On 4-dimensional mapping tori. II Jonathan A. Hillman School of Mathematics and Statistics, The University of Sydney, Sydney, NSW 2006, AUSTRALIA e-mail: jonh@maths.usyd.edu.au

ABSTRACT

We show that a P D_{4}-complex X is homotopy equivalent to the mapping torus of a self homotopy equivalence of a P D_{3}-complex if and only if χ(X) = 0 and π_{1}(X) is an extension of Z by a finitely presentable normal subgroup. AMS Subject Classification (1991): Primary 57P10. Secondary 57N13. Key words and phrases: cup product. 4-manifold. mapping torus. Poincaré duality.

Let M be a closed n-manifold and f : M → S^{1} a map which induces an epimorphism f_{∗} on fundamental groups. Stallings showed that if M is an irreducible aspherical 3-manifold then f is homotopic to a bundle projection if and only if f_{∗} has finitely generated kernel, while Farrell showed that if n ≥ 6 this is so if and only if the homotopy fibre of f is finitely dominated and a torsion invariant τ(f) ∈ W h(π_{1}(M)) is 0 [St62, Fa71]. The corresponding results in dimensions 4 and 5 are constrained by the present limitations of geometric topology in these dimensions. (In fact there are counter-examples to the most natural 4-dimensional analogue of Farrell’s theorem [We87]).

There are however homotopy theoretic analogues which apply in all dimensions. Milnor showed that if the infinite cyclic covering space associated to the kernel of f_{∗} (i.e., the homotopy fibre of f) has finitely generated homology then it satisfies Poincaré duality with simple coefficients of formal dimension n − 1 [Mi68]. Barge extended this to covering spaces associated with other “weak duality groups” [Ba80,80’]. Quinn and Gottlieb obtained a much stronger result, requiring more stringent hypotheses. Namely, the homotopy fibre of a map between Poincaré duality complexes is again a Poincaré duality complex if and only if it is finitely dominated [Go79].

We are interested in providing a 4-dimensional homotopy fibration theorem with hypotheses similar to those of Stallings and a conclusion similar to that of Quinn and Gottlieb. In [Hi98] we gave criteria for a P D_{4}-complex to be homotopy equivalent to the mapping torus of a self homotopy equivalence of a P D_{3}-complex which involved the Euler characteristic, the fundamental group and an injectivity condition on a certain homomorphism. Here we shall show that the latter condition always holds, and so the obvious simple necessary conditions on χ and π_{1} are also sufficient. In §3 we decribe explicitly some of the isomorphisms arising in the special case when ν = 1, and in §4 we consider coverings associated to other P D-groups.

As in our earlier papers on this topic, our argument is prompted by the work of Barge.

§1. Injectivity

We shall begin by summarizing the definitions and notation from [Hi89, Hi98].

Let π be an extension of Z by a normal subgroup ν and fix an element t of π whose image generates π/ν. Let α : ν → ν be the automorphism determined by α(h) = tht^{−1} for all h in ν. This automorphism extends to a ring automorphism (also denoted by α) of the group ring R = ℤ[ν], and the ring S = ℤ[π] is a twisted Laurent extension, S = R_{α}[t, t^{−1}]. Let Λ = ℤ[t, t^{−1}].

If M is a left S-module let M|_{R} denote the underlying R-module, and let M̂ = Hom_{R}(M|_{R}, R). Then M̂ is a right R-module via (fξ)(m) = f(m)ξ for all ξ in R, f in M̂ and m in M. If M = S then Ŝ is also a left S-module via (ϕt^{r}f)(ξt^{s}) = ξα^{−s}(ϕ)f(t^{s−r}) for all f in Ŝ, ϕ, ξ in R and r, s in Z. As the left and right actions commute Ŝ is a (S, R)-bimodule. The homomorphism j : Ŝ → R[[t, t^{−1}]] = {Σ_{n∈Z}t^{n}ϕ_{n} | ϕ_{n} ∈ R} given by j(f) = Σt^{n}f(t^{n}) for all f in Ŝ is a (S, R)-bimodule isomorphism.

