ON FARBER’S INVARIANTS FOR SIMPLE 2q-KNOTS

JONATHAN A. HILLMAN

Abstract. Let K be a simple 2q-knot with exterior X. We show directly how the Farber quintuple (A, Π, α, ℓ, ψ) determines the homotopy type of X if the torsion subgroup of A = π_{q}(X) has odd order. We comment briefly on the possible role of the EHP sequence in recovering the boundary inclusion from the duality pairings ℓ and ψ. Finally we reformulate the Farber quintuple as an hermitean self-duality of an object in an additive category with involution.

In a series of papers in the early 1980s Farber showed that stable high dimensional knots could be classified in terms of stable homotopy theory, and in particular that if q ≥ 4 simple 2q-knots may classified up to isotopy by quintuples (A, Π, α, ℓ, ψ), where A, Π are ℤ[Z]-modules, α is a monomorphism and ℓ and ψ are sesquilinear pairings [6]. Certain significant special cases were dealt with by Kearton [14] and Kojima [15].

Farber’s argument for showing that his invariant determined the knot involves Spanier-Whitehead duality for highly connected Seifert hypersurfaces and Wall’s embedding theorem (to interpolate between different choices of such hypersurfaces). An alternative approach (avoiding Seifert arguments) is suggested by the work of Lashof and Shaneson, who used surgery over π_{1} = Z to show that the exterior X of a high dimensional knot with group Z is determined up to homeomorphism by the homotopy type of the pair (X, ∂X) [16]. There remains a possible ambiguity of order 2, and a further argument is needed to show that the exterior determines the knot if (X, ∂X) is highly connected [20].

In this note we shall provide some evidence for the feasibility of such a homotopytheoretic approach. We show first that the homotopy type of X is determined by the modules A = π_{q}(X) and π_{q+1}(X) and a k-invariant κ(X). We then show that these invariants may be derived algebraically from the Farber quintuple. (In the case of κ(X) we need to assume that the ℤ-torsion submodule T (A) has odd order.) The work of Farber implies that the duality pairings ℓ and ψ determine the inclusion of the boundary. We have not yet been able to verify this independently. However the longitude (i.e., the lift of the boundary inclusion to infinite cyclic covers) has trivial suspension, and the EHP sequence of Chapter XII of [25] suggests a close connection between the longitude and these duality pairings. Further progress appears to depend on having a more direct construction of the pairing ψ (or at least of the version Ψ used below for the case with T (A) odd).

Finally, Quebbemann, Scharlau and Schulte have codified the notion of hermitean pairing in an additive category C with an involution ∗ [19], and we shall show that Farber’s invariant may be described in such terms. This approach has proven useful in studying the factorization of some nontrivial classes of higherdimensional knots into sums of indecomposable knots [8, 10]. It is also of interest in connection with criteria for a knot to be doubly null concordant [7].

1991 Mathematics Subject Classification. 57Q45.

Key words and phrases. EHP sequence, Farber quintuple, F -form, hermitean duality, simple knot.

1. Knots

An n-knot is a locally flat embedding K : S^{n} → S^{n+2}. We shall assume that S^{n} has the standard orientation as the boundary of the unit ball in R^{n+1}, for all n > 0. The exterior of K is X = X(K) = S^{n+2}−N(K), where N(K) ≅ S^{n}×R^{2} is an open regular neighbourhood of the image of K. The orientations of the spheres determine an orientation for X and an isotopy class of homeomorphisms S^{n} × S^{1} ≅ ∂X. The inclusion of S^{1} into X determined by the meridian ∗ × S^{1} ⊂ ∂X induces an isomorphism on integral homology. In particular, there is a canonical generator t for H = H_{1}(X(K); ℤ) ≅ Z, and so we may identify ℤ[H] with ℤ[Z] = Λ = ℤ[t, t^{−1}].

Reattaching S^{n} × D^{2} to X = X(K) along ∂X ≅ S^{n} × S^{1} using the nontrivial twist map of S^{n} × S^{1} gives a homotopy (n + 2)-sphere. The embedding of S^{n} × {0} in this sphere gives an n-knot K^{∗}, called the Gluck reconstruction of K, and any knot with exterior homeomorphic to X is isotopic to K or to K^{∗} (up to change of orientations).

