Simple 4-knots

J. A. HILLMAN School of Mathematics and Statistics, The University of Sydney, Sydney 2006, Australia

C. KEARTON Department of Mathematical Sciences, The University of Durham, Durham, DH1 3LE, United Kingdom

ABSTRACT

We show that the isotopy type of a 1-simple n-knot K is determined by the Postnikov (n − 1)-stage of its exterior X(K), together with the homotopy class of the longitude λ_{K} ∈ π_{n}(X(K)). Moreover any pair (X, j) where X is a 4-dimensional homology circle with π_{1}(X) ≅ Z and j : S^{4} × S^{1} → X is a map such that (X, j) = (MCyl(j), S^{4} × S^{1}) is an orientable P D_{6}-pair is realizable by some simple 4-knot. As a consequence we are able to characterize completely the Artin spins of fibred simple 3-knots.

AMS Subject Classification (1991): Primary 57Q45. Key words: fibred knot, 4-knot, Poincaré duality, Postnikov, simple knot,

Introduction.

In [HK97] we suggested a “homotopy triple” for simple 4-knots which was in some sense an unstable analogue of the Farber quintuple, which is a complete invariant of simple 2q-knots with q ≥ 4 [Fa83]. Here we shall show that a refinement of the homotopy triple provides a complete invariant for simple 4knots. In the fibred case the homotopy triple determines the knot up to a finite ambiguity (proven under rather special additional hypotheses in [HK97]). If moreover π_{2} is torsion free the homotopy invariant is already a complete invariant, and the longitude lies in a canonical quotient of π_{2} ⊗ π_{3}. Our realization result is not yet fully satisfactory, as we have not been able to

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translate the Poincaré duality condition into purely algebraic terms. However we are able to characterize completely the simple 4-knots which are Artin spins of fibred simple 3-knots.

Our approach is motivated by the fact that the exterior X(K) of a 1-simple knot K is determined up to homeomorphism by the homotopy type of the pair (X(K), ∂X(K)) [LS69]. It is an open question whether every such knot is reflexive, that is, is determined by its exterior. This is true for all simple n-knots with n ≠ 4 or 6, and in general at most two knots share a common exterior. Our argument does not address this issue, although it follows easily from the classification of simply connected 5-manifolds that simple 4-knots such that π_{2} is torsion free are reflexive.

The arguments of §2 apply to all 1-simple knots, in any dimension. However the Postnikov data seems only accessible in the low dimensional cases. En route to our main result on simple 4-knots we shall sketch an approach to the classification of simple 3-knots which uses only homotopy theory and the (π, π)-theorem of Wall.

1. Generalities on knots.

An n-knot is a locally flat embedding K : S^{n} → S^{n+2}. If we assume that the spheres have been oriented once and for all, the knot is determined up to ambient isotopy by its image as an oriented submanifold. Composing K with orientation reversing homeomorphisms of range or domain gives the reflected and inverted knots rK, Kρ and −K = rKρ. (See Chapter 1 of [H]). The exterior of K is X(K) = S^{n+2}\intN, where N ≅ S^{n} × D^{2} is a product neighbourhood of the image of K, and the group of K is πK = π_{1}(X(K)).

The orientations of the spheres determine a conjugacy class of meridians, represented by the boundary of an oriented transverse disc, and hence an isomorphism πK/πK′ = H_{1}(X(K); ℤ) = Z. Let t = t_{K} be such a meridian. The n-knot K is simple if πK = Z and π_{i} = 0 for 1 < i ≤ (n − 1)/2. (This concept is only of interest if n ≥ 3, for if n = 1 or 2 any such knot is trivial).

Every n-knot bounds a parallelizable (n + 1)-dimensional submanifold of S^{n+1}; such a submanifold V is called a Seifert hypersurface for K = ∂V. If K is simple then we may assume that V is [(n − 1)/2]-connected and that the natural homomorphism from π_{[(n+1)/2]}(V) to π_{[(n+1)/2]}(X(K)) is injective [Le65]. The knot K is fibred if X(K) fibres over S^{1}, in which case the fibre is a Seifert hypersurface for K. The exterior is then the mapping torus of the monodromy of the fibre F, and its infinite cyclic covering space X(K)′ is homeomorphic to F × R.

