 together with [2]. □

In [2] it was shown that if H is an F P_{2} subnormal subgroup of a P D_{3}-group G then either H is an infinite cyclic normal subgroup or H is a surface group and [G : N_{G}(H)] < ∞ or G is virtually poly-Z. We shall consider next F P subnormal subgroups of P D_{4}-groups.

Theorem 2. Let G be a nontrivial F P subnormal subgroup of infinite index in a P D_{4}-group π. Suppose that G has finitely many ends. Then either

(i) G is a P D_{3}-group, [π : N_{π}(G)] < ∞ and N_{π}(G)/G has two ends; or (ii) c.d.G = 3 and H^{2}(G; Z[G]) is not finitely generated; or (iii) G is a P D_{2}-group, [π : N_{π}(G)] < ∞ and π is virtually the group of a

surface bundle over a surface; or (iv) G is a P D_{2}-group, ζG = 1 and π is virtually the group of the mapping

torus of a self homeomorphism of a surface bundle over the circle; or

(v) c.d.G = 2, χ(G) = 0, H^{2}(G; Z[G]) is not finitely generated and [π : N_{π}(G)] = ∞; or

(vi) G ≅ Z and G ≤ \sqrt{π}, and either \sqrt{π} is abelian of rank ≤ 2 or π is virtually

poly-Z. Proof. Let G = N_{0} < N_{1} < ... < N_{r} = π be a subnormal chain of minimal length. Let j = min{i|[N_{i+1} : G] = ∞}. Then N_{j} is F P and is subnormal in π, and it is easily seen that the theorem holds for G if it holds for N_{j}. Thus we may assume that [N_{1} : G] = ∞. Suppose first that G has one end. Then c.d.G = 2 or 3, since [π : G] = ∞. If c.d.G = 3 and H^{2}(G; Z[G]) is finitely generated then H^{s}(G; Z[G]) = 0 for s ≤ 2, by [5]. It follows immediately from the LHS spectral sequence that H^{s}(N_{1}; W) = 0 for s ≤ 3 and any free Z[N_{1}]-module W. Hence c.d.N_{1} = 4 and so [π : N_{1}] < ∞, by [13]. Hence N_{1} is a P D_{4}-group and (i) follows from Theorem 1. If c.d.G = 3 and H^{2}(G; Z[G]) is not finitely generated (ii) holds.

Suppose next that c.d.G = 2. If G_{1} < G_{2} are two such groups with G_{1} normal in G_{2} then [G_{2} : G_{1}] is finite, by Theorem 8.2 of [1], and χ(G_{1}) = [G_{2} : G_{1}]χ(G_{2}). Moreover if G_{2} is normal in J then [J : N_{J}(G_{1})] < ∞, since G_{2} has only finitely many subgroups of index [G_{2} : G_{1}]. Therefore if χ(G) ≠ 0 we may assume that G is maximal among normal subgroups of N_{1} with cohomological dimension 2. Let n be an element of N_{2} such that nGn^{−1} ≠ G, and let H = G.nGn^{−1}. Then G < H and H is normal in N_{1} so [H : G] = ∞ and c.d._{Q}H = 3. Moreover H is F P and H^{s}(H; Z[H]) = 0 for s ≤ 2, so either N_{1}/H is locally finite or c.d_{Q}N_{1} > c.d._{Q}H, by Theorem 8.2 of [1]. If N_{1}/H is locally finite but not finite then we again have c.d_{Q}N_{1} > c.d._{Q}H, by Theorem 3.3 of [8]. If c.d._{Q}N_{1} = 4 then [π : N_{1}] < ∞, so N_{1} is a P D_{4}-group and (iii) holds, by Theorem 1. Otherwise [N_{1} : H] < ∞ and then c.d.N_{1} = 3, N_{1} is F P and H^{s}(N_{1}; Z[N_{1}]) = 0 for s ≤ 2. Hence N_{1} is a P D_{3}-group by (i), and so (iv) holds.