For each f in M̂ we may define a function T_{M}f : M → Ŝ by the rule (T_{M}f)(m)(t^{n}) = f(t^{−n}m) for all m in M and n in Z. Then T_{M}f is S-linear, and T_{M} : M̂ → Hom_{S}(M, Ŝ) is an isomorphism of right R-modules. Thus we have a natural equivalence T : Hom_{R}(−|_{R}, R) ⇒ Hom_{S}(−, Ŝ) of functors from Mod_{S} to Mod_{R}. If C_{∗} is a chain complex of left S-modules then T induces natural isomorphisms from H^{∗}(C_{∗}|_{R}; R) = H^{∗}(Hom_{R}(C_{∗}|_{R}, R)) to H^{∗}(C_{∗}; Ŝ) = H^{∗}(Hom_{S}(C_{∗}, Ŝ)). In particular, since the forgetful functor −|_{R} is exact and takes projectives to projectives there are isomorphisms from Ext_{R}^{∗}(M|_{R}, R) to Ext_{S}^{∗}(M, Ŝ) which are functorial in M.

If M and N are left S-modules let M ⊗ N denote the tensor product over ℤ with the diagonal left π-action, defined by g(m ⊗ n) = gm ⊗ gn for all m ∈ M, n ∈ N and g ∈ π. The function p_{M} : Λ ⊗ M → M defined by p_{M}(λ ⊗ m) = λ(1)m is then an S-linear epimorphism.

The additive function e_{S} : Λ ⊗ Ŝ → S defined by e_{S}(t^{n} ⊗ f) = t^{n}f(t^{n}) for all f in Ŝ and n in Z is S-linear. We may use e_{S} to define products in cohomology. Let A_{∗} be a Λ-chain complex and B_{∗} a S-chain complex and give the tensor product the total grading A_{∗} ⊗B_{∗} and differential and the diagonal π-action. Let e_{S♯} be the change of coefficients homomorphism induced by e_{S}, and let u ∈ H^{p}(A_{∗}; Λ) and v ∈ H^{q}(B_{∗}; Ŝ). Then u ⊗ v ↦ e_{S♯}(u × v) defines a pairing from H^{p}(A_{∗}; Λ) ⊗ H^{q}(B_{∗}; Ŝ) to H^{p+q}(A_{∗} ⊗ B_{∗}; S).

Now let A_{∗} be the Λ-chain complex concentrated in degrees 0 and 1 with A_{0} and A_{1} free of rank 1, with bases {a_{0}} and {a_{1}}, respectively, and with ∂_{1} : A_{1} → A_{0} given by ∂_{1}(a_{1}) = (t − 1)a_{0}. Let η_{A} : A_{1} → Λ be the isomorphism determined by η_{A}(a_{1}) = 1 and let α_{A} : A_{0} → ℤ be the augmentation determined by α_{A}(a_{0}) = 1. Then [η_{A}] generates H^{1}(A_{∗}; Λ). Let B_{∗} be a projective S-chain complex and let p_{B∗} : A_{∗} ⊗ B_{∗} → B_{∗} be the chain homotopy equivalence defined by p_{Bj}((λa_{0}) ⊗ b_{j}) = λ(1)b_{j} and p_{Bj}((λa_{1}) ⊗ b_{j−1}) = 0, for all λ ∈ Λ, b_{j−1} ∈ B_{j−1} and b_{j} ∈ B_{j}. Let j_{B∗} : B_{∗} → A_{∗} ⊗ B_{∗} be a chain homotopy inverse to p_{B∗}. Let h_{S}([ϕ]) = j_{B}^{∗}e_{S♯}([η_{A}] × [ϕ]) for ϕ : B_{q} → Ŝ such that ϕ∂_{q+1} = 0. If f : C_{∗} → B_{∗} is a chain homomorphism of projective Schain complexes then h_{S}([ϕf_{q}]) = f^{∗}h_{S}([ϕ]). Thus we obtain homomorphisms h_{S} : H^{q}(B_{∗}; Ŝ) → H^{q+1}(B_{∗}; S) which are functorial in B_{∗}. In particular, if B_{∗} is a projective resolution of the S-module M we obtain homomorphisms h_{S} : Ext_{S}^{q}(M, Ŝ) → Ext_{S}^{q+1}(M, S) which are functorial in M. Since cross product with a 1-cycle induces a degree 1 cochain homomorphism, these homomorphisms induce homomorphisms between universal coefficient spectral sequences.