The sum of two n-knots K_{1} and K_{2} may be defined (up to isotopy) as the n-knot K_{1}♯K_{2} obtained as follows. Let D^{n}(±) denote the upper and lower hemispheres of S^{n}. We may isotope K_{1} and K_{2} so that each K_{i}(D^{n}(±)) is contained in D^{n+2}(±), K_{1}(D^{n}(+)) is a trivial n-disc in D^{n+2}(+), K_{2}(D^{n+2}(−)) is a trivial n-disc in D^{n+2}(−) and K_{1}|_{S^{n+1}} = K_{2}|_{S^{n+1}}. Then K_{1}♯K_{2} = K_{1}|_{D^{n}(−)} ∪ K_{2}|_{D^{n}(+)}. It is not hard to see that X(K_{1}♯K_{2})′ ∼ X(K_{1})′ ∨ X(K_{2})′; in particular, the sum of two simple knots is again simple.

We shall write H_{q}(X(K); Λ) for H_{q}(X(K)′; ℤ), considered as modules over Λ = ℤ[t, t^{−1}] = ℤ[πK/πK′] with respect to the natural action of the covering group on X(K)′. These are finitely generated as Λ-modules, and t − 1 acts invertibly; if πK = Z then K is fibred if and only if they are finitely generated as abelian groups [BL66]. If H_{q}(X(K); Λ) is ℤ-torsion free or finite (for all q > 0) the knot is torsion free or finite, respectively. If N is a Λ-module let T (N) be its ℤ-torsion submodule. The conjugate Λ-module \overline{N} has the same underlying abelian group and the Λ-structure obtained by inverting the action of t. Let e^{i}(N) = \overline{Ext_{Λ}^{i}(N, Λ)}. Then e^{1}(N) ≅ \overline{Hom_{Λ}(N, ℚ(t)/Λ)}.

2. Postnikov approximations.

Let K be a 1-simple n-knot and let π_{i} = π_{i}(X(K)), considered as modules over Λ, for i ≥ 2. We shall write π̃_{i} when such groups appear as local coefficient systems. If X is any cell complex we may construct the Postnikov q-stage P_{q}(X) by adding cells of dimension ≥ q + 2 to X, so that π_{j}(P_{q}(X)) = 0 for j > q. The inclusion of X into P_{q}(X) is then (q + 1)- connected. Let HAut_{+}(X) denote the group of self homotopy equivalences of X which induce the identity on π_{1}. If X is n-dimensional HAut_{+}(X) is isomorphic to HAut_{+}(P_{n}(X)), and maps onto HAut_{+}(P_{n−1}(X)).

Lemma 1. Let P be a cell complex with finite n-skeleton such that π_{1}(P) ≅ Z and π_{q}(P) = 0 for all q ≥ n. Suppose also that p.d._{Λ}H_{n−1}(P; Λ) ≤ 1 and H_{n}(P; Λ) = 0. Then there is a finite n-dimensional complex X with an nconnected map to P and such that H_{n}(X; Λ) = 0. Proof. Let U = P^{[n]}. Then the inclusion of U into P is n-connected, and H_{n}(U; Λ) is projective, by Schanuel’s lemma, and hence is free. The relative Hurewicz homomorphism from π_{n+1}(P, U) to H_{n+1}(P, U; Λ) is an isomorphism and so the Hurewicz homomorphism from π_{n}(U) to H_{n}(U; Λ) is onto. We may add (n + 1)-cells to U along representatives for a basis for H_{n}(U; Λ) to obtain a finite complex V with H_{n}(V; Λ) = H_{n+1}(V; Λ) = 0. Since H^{q}(V; Λ) = 0 for all q > n this complex is homotopy equivalent to a finite n-dimensional complex X [Wa65]. □

Lemma 2. Let P be as above. If f : X → P and f_{1} : X_{1} → P are n-connected maps where X and X_{1} are n-dimensional complexes such that H_{n}(X; Λ) = H_{n}(X_{1}; Λ) = 0 then there is a homotopy equivalence h : X → X_{1} such that f ∼ f_{1}h. Proof. Up to homotopy we may assume that f_{1} is an inclusion of a subcomplex. There are no obstructions to homotoping f to a map h : X → X_{1} (so that f ∼ f_{1}h). Since π_{q}(h) is an isomorphism for all q < n and H_{n}(X; Λ) = H_{n}(X_{1}; Λ) = 0 the lift of h to the universal covers is a homotopy equivalence, by Whitehead’s theorem, and so h itself is a homotopy equivalence. □

If K : S^{n} → S^{n+2} is an n-knot with n > 1 then its normal bundle has an essentially unique framing. Let j_{K} be the homotopy class in [S^{n} × S^{1}, X] of the homeomorphism onto ∂X determined by such a framing. The longitude λ_{K} of K is the image of j_{K} in π_{n} under restriction to S^{n} × {∗}.