Let p : X′ → X be the infinite cyclic cover space corresponding to the commutator subgroup π′. The modules H_{∗}(X; Λ) = H_{∗}(X′; ℤ) are finitely generated, since X is compact and Λ is noetherian. Moreover t − 1 acts invertibly on H_{∗}(X; Λ), since X is an homology circle, and so these are torsion modules. The longitude λ_{K} ∈ π_{n}(X) is the class of the inclusion S^{n} × {1} → S^{n} × S^{1} ≅ ∂X ⊂ X. If V is a Seifert hypersurface for K then λ_{K} factors though the inclusion of ∂V ≅ S^{n} into V. Let k : V/∂V → S^{n+1} be the cofibre of the natural map from V to V/∂V. Then the Pontrjagin-Thom collapse of S^{n+2} onto S(V/∂V) provides a section to Sk. It follows easily that the suspension of λ_{K} is trivial [22].

The knot K is r-simple if π_{1}(X) ≅ Z and π_{j}(X) = 0 for 1 < j ≤ r. The sum K#K̃ of r-simple n-knots K and K̃ is r-simple. If K is 1-simple then X is determined up to homeomorphism by the homotopy type of the pair (X, ∂X) [16]. The knot K is simple if it is [(n − 1)/2]-simple and n ≥ 3.

It is a cosequence of Farber’s work on stable knots that an r-simple n-knot K is determined by its exterior (i.e., K ≅ K^{∗}) if 3r > n. In [20] Richter shows that if 3r ≥ n and K is an r-simple n-knot the twist map of ∂X extends to a self homotopy equivalence of X and hence that K is determined by its complement, as a straightforward consequence of Williams’ Poincaré Embedding Theorem. Richter’s approach is close to the one taken here, in that he strives to work as far as possible in the homotopy category, before appealing to geometric topology through surgery. In particular, he subsequently gave a purely homotopy theoretic proof of William’s theorem [21]. (The application to knot theory does however use the fact that K has a highly connected Seifert hypersurface, in an essential way.) Is there an ad hoc simplification for the case of simple knots?

2. Modules and extensions

If M is a finitely generated left Λ-module let T (M) be the maximal finite submodule of M. In the cases of interest here T (M) is the ℤ-torsion submodule of M. The quotient M/T (M) is ℤ-torsion free, and has projective dimension at most 1 (p.d._{Λ}M/T (M) ≤ 1). If R is a right Λ-module let \overline{R} denote the left module with the conjugate structure determined by t.r = rt^{−1} for all r ∈ R.

The extension modules Ext_{Λ}^{i}(M, Λ) are naturally right modules. Let e^{i}M = \overline{Ext_{Λ}^{i}(M, Λ)}. If M is a finite Λ-module then e^{0}M = e^{1}M = 0, and so the inclusion T (M) ≤ M induces isomorphisms e^{i}M ≅ e^{i}(M/T (M)) for i = 0, 1 and e^{2}M ≅ e^{2}T (M). More generally, Ext_{Λ}^{2}(M, N) ≅ Ext_{Λ}^{2}(T (M), N), for any Λmodule N. If p.d._{Λ}M = d then Ext_{Λ}^{d}(M, N) and Ext_{Λ}^{d}(M, Λ)⊗_{Λ}N are naturally isomorphic, for if P_{∗} is a finitely generated free resolution of M then Hom_{Λ}(P_{∗}, N) ≅ Hom_{Λ}(P_{∗}, Λ) ⊗_{Λ} N, and − ⊗_{Λ} N is right exact.

If P_{∗} is a chain complex of finitely generated free left Λ-modules let D(P)_{j} = \overline{Hom_{Λ}(P_{−j}, Λ)} be the dual complex, and let P [n]_{∗} be the n-fold suspension, with P [n]_{j} = P_{j−n} for all j. Note that D(D(P))_{∗} is naturally isomorphic to P_{∗}. If H_{∗}(P) = 0 for j ≠ q or q + 1 then P is determined up to chain homotopy equivalence by the modules A = H_{q}(P_{∗}) and B = H_{q+1}(P_{∗}) and a k-invariant κ(P_{∗}) in Ext_{Λ}^{2}(A, B) [4]. This may be identified with the class of the sequence

0 → B → P_{q+1}/∂P_{q+2} → Ker(∂_{q}) → A → 0

in Ext_{Λ}^{2}(A, B) ≅ Ext_{Λ}^{2}(T (A), B). Every such class is realizable by a free complex concentrated in degrees [q, q + 2].

A simple Λ-complex concentrated in degrees [q, q + 2] is a finitely generated free Λ-complex C_{∗} whose homology modules are Λ-torsion modules and are 0 except in degrees q and q + 1. Let A = H_{q}(C_{∗}), T = T (A) and B = H_{q+1}(C_{∗}) and let k(A) ∈ Ext_{Λ}^{1}(A/T, T) ≅ \overline{e^{1}A} ⊗ T and κ(C_{∗}) ∈ Ext_{Λ}^{2}(A, B) ≅ \overline{e^{2}T} ⊗ B be the associated extension classes.