Lemma 1. Let M be an S-module such that M|_{R} is finitely generated as an R-module. Then h_{S} : Hom_{S}(M, Ŝ) → Ext_{S}^{1}(M, S) is injective. Proof. Let B_{∗} be a projective resolution of the S-module M and let q : B_{0} → M be the defining epimorphism (so that q∂_{1} = 0). We may use composition with q to identify Hom_{S}(M, Ŝ) with the submodule of 0-cocycles in Hom(B_{∗}, Ŝ), and we set h_{S}(ϕ) = h_{S}([ϕq]) for all ϕ : M → Ŝ.

Let ϕ : M → Ŝ be a homomorphism and let g = ϕq : B_{0} → Ŝ. We may write g(b) = Σt^{n}g_{n}(b) = Σt^{n}g_{0}(t^{−n}b), where g_{0} : B_{0} → R is R-linear (and g_{0}∂_{1} = 0). If h_{S}(ϕ) = 0 then e_{♯}([η_{A}] × [g]) = δf for some S-linear homomorphism f : A_{0} ⊗ B_{0} → S. Therefore g_{0}(b) = f((t − 1)a_{0} ⊗ b) for all b ∈ B_{0}, while f(1 ⊗ ∂_{1}) = 0. Let k(b) = f(a_{0} ⊗ b) for b ∈ B_{0}. Then k : B_{0} → S is R-linear, and k∂_{1} = 0, so k factors through M. In particular, k(B_{0}) is finitely generated as an R-submodule of S. But as S = ⨁ t^{n}R and g_{0}(b) = tk(t^{−1}b) − k(b) for all b ∈ B_{0}, this is only possible if k = g_{0} = 0. This proves the lemma. □

§2. The main result

Suppose now that B_{∗} is a projective S-chain complex such that B_{j} = 0 for j < 0, H_{0}(B_{∗}) ≅ ℤ and H_{1}(B_{∗}) = 0. Then the quotient epimorphism from B_{0} to H_{0}(B_{∗}) factors through Λ and we may choose a chain homomorphism ϵ_{B∗} : B_{∗} → A_{∗} which induces an isomorphism H_{0}(B_{∗}) ≅ H_{0}(A_{∗}). Let α_{B} = α_{A}ϵ_{B_{0}} : B_{0} → ℤ and η_{B} = η_{A}ϵ_{B_{1}} : B_{1} → Λ. Then α_{B} is a generator of H^{0}(B_{∗}; ℤ) and [η_{B}] = ϵ_{B}^{∗}([η_{A}]) is a generator of H^{1}(B_{∗}; Λ).

Lemma 2. Let X be a connected space with π_{1}(X) ≅ π and let B_{∗} = C_{∗}(X̃) be the singular chain complex of the universal cover of X. Then h_{S}([ϕ]) = [η_{B}] ∪ [ϕ]. Proof. The Alexander Whitney diagonal approximation d_{∗} : B_{∗} → B_{∗} ⊗ B_{∗} is π-equivariant, if the tensor product is given the diagonal left π-action, and we may take j_{B∗} = (ϵ_{B} ⊗ 1)d_{∗} as a chain homotopy inverse to p_{B∗}. Therefore h_{S}([ϕ]) = d^{∗}e_{S♯}([η_{B}] × [ϕ]) = [η_{B}] ∪ [ϕ]. □

Thus we may identify the homomorphisms h_{S} with the cup-products involved in our description of duality.