Lemma 3. Let X be an n-dimensional homology circle with π = π_{1}(X) ≅ Z and such that H_{n}(X; Λ) = 0. If two maps j_{1}, j_{2} : S^{n} × S^{1} → X induce isomorphisms on π and agree on Y = S^{n} ∨ S^{1} then there is a self homotopy equivalence f : X → X such that fj_{1} ∼ j_{2}. Proof. Let [S^{n} × S^{1}, X]_{S^{1}} be the set of homotopy classes of maps from S^{n} ×S^{1} to X which induce the isomorphism on π which sends the inclusion of S^{1} as {∗} × S^{1} to t ∈ π. As S^{n} × S^{1} is the cofibre of the Whitehead product w = [ι_{n}, ι_{1}] : S^{n} → Y = S^{n}∨S^{1} and Σw is trivial (see page 485 of [Wh]), there is an exact sequence of pointed sets Z ⊗_{Λ} π_{n+1} → [S^{n} × S^{1}, X]_{S^{1}} → π_{n}^{π} → 0, where the right hand map is induced by restriction to S^{n} × {1}, and maps onto the subgroup of π_{n} fixed by the action of π = Z.

The inclusion of X into (X, Y) induces isomorphisms of H_{q}(X; Λ) with H_{q}(X, Y; Λ) for q ≤ n, while H_{n+1}(X, Y; Λ) ≅ H_{n}(Y; Λ) ≅ Λ. Hence H^{n+1}(X, Y; π̃_{n+1}) ≅ π_{n+1}. For any α ∈ π_{n+1} there is a self map f_{α} : X → X which is the identity on the (relative) n-skeleton of (X, Y) and such that d(f_{α}, id_{X}) = α ∈ H^{n+1}(X, Y; π̃_{n+1}). Clearly f is a self homotopy equivalence of X. Moreover d(f_{α}j, j) is the image of α in H^{n+1}(S^{n} × S^{1}, Y; π̃_{n+1}) ≅ Z ⊗_{Λ} π_{n+1}, and so composition with such self homotopy equivalences induces the action of Z ⊗_{Λ} π_{n+1} in the above sequence. □

If j : S^{n} × S^{1} → X is a map let (X, j) denote the pair (MCyl(j), S^{n} × S^{1}), where MCyl(j) is the mapping cylinder of j. A self homotopy equivalence h : X → X induces a homotopy equivalence of pairs (X, j) ≃ (X, hj).

Lemma 4. Let X be an homology circle with π_{1}(X) ≅ Z and j : S^{n}×S^{1} → X be a map such that π_{1}(j) is an isomorphism and (X, j) is an orientable P D_{n+2}- pair whose Spivak normal bundle admits a TOP reduction. Then there is a 1-simple n-knot K such that (X(K), j_{K}) ≃ (X, j). Proof. We may assume that n ≥ 3, since otherwise X ≃ S^{1} and we may take K to be a trivial knot. Since the Spivak normal bundle has a TOP reduction there is a degree 1 normal map F : (W, ∂W) → (X, j), where W is a compact (n + 2)-manifold. Since π_{1}(j) is an isomorphism we may surger the map F to obtain a homotopy equivalence of pairs F_{1} : (W_{1}, ∂W_{1}) → (X, j), by Theorem 3.3 of [W]. The map j^{−1}F_{1}|_{∂} is homotopic to a homeomorphism, h say. Hence Σ = W_{1} ∪_{h} (S^{n} × D^{2}) is then a simply connected homology sphere, and so is homeomorphic to S^{n+2}. The submanifold S^{n} × {0} is an n-knot with exterior W_{1}. □

Moreover the knot K is unique up to isotopy.

Theorem 5. Two 1-simple n-knots K_{1} and K_{2} are isotopic if and only if there is a homotopy equivalence f : X(K_{1}) ≃ X(K_{2}) such that f_{∗}t_{K_{1}} = t_{K_{2}} and fλ_{K_{1}} ∼ λ_{K_{2}}. Moreover the following are equivalent:

(i) X(K_{1}) is homotopy equivalent to X(K_{2}); (ii) there is an (n − 1)-connected map from X(K_{1}) to X(K_{2}); (iii) P_{n−1}(X(K_{1})) ≃ P_{n−1}(X(K_{2})).

Proof. The first assertion follows from Lemma 3 and [LS69]. (Note that j_{K} determines the attachment of S^{n} × D^{2}).