Theorem 1. Let C_{∗} be a simple Λ-chain complex concentrated in degrees [q, q + 2].

(1) C is determined up to chain homotopy equivalence by the quintuple (A/T (A), T (A), B, k(A), κ(C_{∗}));

(2) any such system of invariants with A and B finitely generated Λ-torsion modules and p.d._{Λ}B ≤ 1 is realized by a simple Λ-complex;

(3) D_{∗} = D(C)_{∗} is a simple complex concentrated in degrees [−q − 2, −q], with invariants (e^{1}B, e^{2}A, e^{1}A, k_{D}, κ(D_{∗})) where k_{D} and κ(D_{∗}) are determined by κ(C_{∗}) and k(A), respectively.

Proof. The module A is determined by T = T (A), A/T and k(A), and so the first assertion follows from the result of [4] cited above. Since A is a torsion module D_{∗} is easily seen to be a simple Λ-complex. If A_{D} = H_{−q−2}(D_{∗}) and B_{D} = H_{−q−1}(D_{∗}) then B_{D} ≅ e^{1}A, T_{D} = T (A_{D}) ≅ e^{2}A = e^{2}T and A_{D}/T_{D} ≅ e^{1}B.

There are natural isomorphisms T ≅ e^{2}e^{2}A = e^{2}T_{D} and B ≅ e^{1}A_{D}. Hence the transpositions τ_{1} : \overline{e^{1}A} ⊗ T ≅ \overline{T} ⊗ e^{1}A and τ_{2} : \overline{e^{2}T} ⊗ B ≅ \overline{B} ⊗ e^{2}T given by interchange of factors induce natural isomorphisms \overline{e^{1}A} ⊗ T ≅ \overline{e^{2}T_{D}} ⊗ B_{D} and \overline{e^{2}T} ⊗ B ≅ \overline{e^{1}A_{D}} ⊗ T_{D} between the abelian groups in which the extension classes for A and D_{∗} and for C_{∗} and A_{D} lie.

To verify (2) and see that the extension classes correspond as asserted we shall construct an explicit representative for the chain homotopy type determined by the invariants (A/T, T, B, k(A), κ(C_{∗})). We shall assume for simplicity of notation that q = 0. Let P_{1} → P_{0}, Q_{1} → Q_{0} and E_{2} → E_{1} → E_{0} be finite free resolutions of A/T, B and T, respectively, and let k : P_{1} → E_{0} and κ : E_{2} → Q_{0} be homomorphisms representing k(A) and κ(C_{∗}), respectively. Then we may using a mapping cone construction (twice) as in Satz 7.6 of [4] to construct a complex

(1)  E_{2} ⊕ Q_{1} ⟶^{Δ_{2}(κ)} P_{1} ⊕ E_{1} ⊕ Q_{0} ⟶^{Δ_{1}(k)} P_{0} ⊕ E_{0}

with these invariants. Here Δ_{2}(κ) and Δ_{1}(k) are matrices whose nonzero entries involve the differentials of the constituent complexes and the homomorphisms k and κ. It is clear that the dual complex is isomorphic to the complex obtained by taking e^{0}κ and e^{0}k as representatives for k_{D} and κ(D_{∗}), respectively. □

3. Duality for finite 𝔽Λ-modules

Let 𝔽 be a field and let 𝔽Λ = 𝔽[t, t^{−1}]. If M is a finite dimensional 𝔽Λ-module let M^{∗} = \overline{Hom_{𝔽Λ}(M, 𝔽(t)/𝔽Λ)}, E(M) = \overline{Ext_{𝔽Λ}^{1}(M, 𝔽Λ)} and F (M) = Hom_{𝔽}(M, 𝔽). (The left Λ-module structure on the latter group is given by (tϕ)(m) = ϕ(t^{−1}m) for all m ∈ M and ϕ : M → 𝔽.) These modules are each non-canonically isomorphic to M, since 𝔽Λ is a PID. The functors (−)^{∗}, E(−) and F (−) each define an involution on the category of finite dimensional 𝔽Λ-modules. They are in fact naturally equivalent. We shall give a simple proof of this for the case when 𝔽 is finite.