We are now able to prove the theorem.

Theorem 3. A finite P D_{4}-complex M with fundamental group π is homotopy equivalent to the mapping torus of a self homotopy equivalence of a P D_{3}-complex if and only if χ(M) = 0 and π is an extension of Z by a finitely presentable normal subgroup ν. Proof. By Theorem 4 of [Hi98] it is sufficient to observe that if Π = H^{2}(π; ℤ[π]) then Π|_{ν} ≅ H^{1}(ν; ℤ[ν]) is finitely generated as a ℤ[ν]-module, and so the map from Hom_{ℤ[ν]}(Π|_{ν}, ℤ[ν]) to Ext_{ℤ[π]}^{1}(Π, ℤ[π]) induced by h_{S} is a monomorphism, by Lemma 1. □

A rather different proof of this theorem could be given using Ranicki’s criterion for an infinite cyclic cover to be finitely dominated [Ra95] and the Quinn-Gottlieb theorem, if finitely generated stably free modules of rank 0 over the Novikov rings A_{±} = ℤ[ν]_{α}((t^{±1})) are trivial. (For H_{∗}(A_{±} ⊗_{π} C_{∗}) is a stably free module of rank 0, concentrated in degree 2, by the argument of Lemma 2.3 of [Wl]. See also Chapter II of [Hi]).

The hypothesis that ν be almost finitely presentable (F P_{2}) suffices to show that M_{ν} satisfies Poincaré duality with local coefficients. Finite presentability is used only to show that M_{ν} is finitely dominated. In view of the fact that 3-manifold groups are coherent, we might hope that the condition on ν could be weakened still further to require only that it be finitely generated.

Current technology does not enable us to decide whether M_{ν} is homotopy equivalent to a closed 3-manifold. However we may use Farrell’s theorem to obtain a partial reduction to the case of a cartesian product. The projection of π onto ℤ may be realized by a map f : M → S^{1} with homotopy fibre M_{ν}. The composite of f with the projection of M × S^{1} × S^{1} onto M defines a map g : M × S^{1} × S^{1} → S^{1} with homotopy fibre M_{ν} × S^{1} × S^{1} which is again finitely dominated (in fact, homotopy finite). The homotopy equivalence associated to g by Farrell is obtained from the one associated to f by product with id_{S^{1}×S^{1}} and so has torsion τ(g) = τ(f)χ(S^{1} × S^{1}) = 0 in W h(π × Z^{2}). Therefore if M is a closed 4-manifold g is homotopic to a bundle projection and so M_{ν} × S^{1} × S^{1} is homotopy equivalent to a closed 5-manifold.

There are however 4-manifolds which are simple homotopy equivalent to S^{1} × RP^{3} (and thus satisfy the hypotheses of our theorem) but which are not homeomorphic to mapping tori [We87].

§3. The case ν = 1

When ν = 1 (so R = ℤ and S = Λ) we can relate h_{Λ} to other more explicit homomorphisms. Let M be a Λ-module which is finitely generated as an abelian group, and let N be its maximal finite submodule. Let λ_{M} be the minimal polynomial of t, considered as an automorphism of (M/N)|_{ℤ}. Then Ann_{Λ}(M/N) = (λ_{M}) [Hi81]. Let j_{λ} : Λ/(λ) → ℚ(t)/Λ be the homomorphism sending the class of g to \frac{g}{λ}. Then j_{λ_{M}} induces an isomorphism Hom_{Λ}(M, Λ/(λ_{M})) ≅ D(M) = Hom_{Λ}(M, ℚ(t)/Λ). We shall show that D(M) is isomorphic to each of D̂(M) = Hom_{Λ}(M, Λ̂), E(M) = Ext_{Λ}^{1}(M, Λ) and F (M) = Hom_{ℤ}(M|_{ℤ}, ℤ).