If f : X(K_{1}) → X(K_{2}) is a homotopy equivalence it is (n − 1)-connected, and determines a homotopy equivalence of the Postnikov (n − 1)-stages. Conversely, if f is an (n − 1)-connected map the modules H_{n−1}(X(K_{1}); Λ) and H_{n−1}(X(K_{2}); Λ) are isomorphic (as they are each isomorphic to e^{1}(π_{2})). As these modules are noetherian and H_{n−1}(f; Λ) is an epimorphism it is an isomorphism. Hence f is a homotopy equivalence, by Whitehead’s theorem. An isomorphism of the Postnikov data determines a homotopy equivalence g : P_{n−1}(X(K_{1})) ≃ P_{n−1}(X(K_{2})), and so X(K_{1}) and X(K_{2}) are homotopy equivalent, by Lemma 2. □

In low dimensions the homotopy type of the Postnikov (n − 1)-stage and hence of X may be described in algebraic terms. We shall consider below the cases n = 3 and n = 4. We recall here the “certain exact sequence” of J.H.C.Whitehead for a simply-connected complex Y:

⋯ → π_{4}(Y) → H_{4}(Y; ℤ) → Γ(π_{2}(Y)) → π_{3}(Y) → H_{3}(Y; ℤ) → 0.

Here the extreme maps are Hurewicz homomorphisms and Γ is the universal quadratic functor of Whitehead. In particular, H_{3}(K(A, 2); ℤ) = 0 and H_{4}(K(A, 2); ℤ) ≅ Γ(A). (See Chapter 1 of [Ba]).

3. Simple 3-knots.

It is well known that simple odd-dimensional knots may be classified by means of their Blanchfield pairings [Ke75, Hi84]. We shall sketch here a homotopy-theoretic approach to this classification when n = 3, the lowest dimension in which there are nontrivial simple knots. In this case X = X(K) is homotopy equivalent to a 3-dimensional complex and X′ ≃ M(π_{2}, 2), the Moore space with nontrivial homology in degree 2 only. In particular, π_{3} ≅ Γ(π_{2}). Therefore the homotopy type of X is determined by the Λ-module π_{2}(X), and HAut_{+}(X) ≅ HAut_{+}(P_{3}(X)) maps onto Aut_{Λ}(π_{2}). Poincaré duality gives rise to a “skew-hermitean” Blanchfield pairing, which is an isomorphism β_{λ} : e^{1}(π_{2}) → π_{2} such that e^{1}(β_{λ}) = −β_{λ}. Hence p.d._{Λ}π_{2} ≤ 1.

Let M = X ∪ D^{4} × S^{1} be the closed 5-manifold obtained by surgery on the knot. Then M′ ≃ X′ ∪_{λ} D^{4} has a fundamental class [M′] ∈ H_{4}(M′; ℤ), and is an orientable P D_{4}-complex over ℚ. Cup product on H^{2}(M′; ℚ) defines a pairing B(x, y) = (x∪y)∩[M′], for which t acts isometrically, and the pairing and isometry together are essentially equivalent to the rational Blanchfield pairing.

The relative Hurewicz homomorphism from π_{4}(M′, X′) to H_{4}(M′, X′; ℤ) is an isomorphism. The composition of the homomorphism induced by the inclusion of M′ into (M′, X′) with the inverse of this Hurewicz isomorphism and the connecting map in the homotopy exact sequence of this pair gives a homomorphism from H_{4}(M′; ℚ) to π_{3} ⊗ ℚ which carries [M′] to the image of λ. Considering instead the pair (K(π_{2}, 2), X′) we see that the corresponding map from H_{4}(K(π_{2}, 2); ℚ) to π_{3} ⊗ ℚ is an isomorphism, since X′ ≃ M(π_{2}, 2) is a Moore space. Let c : M′ → K(π_{2}, 2) be the classifying map. As H_{4}(K(A, 2); ℚ) ≅ Γ(A) ⊗ ℚ is isomorphic to the space of symmetric bilinear forms : A^{∗}×A^{∗} → ℚ the Blanchfield pairing determines c_{∗}[M′]. Since π_{3} is ℤ-torsion free this in turn determines λ.

As HAut_{+}(X) maps onto Aut_{Λ}(π_{2}) an isometry of pairings on π_{2} corresponding to longitudes λ_{i} is induced by a self homotopy equivalence f : X → X such that β_{f_{∗}λ_{1}} = f_{∗}β_{λ_{1}}f^{∗} = β_{λ_{2}}. Thus the orbit of the longitude under HAut_{+}(X) is determined by the isometry class of the Blanchfield pairing, and so the isotopy type of the knot is determined by the isomorphism class of the pair (π_{2}, β_{λ}).