If f ∈ 𝔽Λ then Res(f \frac{dt}{t}, ∞) + Res(f \frac{dt}{t}, 0) = 0, since the only poles of f \frac{dt}{t} are at 0 and ∞. Thus we may define an 𝔽-linear function R : 𝔽(t)/𝔽Λ → 𝔽 by setting R(q + 𝔽Λ) = Res(q \frac{dt}{t}, ∞) + Res(q \frac{dt}{t}, 0) for all q ∈ 𝔽(t). In particular, if q = \frac{f}{t^{n}−1} where f ∈ 𝔽[t] has degree < n then R(q + 𝔽Λ) = f(0).

Theorem 2. There are natural equivalences M^{∗} ≅ E(M) ≅ F (M) on the category of finite dimensional 𝔽Λ-modules, if 𝔽 is a finite field.

Proof. Let 0 → P_{1} → P_{0} → M → 0 be a free resolution of M. If ϕ ∈ M^{∗} let ϕ_{0} : P_{0} → 𝔽(t) be a lift of ϕ. Then ϕ_{0}∂_{1} has image in 𝔽Λ, and so defines a homomorphism ϕ_{1} : P_{1} → 𝔽Λ such that ϕ_{1}∂_{2} = 0. Consideration of the short exact sequence of complexes

0 → Hom_{𝔽Λ}(P_{∗}, 𝔽Λ) → Hom_{𝔽Λ}(P_{∗}, 𝔽(t)) → Hom_{𝔽Λ}(P_{∗}, 𝔽(t)/𝔽Λ) → 0

shows that δ_{M}(ϕ) = [ϕ_{1}], where δ_{M} : M^{∗} → E(M) is the Bockstein homomorphism associated to the coefficient sequence. (The extension corresponding to δ_{M}(ϕ) is the pullback over ϕ of the sequence 0 → 𝔽Λ → 𝔽(t) → 𝔽(t)/𝔽Λ → 0.) Since M is a torsion 𝔽Λ-module and 𝔽(t) is divisible δ_{M} is an isomorphism.

Define χ_{M} : M^{∗} → F (M) by χ_{M}(ϕ)(m) = R(ϕ(m)) for all ϕ ∈ M^{∗} and m ∈ M. Then χ_{M} is 𝔽Λ-linear. Suppose now that M has finite order n. If g ∈ F (M) let L_{M}(g)(m) = \frac{ĝ(m)}{t^{n!}−1} + 𝔽Λ, where ĝ(m) = Σ_{0≤k<n!}g(t^{k}m)t^{−k} for all m ∈ M. Then L_{M}(g) is 𝔽Λ-linear, since (t^{n!} − 1)M = 0, and χ_{M}(L_{M}(g)) = g. (This is the only point at which we need 𝔽 and M to be finite). Since M^{∗} and F (M) have the same dimension and χ_{M} is 𝔽Λ-linear χ_{M} and L_{M} are mutually inverse isomorphisms. □

Let τ_{M} = χ_{M}δ_{M}^{−1} : E(M) → F (M) be the composite isomorphism.

Theorem 2 holds also for infinite fields, but verifying that χ is an equivalence involves slighly more work. See [23] for the case 𝔽 = ℚ.

There is also a corresponding result over Λ, i.e., with integer coefficients. Levine showed that if M is a finite Λ-module such that (t^{k} − 1)M = 0 then e^{2}M ≅ \overline{Ext_{Λ}^{1}(M, Λ/(t^{k} − 1))} ≅ \overline{Hom_{Λ}(M, ℚ/ℤ ⊗ Λ/(t^{k} − 1))} (via appropriate Bockstein homomorphisms), which is in turn isomorphic to Hom_{ℤ}(M, ℚ/ℤ) (via an explicit homomorphism similar to L_{M}) [17]. In particular, if M has exponent n as an abelian group then e^{2}M ≅ Hom_{ℤ}(M, Z/nZ). If A is a finitely generated Λ-module let σ_{A} : e^{2}A → Hom_{ℤ}(T (A), ℚ/ℤ) be the composite of these isomorphisms with the natural isomorphism e^{2}A = e^{2}T (A).

4. Algebraic invariants for the exterior of a simple 2q-knot

Let K be a simple 2q-knot with q > 2. Then π_{1}(X) = Z and H_{i}(X; Λ) = 0 for 0 < i < q. Let A = H_{q}(X; Λ) and B = H_{q+1}(X; Λ). As observed in §1, these are torsion Λ-modules on which t − 1 acts invertibly. The Universal Coefficient spectral sequence and Poincaré duality give isomorphisms e^{1}A = e^{1}(A/T (A)) ≅ B and e^{2}A ≅ T (\overline{H^{q+2}(X; Λ)}) ≅ T (A), while H_{i}(X; Λ) = 0 for i > q +1. In particular, there is a nonsingular pairing ℓ : T (A) × T (A) → ℚ/ℤ and p.d._{Λ}B ≤ 1. Since B = (t − 1)B it follows that B is ℤ-torsion free [17]. The exterior X fibres over S^{1} if and only if A is finitely generated as an abelian group [3].