Since M|_{ℤ} is finitely generated the leading coefficient and constant term of λ_{M} are ±1, and we may define a homomorphism ρ_{M} : D(M) → F (M) by ρ_{M}(ϕ)(m) = −Res(ϕ(m)\frac{dt}{t}, 0) − Res(ϕ(m)\frac{dt}{t}, ∞), where Res(ω, a) is the residue of the differential form ω at a. (Note that if u = t^{−1} then \frac{du}{u} = −\frac{dt}{t}, and so Res(f \frac{dt}{t}, ∞) = −Res(f̄\frac{du}{u}, 0), where f̄(u) = f(u^{−1})). Let T_{M} : F (M) → D̂(M) be the isomorphism defined as in [Ba80’] (and as above) by T_{M}f(m) = Σt^{n}f(t^{−n}m). Then T_{M}ρ_{M}(ϕ)(m) = Lexp(ϕ(m), ∞) − Lexp(ϕ(m), 0), where Lexp(f, a) is the Laurent expansion of the rational function f about a, since Lexp(f, 0) = Σt^{n}Res(t^{−n}f \frac{dt}{t}, 0) and Lexp(f, ∞) = −Σt^{n}Res(t^{−n}f \frac{dt}{t}, ∞).

Let B_{∗} be a projective resolution of M. If ϕ : M → ℚ(t)/Λ let ϕ_{0} : B_{0} → ℚ(t) be a lift of ϕ. Then ϕ_{0}∂_{1} has image in Λ, and so defines a homomorphism ϕ_{1} : B_{1} → Λ such that ϕ_{1}∂_{2} = 0. Consideration of the short exact sequence of complexes 0 → Hom_{Λ}(B_{∗}, Λ) → Hom_{Λ}(B_{∗}, ℚ(t)) → Hom_{Λ}(B_{∗}, ℚ(t)/Λ) → 0 shows that δ(ϕ) = [ϕ_{1}], where δ : D(M) → E(M) is the Bockstein homomorphism associated to the coefficient sequence. (The extension corresponding to δϕ under the isomorphism of Theorem III.2.4 of [HS] is the pullback over ϕ of the sequence 0 → Λ → ℚ(t) → ℚ(t)/Λ → 0).

Lemma 4. The natural transformations h_{Λ} and ρ are equivalences, and h_{Λ}T ρ = δ. Proof. As every homomorphism from M to ℚ(t)/Λ factors through Λ/(λ_{M}) it suffices to check that h_{Λ}T_{M}ρ_{M}(j_{λ}) = δ(j_{λ}) for λ = λ_{M}. Moreover we may extend coefficients to ℂ, and so we may reduce to the case λ = (t − α)^{n}.

We may assume that B_{1} and B_{0} are freely generated by b_{1} and b_{0}, respectively, and that ∂(b_{1}) = λb_{0}. The chain homotopy equivalence j_{B∗} may be defined by j_{0}(b_{0}) = a_{0} ⊗ b_{0} and j_{1}(b_{1}) = a_{0} ⊗ b_{1} + Σβ_{pq}(t^{p}a_{1}) ⊗ (t^{q}b_{0}), where Σβ_{pq}x^{p}y^{q} = (λ(xy) − λ(y))/(x − 1) = yΣ_{0≤r<n}(xy − α)^{r}(y − α)^{n−r−1}. (This formula arises naturally if we identify Λ ⊗_{ℤ} Λ with ℤ[x, y, x^{−1}, y^{−1}], with t ∈ Λ acting via xy). Note that δ(j_{λ})(b_{1}) = β_{0n} = 1 and β_{pq} = 0 unless 0 ≤ m < q ≤ n.