Conversely, every pair (A, b) where A is a finitely generated Λ-module such that p.d._{Λ}A ≤ 1 and (t − 1)A = A and b : e^{1}(A) → A is an isomorphism such that e^{1}(b) = −b arises from some TOP locally flat 3-knot [Hi84]. (In the fibred case this can be seen as follows. There is a Postnikov 2-stage P with π_{1}(P) ≅ Z and π_{2}(P) ≅ A. Since p.d._{Λ}A ≤ 1 and H_{3}(P; Λ) = H_{3}(K(A, 2); ℤ) = 0 there is a finite 3-dimensional homology circle X with π_{1}(X) ≅ Z and π_{2}(X) ≅ A, by Lemma 1. The isomorphism b determines a longitude λ in π_{3}(X) such that (t − 1)λ = 0, by the integral version of the above argument. Hence it determines an orientable P D_{5}-pair (X, j). The only obstruction to a TOP reduction of the Spivak normal bundle ν is in H^{i}(X, j; π_{5}(BG/T OP)) ≅ Z, and its image in H^{5}(X, j; ℚ) is (θν)^{∗}k_{5}, where θ : BSG → BG/T OP is the canonical map and k_{5} generates H^{5}(BG/T OP; ℚ). Since H^{5}(BSG; ℚ) = 0 the obstruction is 0, and so the pair (A, b) is realized by some simple 3-knot. Can this argument be made to work in general?)

4. Simple 4-knots.

When n = 4 the homotopy type of X = X(K) is determined by the equivalence class of its “Postnikov triple” (π_{2}, π_{3}, κ), where κ is the k-invariant in H^{4}(P_{2}(X); π̃_{3})), and the Postnikov 2-stage P_{2}(X) is determined by the Λmodule π_{2}. Two such triples (π_{2}, π_{3}, κ) and (π_{2}^{∘}, π_{3}^{∘}, κ^{∘}) are equivalent if there are isomorphisms θ_{2} : π_{2} → π_{2}^{∘} and θ_{3} : π_{3} → π_{3}^{∘} such that θ_{2}^{∗}(κ^{∘}) = θ_{3♯}(κ), where θ_{3♯} : H^{4}(P_{2}(X); π_{3}) → H^{4}(P_{2}(X); π_{3}^{∘}) is the change of coefficients homomorphism.

The Wang sequence for the classifying map from P_{2}(X) to P_{1}(X) = S^{1} gives rise to an exact sequence

0 → H^{3}(K(π_{2}, 2); π_{3})/(t − 1) → H^{4}(P_{2}(X); π̃_{3}) → H^{4}(K(π_{2}, 2); π_{3})^{⟨t⟩} → 0

in which the right hand map is onto the subgroup of H^{4}(K(π_{2}, 2); π_{3}) fixed by the action of πK = ⟨t⟩. Since H_{3}(K(A, 2); ℤ) = 0 and H_{4}(K(A, 2); ℤ) ≅ Γ(A) we have isomorphisms H^{3}(K(A; 2); B) ≅ Ext_{ℤ}(A, B) and H^{4}(K(A, 2); B) ≅ Hom_{ℤ}(Γ(A), B). The image of κ is the first nontrivial k-invariant of the universal cover X′, and corresponds to the inclusion of Γ(π_{2}) into π_{3} under the latter isomorphism (see Chapter 1 of [Ba]). Thus κ determines the Whitehead product and Hopf map η on the truncated homotopy algebra L = ℤ⊕π_{2} ⊕π_{3}, which was part of the “homotopy triple” invariant (L, η, [λ]) of [HK97]. Every such algebra (L, η) with an automorphism t such that L_{1} and L_{2} are finitely generated as Λ-modules and t − 1 acts invertibly on L_{1} and L_{2}/L_{1}^{2} + η(L_{1}) determines a k-invariant in H^{4}(K(L_{1}, 2); L_{2}) which is invariant under t, and so may be lifted to such a Postnikov triple.

A presentation for π_{4} in terms of π_{2} and π_{3} may be extracted from the discussion in Part III.§2 of [Ba85]. In particular, the subgroup [π_{2}, π_{3}] generated by Whitehead products is isomorphic to the quotient of π_{2} ⊗ π_{3} by the relations determined by the Jacobi identities, commutativity and the vanishing of [α ∘ η, α] for all α ∈ π_{2}. (This uses the fact that Γ(π_{2}) is a subgroup of π_{3}). The longitude is in the kernel of the suspension homomorphism E : π_{4} = π_{4}(X′) → π_{4}^{st}(X′) [Su92]). In the torsion free case kerE = [π_{2}, π_{3}] (see page 533 of [Wh]), and it is easy to see directly that λ_{K} ∈ [π_{2}, π_{3}]. Note that HAut_{+}(X) acts naturally on [π_{2}, π_{3}] through HAut_{+}(P_{3}(X)), which is the subgroup of Aut_{Λ}(π_{2}) × Aut_{Λ}(π_{3}) that fixes κ.

Since X is 4-dimensional every automorphism of the Postnikov triple is realizable by a self homotopy equivalence of X. The equivalence class of the Postnikov triple and the orbit of λ_{K} under the action of HAut_{+}(X) together determine the pair (X(K), j_{K}), and hence the knot, by Theorem 5.