Suppose that χ(G) = 0 and that G is a P D_{2}-group. Then G ≅ Z^{2} or Z ×_{−1} Z, so h(\sqrt{π}) ≥ 2 and χ(π) = 0. We may assume that π is orientable, so Hom(π, Z) ≠ 0. If h(\sqrt{π}) > 2 then π is virtually poly-Z, by Theorem 8.1 of [9]. Therefore we may also assume that h(\sqrt{π}) = 2. In this case \sqrt{π} ≅ Z^{2} and π is virtually the group of a torus bundle over a surface, by Theorem 9.2 of [9]. Since [\sqrt{π} : G] < ∞ it follows also that [π : N_{π}(G)] < ∞ and so (iii) holds. If G has one end and c.d.G = 2 but G is not a P D_{2}-group then H^{2}(G; Z[G]) is not finitely generated [6] and [π : N_{π}(G)] = ∞, and so (v) covers the remaining possibilities.

Finally, if G has two ends then G ≅ Z, so G ≤ \sqrt{π}. If h = h(\sqrt{π}) > 2 then π is virtually poly-Z. If h ≤ 2 then \sqrt{π} is abelian of rank h. □

To what extent can the hypotheses be relaxed? Are all F P subnormal subgroups P D-groups? If so then cases (ii) and (vi) cannot arise. (This is certainly so if there is a subnormal chain consisting of F P subgroups). If G is F P and c.d.G = 3 then G has one end (cf [2]). Can a finitely generated noncyclic free group be a subnormal subgroup of a P D_{4}-group?

Examples. 1. Let π be the semidirect product of S = ⟨a, b, c, d | [a, b][c, d] = 1⟩ (the genus 2 surface group) with the rank 2 free abelian normal subgroup G generated by x and y, with the action of S on G given by axa^{−1} = xy^{2}, cyc^{−1} = x^{2}y, b, c, d commute with x and a, b, d commute with y. Then \sqrt{π} = G and C_{π}(G) ≅ Z^{2} ×F (∞). In particular, C_{π}(\sqrt{π}) need not be finitely generated.

2. Let G be a P D_{2}-group such that ζG = 1. Let θ : G → G have infinite order in Out(G), and let λ : G → Z be an epimorphism. Let π = (G×Z)×_{ϕ}Z where ϕ(g, n) = (θ(g), λ(g) + n) for all g ∈ G and n ∈ Z. Then G is subnormal in π but this group is not virtually the group of a surface bundle over a surface.

3. Any group with a finite 2-dimensional Eilenberg - Mac Lane complex is the fundamental group of a compact aspherical 4-manifold with boundary, obtained by attaching 1- and 2-handles to D^{4}. On applying the orbifold hyperbolization technique of Gromov, Davis and Januszkiewicz to the boundary we see that each such group embeds in a P D_{4}-group. (See [10]). (Conjecturally such groups are exactly the finitely presentable groups of cohomological dimension 2). The simplest such groups G with χ(G) = 0 which are not P D_{2}- groups are the Baumslag-Solitar 1-relator groups G_{p,q} = ⟨a, t | ta^{p}t^{−1} = a^{q}⟩ with |pq| > 1. Can they be realised as subnormal subgroups of P D_{4}-groups?

§2. Closed 4-manifolds

In this section we shall investigate closed 4-manifolds whose fundamental groups have subnormal subgroups and whose homology is constrained in other ways.

Theorem 3. Let M be a closed 4-manifold with fundamental group π and let p : M̂ → M be a covering projection such that M̂ is finitely dominated and such that G = π_{1}(M̂) is a nontrivial subnormal subgroup of infinite index in π. Suppose also that G is F P_{3}. Then

(i) if G is finite then the universal covering space M̃ is homotopy equivalent to S^{2} or S^{3} and [π : N_{π}(G)] is finite;

(ii) if G has one end then M is aspherical; (iii) if G has two ends then either M is aspherical or it is finitely covered by

S^{2} × S^{1} × S^{1} or h(\sqrt{π}) = 1 and H^{2}(π; Z[π]) is not finitely generated; (iv) if G has infinitely many ends and is subnormal in N where N is an F P_{2}

normal subgroup of infinite index in π then either M has a finite covering space which is homotopy equivalent to the mapping torus of a self homotopy equivalence of a P D_{3}-complex and [π : N_{π}(G)] is finite or M is aspherical and N is not F P_{3}.