Now h_{Λ}T_{M}ρ_{M}(j_{λ})(b_{1}) = e_{♯}(η_{A} × T_{M}ρ_{M}(j_{λ}))(j_{∗}(b_{1})) = Σβ_{pq}t^{p}ψ_{p−q}, where ψ_{−r} is the coefficient of t^{−r} in Lexp(λ^{−1}, ∞). Clearly ψ_{r} = 0 if −n < r < 0 and ψ_{−n} = 1, since λ^{−1} = t^{−n}(1 − αt^{−1})^{−n}. Hence h_{Λ}T_{M}ρ_{M}(j_{λ})(b_{1}) = β_{0n} = δ(j_{λ})(b_{1}), and so h_{Λ}T ρ = δ, by linearity and functoriality.

Since δ and T are natural equivalences and h_{Λ} is injective, by Lemma 1, ρ and h_{Λ} are also natural equivalences. □

Barge used a homomorphism similar to T ρ in [Ba80] to embed ℚ(t)/Λ in ℚ[[t, t^{−1}]]. The definition of ρ was influenced by the “trace” function used by Trotter to relate Seifert forms and Blanchfield pairings on a knot module M [Tr78].

§4. Other covering groups

The argument of Theorem 1 of [Hi98], using Shapiro’s lemma and the finiteness criteria of Brown, readily extends to show that if X is a P D_{n}- complex then the homotopy fibre of a map f : X → S^{1} which induces an epimorphism on fundamental groups is finitely dominated (and hence is a P D_{n−1}-complex) if and only if it is homotopy equivalent to a complex with finite m-skeleton, where m ≥ max{2, [n/2]}. A simple induction extends this further to show that the cover associated to an epimorphism onto a virtually poly-Z group of Hirsch length r is a P D_{n−r}-complex, under the same finiteness hypothesis. However this hypothesis is a priori apparently stronger than that of Theorem 3. Can aspects of these arguments be combined to show that it suffices to assume χ(X) = 0 and, say, m ≥ [(n − 1)/2]? In particular, suppose that M is a closed 4-manifold with χ(M) = 0 and π is an extension of Z^{r} by a finitely generated infinite normal subgroup K, for some r > 1. Is M aspherical?

For more general quotient groups we must at present assume that X is itself aspherical, as in the next result.

Theorem 5. Let π be a P D_{n}-group with an F P_{r} normal subgroup K such that G = π/K is a P D_{n−r} group and 2r ≥ n − 1. Then K is a P D_{r}-group. Proof. It shall suffice to show that H^{s}(K; F) = 0 for any free ℤ[K]-module F and all s > r, for then c.d.K = r and the result follows from Theorem 9.11 of [Bi]. Let W = Hom_{ℤ[K]}(ℤ[π], F) be the ℤ[π]-module coinduced from F. Then H^{s}(K; F) ≅ H^{s}(π; W) ≅ H_{n−s}(π; \overline{W}), by Shapiro’s lemma and Poincaré duality. As a ℤ[K]-module \overline{W} ≅ F^{G} (the direct product of |G| copies of F), and so H_{q}(K; \overline{W}) = 0 for 0 < q ≤ r (since K is F P_{r}), while H_{0}(K; \overline{W}) ≅ A^{G}, where A = H_{0}(K; F). Moreover A^{G} ≅ Hom_{ℤ}(ℤ[G], A) as a ℤ[G]-module, and so is coinduced from a module over the trivial group. Therefere if n − s ≤ r the LHSSS gives H^{s}(K; F) ≅ H_{n−s}(G; A^{G}). Poincaré duality for G and another application of Shapiro’s lemma now give H^{s}(K; F) ≅ H^{s−r}(G; A^{G}) ≅ H^{s−r}(1; A) = 0, if s > r. □

Can a similar argument be applied in the non-aspherical case? The difficulty appears to be in computing H_{∗}(G; A^{G}) where A is a nontrivial G-module and A^{G} has the “diagonal” structure.