We also have a realization theorem, given appropriate boundary data.

Theorem 6. Let (Π_{2}, Π_{3}, κ) be a Postnikov triple such that κ induces a monomorphism from Γ(Π_{2}) to Π_{3}, Π_{2} and H_{3} = Π_{3}/Γ(Π_{2}) are finitely generated as Λ-modules, t − 1 acts invertibly on these and p.d._{Λ}H_{3} ≤ 1. Then there is a finite 4-dimensional homology circle X with π_{1}(X) ≅ Z realizing this Postnikov triple. If j : S^{4} × S^{1} → X is a map such that (X, j) is an orientable P D_{6}-pair there is a simple 4-knot K such that (X(K), j_{K}) ≃ (X, j). Proof. The Postnikov triple may be realized by a complex P with π_{1}(P) ≅ Z and π_{q}(P) = 0 for q > 3. Since H_{q}(P; Λ) is finitely generated for all q we may assume that P has finite skeleta. Since the map from Γ(Π_{2}) ≅ H_{4}(K(Π_{2}, 2); ℤ) to Π_{3} = H_{3}(K(Π_{3}, 3; ℤ) induced by κ is a monomorphism H_{4}(P; Λ) = H_{4}(P′; ℤ) = 0, by the Leray-Serre spectral sequence for the classifying map from P′ to K(Π_{2}, 2). Hence there is a finite 4-dimensional cell complex X such that H_{4}(X; Λ) = 0 which realizes this data, by Lemma 1. It is an immediate consequence of the Wang sequence for the classifying map X → S^{1} that X is an homology circle.

Since S^{4} × S^{1} is s-parallelizable the obstructions to trivializing the Spivak normal fibration ν : X → BSG lie in H^{q}(X, j; π_{q}(BSG)). (The coefficients are simple as BSG is an H-space). By duality, we need consider only q = 5 or 6. Since π_{5}(BSG) = π_{6}(BSG) = 0 there are no obstructions. The result now follows from Lemma 3. □

We shall see in the next section that the duality condition can be described more explicitly in the fibred case.

If π_{2} is ℤ-torsion free K has a 1-connected Seifert hypersurface with torsion free homology. As this hypersurface is parallelisable and has boundary S^{4}, it is homeomorphic to a once-punctured connected sum of copies of S^{2} × S^{3} [Sm62]. Hence it admits S^{1}-actions with nonempty, codimension-2 fixed point set, and so K is reflexive (see §17 of [Gl62]). The homotopy class of the inclusion of S^{4} as the boundary of S^{2} × S^{3}\intD^{5} is the basic Whitehead product [ι_{2}, ι_{3}]. Therefore λ_{K} is in [π_{2}, π_{3}]. We shall modify the homotopy triple q(K) = (L, η, [λ_{K}]) for such knots by taking L as the commutative, graded ring with L_{0} = Z, L_{1} = π_{2}, L_{2} = π_{3} and L_{3} = [π_{2}, π_{3}] (the quotient of L_{1} ⊗ L_{2} by the relations stated above), and [λ_{K}] to be the orbit of λ_{K} under the group {(θ_{1}, θ_{2}) ∈ Aut_{Λ}(L_{1}) × Aut_{Λ}(L_{2}) | θ_{2}|_{Γ(L_{1})} = Γ(θ_{1})} (rather than the group G(X′)). The quadratic function η : L_{1} → L_{2} determined by composition with the Hopf map in π_{3}(S^{2}), and the automorphism induced by t ∈ πK are the other ingredients of this invariant.

The sum of two homotopy triples is q♯q^{∘} = (L♯L^{∘}, η♯η^{∘}, [λ + λ^{∘}]), where L♯L^{∘} = L ⊗ L^{∘}/(degree > 3) is the truncation of the graded tensor product (with the diagonal action of the group ⟨t⟩), and η♯η^{∘}(x, x^{∘}) = η(x) + x ⊗ x^{∘} + η^{∘}(x^{∘}). The orbit [λ + λ^{∘}] is well defined, and so the sum is well defined.