Proof. Let G = N_{0} < N_{1} < ... < N_{r} = π be a subnormal chain. Suppose first that G is finite. Then M̃ is also finitely dominated. Since π has nontrivial torsion M cannot be aspherical, so is homotopy equivalent to S^{2} or S^{3}, by Theorem 3.9 of [9]. If M̃ ≃ S^{2} then the kernel of the natural homomorphism from π to Aut(π_{2}(M)) is torsion free. Hence G = Z/2Z and so G is central in N_{1}. Moreover as it is the torsion subgroup of ζN_{1} it is characteristic in N_{1}, and hence normal in N_{2}. A finite induction now shows that G is normal in π. If M̃ ≃ S^{3} then π has two ends, and so [π : N_{π}(G)] is finite.

If G is infinite then a finite induction using the LHS spectral sequence shows that π has one end, and that if moreover G has one end then H^{2}(π; Z[π]) = 0. Since G is F P_{3} and M̂ is finitely dominated π_{2}(M) = π_{2}(M̂) is finitely generated as a Z[G]-module, and so Hom_{π}(π_{2}(M), Z[π]) = 0. Therefore π_{2}(M) ≅ \overline{H^{2}(π; Z[π])}, by Lemma 3.3 of [9], and so M is aspherical if and only if H^{2}(π; Z[π]) = 0. In particular, M is aspherical if G has one end.

If G has two ends then it has an infinite cyclic normal subgroup of finite index, and so we may assume without loss of generality that G ≅ Z. A finite induction then shows that G ≤ \sqrt{π}. If h(\sqrt{π}) > 2 then H^{2}(π; Z[π]) = 0, by Theorem 1.16 of [9], and so M is aspherical. (In fact M is then homeomorphic to an infrasolvmanifold, by Theorem 8.1 of [9]). If h(\sqrt{π}) = 2 and \sqrt{π} has infinite index in π then we again have H^{2}(π; Z[π]) = 0 and so M is aspherical. (If \sqrt{π} is finitely generated it is nilpotent, hence F P, and the vanishing of H^{2}(π; Z[π]) follows immediately from an LHS spectral sequence argument. If \sqrt{π} is not finitely generated then it is the increasing union of finitely generated subgroups of Hirsch rank 2, and we may apply Theorem 3.3 of [7] to conclude that H^{s}(\sqrt{π}; Z[π]) = 0 for s ≤ 2). If h(\sqrt{π}) = 2 and \sqrt{π} has finite index in π then π is virtually Z^{2}. We may then assume that π ≅ Z^{2} and π/G ≅ Z. Since H_{∗}(M̂; Q) is finitely generated it follows from the Wang sequence for the projection of M̂ onto M that χ(M) = 0. Hence M is finitely covered by S^{2} × S^{1} × S^{1}, by Theorem 10.10 of [9].

Suppose that h(\sqrt{π}) = 1 and let \sqrt{M} be the associated covering space. Since h(G) = h(\sqrt{π}) the stages of a subnormal chain between G and \sqrt{π} are locally finite, and so the rational homology spectral sequences between the corresponding covering spaces collapse, to show that H_{∗}(\sqrt{M}; Q) is finitely generated and χ(\sqrt{M}) = χ(M̂). In particular, π/\sqrt{π} has finitely many ends, since H_{3}(\sqrt{M}; Q) is finite dimensional.