Corollary. Let π be a P D_{n}-group with an F P_{r} normal subgroup K such that G = π/K is a P D_{n−r} group and 2r ≥ n−1. Then K is a P D_{r}-group. □

This corollary may be used to reconcile the hypotheses of [Ka98] with those of [Ko99] and [Hi00] in their homotopy characterizations of complex surfaces which admit holomorphic submersions to complex curves. (Kapovich assumes the surface S is aspherical and that π_{1}(S) is an extension of an orientable P D_{2}-group by a finitely presentable normal subgroup, while Kotschick and

Hillman assume that π_{1}(S) is an extension of one orientable P D_{2}-group by

another such and that χ(S) is the product of the Euler characteristics of these

groups).

References

[Ba80] Barge, J. An algebraic proof of a theorem of J.Milnor, in Proceedings of the Topology Symposium, Siegen, 1979, (edited by U. Koschorke and W.D.Neumann), Lecture Notes in Mathematics 788, Springer-Verlag, Berlin - Heidelberg - New York (1980), 396-398.

[Ba80’] Barge, J. Dualité dans les revêtements galoisiens,

Inventiones Math. 58 (1980), 101-106.

[Bi] Bieri, R. Homological Dimension of Discrete Groups,

Queen Mary College Mathematics Notes, London (1976).

[Br75] Brown, K.S. A homological criterion for finiteness, Comment. Math. Helvetici 50 (1975), 129-135.

[Fa71] Farrell, F.T. The obstruction to fibering a manifold over a circle, Indiana Univ. J. Math. 21 (1971), 315-346.

[Hi81] Hillman, J.A. Alexander Ideals of Links, Lecture Notes in Mathematics 895, Springer-Verlag, Berlin - Heidelberg - New York (1981).

[Hi89] Hillman, J.A. A homotopy fibration theorem in dimension 4,

Top. Appl. 33 (1989), 151-161.

[Hi] Hillman, J.A. The Algebraic Characterization of Geometric 4-Manifolds, London Mathematical Society Lecture Note Series 198, Cambridge University Press, Cambridge - New York - Melbourne (1994).

[Hi98] Hillman, J.A. On 4-dimensional mapping tori and product geometries, J. London Math. Soc. 58 (1998), 229-238.

[Hi00] Hillman, J.A. Complex surfaces which are fibre bundles, Top. Appl. 100 (2000), 187-191.

[HS] Hilton, P.J. and Stammbach, U. A Course in Homological Algebra, second edition, Graduate Texts in Mathematics 4, Springer-Verlag, Berlin - Heidelberg - New York (1997).

[Ka98] Kapovich, M. On normal subgroups in the fundamental groups of complex surfaces, preprint, the University of Utah (August, 1998).

[Ko99] Kotschick, D. On regular fibred complex surfaces, in Proceedings of the Kirbyfest (edited by J. Hass and M.Scharlemann), Geometry and Topology Monographs, Volume 2 (1999), 291-298.

[Ra95] Ranicki, A. Finite domination and Novikov rings, Topology 34 (1995), 619-632.

[St62] Stallings, J.R. On fibering certain 3-manifolds, in Topology of 3-Manifolds and Related Topics (edited by M.K.Fort, Jr), Prentice-Hall, Englewood Cliffs, N.J.(1962), 95-100.

[Tr78] Trotter, H.F. Knot modules and Seifert matrices, in Knot Theory, Proceedings, Plans-sur-Bex, Switzerland 1977, (edited by J.-C.Hausmann), Lecture Notes in Mathematics 685, Springer-Verlag, Berlin - Heidelberg - New York (1978),291-299.

[Wl] Wall, C.T.C. Surgery on Compact Manifolds,

Academic Press, New York - London (1970).

[We87] Weinberger, S. On fibering four- and five-manifolds, Israel J. Math. 59 (1987), 1-7.