Theorem 7. Let K and K^{∘} be torsion free simple 4-knots. Then

(i) q(K♯K^{∘}) = q(K)♯q(K^{∘}); (ii) q(rK) = (L̄, η̄, [λ]), q(Kρ) = (L̄, η̄, [−λ]) and q(−K) = (L, η, [−λ]). Proof. Since X(K♯K^{∘})′ ∼ X(K)′ ∨ X(K^{∘})′ it follows from the Hilton-Milnor theorem that L(X(K♯K^{∘})′) = L(X(K)′)♯L(X(K^{∘})′) and η_{X(K♯K^{∘})′} = η_{X(K)′}♯η_{X(K^{∘})′}. If V and V^{∘} are Seifert hypersurfaces for K and K^{∘} respectively then V ♮V^{∘} is a Seifert hypersurface for K♯K^{∘}. Hence the sum of the longitudes is the longitude of the sum. This proves the first assertion. The second assertion follows from the facts that if the orientation of the knot is changed the longitude is inverted, while if the transverse orientation of the knot is changed the meridian is inverted. □

How should one define the sum of Postnikov triples in order to obtain an additive invariant in general?

Can we compute this invariant from Seifert data or handlebody presentations of K?

5. Fibred simple 4-knots.

A simple 4-knot K is fibred if and only if π_{2} is finitely generated as an abelian group, for then H_{3}(X′; Z) is also finitely generated, and the higher homology groups are 0, by Poincaré duality. In this case there are natural isomorphisms H^{q}(X′; ℤ) ≅ H^{q+1}(X; Λ), and so the duality isomorphism D : H^{4}(X; Λ) → H_{2}(X; Λ) is detected by the corresponding duality for X′.

Let β^{∗} : H^{2}(X′; ℚ/ℤ) → H^{3}(X′; ℤ) be the Bockstein homomorphism and let Sη ∈ π_{4}(S^{3}) be the suspension of the Hopf map. Let W = [π_{2}, π_{2}] ∘ Sη + [π_{2}, [π_{2}, π_{2}]] and F = {f ∈ Hom_{ℤ}(H^{3}, H_{2}) | ⟨x, f(β^{∗}(x))⟩ = 0 ∀x ∈ H^{2}(X′; ℚ/ℤ)}. We may identify W with [π_{2}, Γ(π_{2})] by means of the Barcus-Barratt formula [α, β] ∘ Sη = [α ∘ η, β] + [[α, β], α] [BB58]. There is an exact sequence 0 → W → kerE → F → 0, by Proposition 7.2 of [St82]. Then (X′, λ) is a P D_{5}-pair (and so (X, j) is a P D_{6}-pair) if and only if the image D(λ) of λ in F is an isomorphism.

If f : X → X is a self homotopy equivalence then D(π_{1}(f)(λ)) = f_{∗}D(λ)f^{∗}, where f_{∗} and f^{∗} are the induced homomorphisms on (co)homology. In general, we shall say that two isomorphisms D_{i} : H^{3}(X′; ℤ) ≅ H^{4}(X; Λ) → H_{2}(X; Λ) = H_{2}(X′; ℤ) (for i = 1, 2) are equivalent if D_{2} = f_{∗}D_{1}f^{∗} for some self homotopy equivalence f. If D(λ_{1}) = D(λ_{2}) there is a self homotopy equivalence f_{u} : X′ → X′ which induces the identity on homology and such that f_{u∗}λ_{1} = λ_{2}, by Proposition 7.13 of [St82]. Here u ∈ H^{3}(X′; Γ(π_{2})). As tλ_{1} = λ_{1} and tλ_{2} = λ_{2} we may assume that tu = u and hence that π_{3}(f_{u}) is a Λ-homomorphism. Hence f_{u} is the lift of a self homotopy equivalence f : X → X such that H_{∗}(f; Λ) = id and fj_{1}|_{S^{4}∨S^{1}} ∼ j_{2}|_{S^{4}∨S^{1}}. Thus the orbit of the longitude and the equivalence class of the duality isomorphism determine each other.

The truncated homotopy algebra and the orbit of the longitude determine the isotopy class of the knot up to a finite ambiguity, for as π_{2} and π_{3} are finitely generated Ext_{ℤ}(π_{2}, π_{3}) ≅ Ext_{ℤ}(T (π_{2}), π_{3}) and is finite. The homotopy triple (as defined in §4) is a complete invariant for torsion free fibred simple 4-knots, and is additive with respect to sum of such knots.

The duality isomorphism gives rise to the Farber-Levine pairing, which is a nondegenerate skew-symmetric bilinear pairing ℓ_{K} : T (π_{2}) × T (π_{2}) → ℚ/ℤ on which t acts isometrically. If π_{2} is finite ℓ_{K} determines the duality isomorphism and the homotopy triple, and every such pairing is realized by some finite simple 4-knot. In dimensions 2q ≥ 8 the analogous pairing is a complete invariant for finite simple 2q-knots with |π_{2}| odd [Ko79,Fa83]. However this may not be correct for such simple 4-knots. For instance, if T = (Λ/(5, t − 2)) ⊕ (Λ/(5, t − 3)) then T^{2} supports a skew-symmetric Farber-Levine pairing, and so may be realized by some odd finite simple 4-knot. However it may be shown that t−1 does not act invertibly on Ext_{ℤ}(T^{2}, Γ(T^{2})).