If [π : \sqrt{π}] is finite then \sqrt{π} is finitely generated. But then [\sqrt{π} : G] < ∞ and so [π : G] < ∞, contrary to hypothesis.

If π/\sqrt{π} has two ends then we may assume that π/\sqrt{π} ≅ Z. But then π is an ascending HNN construction over a finitely generated base, and so the torsion subgroup T of \sqrt{π} is finite, while \sqrt{π}/T is abelian. Therefore \sqrt{π} has a finitely generated infinite normal subgroup and so H^{2}(π; Z[π]) is free abelian [11]. Since H_{∗}(\sqrt{M}; Q) is finitely generated \sqrt{M} satisfies Poincaré duality with simple coefficients Q and formal dimension 3 [12] and so χ(\sqrt{M}) = 0. Hence χ(M̂) = 0. This in turn implies that π_{2}(M̂) is a torsion Z[G]- module. Since Z[G] ≅ Z[t, t^{−1}] and π_{2}(M̂) = π_{2}(M) ≅ H^{2}(π; Z[π]) is free abelian it must be finitely generated. Since π has elements of infinite order H^{2}(π; Z[π]) must therefore be 0 or Z, by Corollary 5.2 of [5]. But M cannot be aspherical as c.d._{Q}(π) ≤ c.d._{Q}\sqrt{π} + c.d._{Q}Z = 2. Therefore M̃ ≃ S^{2}. As π is elementary amenable it must be virtually Z^{2}, by Theorem 10.10 of [9]. But this contradicts the assumption that h(\sqrt{π}) = 1. Therefore π/\sqrt{π} has one end. As we may again exclude the possibility that H^{2}(π; Z[π]) ≅ Z, either M is aspherical or H^{2}(π; Z[π]) is not finitely generated.

Suppose that G has infinitely many ends and is subnormal in N where N is an F P_{2} normal subgroup of infinite index in π. If [N : G] is finite then N has infinitely many ends and the regular covering space associated to N is finitely dominated, so π/N has two ends and the covering space associated to N is a P D_{3}-complex, by Theorem 3.9 of [9]. If [N : G] = ∞ then H^{s}(π; Z[π]) = 0 for s ≤ 2 and so M is aspherical, as before. This cannot happen if N is F P_{3}, by the corollary to Theorem 2. □

What happens if we drop the subnormality hypothesis? It can be shown that a closed 3-manifold has a finitely dominated infinite covering space if and only if its fundamental group has one or two ends. Does this condition remain necessary in dimension 4? The hypothesis that G be F P_{3} is automatic if π is finite or has two ends. It is used only to ensure that Hom_{π}(π_{2}(M), Z[π]) = 0. Can it be relaxed to F P_{2} in general? The final possibility in case (iii) surely never occurs. Mapping tori of self homeomorphisms of 3-manifolds whose fundamental group is a nontrivial free product give examples in which G is F P_{3}, has infinitely many ends and is normal in π.

Corollary. If π is virtually torsion free and G has finitely many ends then either M is aspherical or its universal covering space is homotopy equivalent to S^{2} or S^{3}. Proof. It is sufficient to note that if \sqrt{π} is torsion free and h(\sqrt{π}) = 1 then \sqrt{π} is abelian and has a finitely generated infinite normal subgroup. Hence H^{2}(π; Z[π]) = 0, by [7] and [11], and so M is aspherical. □

Conversely, if M̃ is finitely dominated then π is virtually torsion free, by Theorems 3.9, 10.1 and 11.1 of [9]. This also holds if M is finitely covered by the mapping torus of a self homotopy equivalence of a P D_{3}-complex [4].

If M is a closed 4-manifold with χ(M) = 0 and such that π = π_{1}(M) has a subnormal subgroup G of infinite index which is a P D_{2}-group then M is aspherical and either has a finite regular covering space which is homotopy equivalent to the total space of a torus bundle over an aspherical closed surface or has a finite covering space which is homotopy equivalent to a mapping torus. See Chapters 4 and 5 of [9].

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