What can be said about the factorization of finite simple 4-knots? (See [Hi86] for the factorization of finite simple 2q-knots with q ≥ 4).

6. Artin spins of fibred simple 3-knots.

The Artin spin σk of a simple 3-knot k is simple, and is torsion free, as π_{2}(X(σk)) ≅ π_{2}(X(k)) (see [Ke84]). Thus such spun simple knots are determined by their homotopy triples. In this case an additional feature enables us to give a purely algebraic characterization of such knots.

Lemma 8. If π_{2} is finitely generated and ℤ-torsion free and the inclusion of Γ(π_{2}) into π_{3} splits as a Λ-homomorphism then any automorphism θ of H_{3}(X; Λ) is induced by a self homotopy equivalence f : X → X such that π_{i}(f) = id for i ≤ 2. In particular, any two duality isomorphisms D_{i} : H^{4}(X; Λ) → H_{2}(X; Λ) are equivalent. Proof. Since the inclusion splits, θ extends to a Λ-automorphism of π_{3} which is the identity on Γ(π_{2}). Since π_{2} is torsion free κ is determined by the inclusion of Γ(π_{2}) into π_{3}, and so (id_{π_{2}}, θ) determines an isomorphism of the Postnikov data. Therefore there is a self homotopy equivalence f : X → X such that π_{i}(f) = id for i ≤ 2 and π_{3}(f) = θ.

Now suppose that D_{i} : H^{4}(X; Λ) → H_{2}(X; Λ) are two isomorphisms. Then D_{2}^{−1}D_{1} is an automorphism of H^{4}(X; Λ) ≅ H^{3}(X′; ℤ). Let Θ be the transposed automorphism of H_{3}(X; Λ) = H_{3}(X′; ℤ). Hence there is a self homotopy equivalence f : X → X such that H_{2}(f; Λ) = id and H^{4}(f; Λ) = D_{2}^{−1}D_{1}. Hence D_{1} = f_{∗}D_{2}f^{∗}. □

Theorem 9. Let K be a simple 4-knot. Then K is the Artin spin of a fibred simple 3-knot if and only if π_{2} is finitely generated, ℤ-torsion free, admits a skew-hermitean Blanchfield pairing and the inclusion of Γ(π_{2}) into π_{3} splits over Λ. Moreover K is determined up to isotopy by π_{2}. Proof. Let k be a simple 3-knot; then the spun knot σk is a simple 4-knot, and π_{2}(X(k)) ≅ π_{2}(X(σk)), so σk is torsion free and admits a skew-hermitean Blanchfield pairing. As X(σk) ≃ (X(k) × S^{1}) ∪_{S^{1}×S^{1}} (S^{1} × D^{2}) [Ke84] and projection onto the first factor induces a retraction of the latter space onto X(k), the inclusion of Γ(π_{2}) into π_{3}(X(σk)) splits over Λ. Conversely, if π_{2} is finitely generated and torsion free, admits a skew-hermitean Blanchfield pairing and the inclusion of Γ(π_{2}) into π_{3} splits then K is the spin of any simple 3-knot k with π_{2}(X(k)) ≅ π_{2}. □

Let Δ(t) = t^{3} − at^{2} + (a + 1)t − 1 and let A = Λ/(Δ(t)). Then e^{1}(A) ≅ \overline{Hom_{Λ}(A, ℚ(t)/Λ)} ≅ \overline{A} = Λ/(Δ(t^{−1})). As Δ is not palindromic (up to sign), \overline{A} is not isomorphic to A, and so A does not admit a Blanchfield pairing. Therefore no simple 4-knot with π_{2} ≅ A can be a spun 3-knot.

Is it generally true that Artin spins of simple 3-knots are determined by their knot modules π_{2} (cf. [Ke84])?

All spun knots are DNC (doubly null concordant). If a simple 4-knot is DNC then its homotopy algebra admits maps i_{±} to algebras W_{±} such that (i_{+}, i_{−}) induces isomorphisms π_{2} ≅ W_{+2} ⊕ W_{−2} and H_{3} ≅ H_{+3} ⊕ H_{−3}, where H_{±3} = W_{±3}/Γ(W_{±2}) and the duality behaves appropriately. Does this condition characterize DNC simple 4-knots, at least in the torsion free case? (A related criterion was established for simple 2q-knots with q ≥ 4 in [Ke84’]). Acknowledgment. This work was done while the second author was visiting the University of Sydney under a research grant from the Engineering and Physical Sciences Research Council.

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