We may identify A with the Λ-module π_{q}(X) = π_{q}(X′), by the Hurewicz Theorem. Since π_{1}(X) ≅ Z and π_{j}(X) = 0 for 1 < j < q the Postnikov q-stage of X is the generalized Eilenberg-Mac Lane space L_{Z}(A, q) determined by the Λ-module A. (See page 214 of [2].) Let π̃_{q+1} = π_{q+1}(X), considered as a Λ-module, and let κ(X) ∈ H^{q+2}(L_{Z}(A, q); π̃_{q+1}) be the k-invariant for the next stage. Let η ∈ π_{3}(S^{2}) be the Hopf map. Then η_{q} = Σ^{q−2}η generates π_{q+1}(S^{q}) = π_{1}^{st} = Z/2Z for all q > 2.

Since X′ is (q − 1)-connected there is an exact sequence

(2) π_{q+2}(X′) → H_{q+2}(X′; ℤ) ⟶^{b} A/2A ⟶^{η_{q}^{∗}} π̃_{q+1} → H_{q+1}(X′; ℤ) → 0,

where η_{q}^{∗} is induced by composition with η_{q} and the unlabeled homomorphisms are Hurewicz homomorphisms. (See Theorem XII.3.12 of [25].) In our situation H_{q+2}(X′; ℤ) = 0 and the right-hand portion of this sequence reduces to a short exact sequence of Λ-modules. Thus π̃_{q+1} is determined by A and a class in Ext_{Λ}^{1}(B, A/2A) = Ext_{Λ}^{1}(e^{1}(A/T (A)), A/2A). (See Theorem XII.4.16 of [25] for more on the corresponding class in Ext_{ℤ}^{1}(B, A/2A).) There is an analogous sequence when q = 2, involving the universal quadratic functor Γ(A) instead of A/2A. In this case π̃_{q+1} = π̃_{3} is not a stable homotopy group, there are nontrivial Whitehead products in the image of Γ(A) and η has infinite order.

The chain complex of the covering space X′ is naturally a complex of Λ-modules, and is chain homotopy equivalent to a finitely generated free complex C_{∗}. As H_{j}(C_{∗}) = 0 if j > q + 2 and p.d._{Λ}H_{q+1}(C_{∗}) ≤ 1 we may assume that C_{∗} is the direct sum of the standard resolution of the augmentation Λ-module ℤ and a simple Λ-complex C_{∗}^{rel} concentrated in degrees [q, q + 2]. (This is homotopy equivalent to the equivariant chain complex of the pair (X, S^{1}) where S^{1} is the meridian.) Therefore X is homotopy equivalent to a CW complex of dimension q + 2 [24]. Poincaré duality for (X, ∂X) reduces to a chain homotopy equivalence D(C^{rel})[−2q − 2]_{∗} ≃ C_{∗}^{rel}, and so it follows from Theorem 1 that C_{∗}^{rel} and C_{∗} are determined up to chain homotopy equivalence by A.

The universal covering space of L_{Z}(A, q) is K(A, q), and the spectral sequence for the classifying map from L_{Z}(A, q) to S^{1} = K(Z, 1) gives an exact sequence

(3) 0 → Ext_{Λ}^{2}(A, π̃_{q+1}) → H^{q+2}(L_{Z}(A, q); π̃_{q+1}) → Hom_{Λ}(A/2A, π̃_{q+1}) → 0.

The image of κ(X) in Hom_{Λ}(A/2A, π̃_{q+1}) is the homomorphism η_{q}^{∗}. (It suffices to check the corresponding assertion for κ(X′), which follows from Lemma 4.15 of Chapter XII of [25], since Hom_{Λ}(A/2A, π̃_{q+1}) ≤ Hom_{ℤ}(A/2A, π̃_{q+1}).) The corresponding sequence with coefficients B reduces to an isomorphism Ext_{Λ}^{2}(A, B) ≅ H^{q+2}(L_{Z}(A, q); B), since B is ℤ-torsion free, and the image of κ(X) under the coefficient homomorphism induced by h is κ(C_{∗}).

Theorem 3. Let A and π̃_{q+1} be finitely generated Λ-torsion modules and κ ∈ H^{q+2}(L_{Z}(A, q); π̃_{q+1}). Suppose that the image of κ in Hom_{Λ}(A/2A, π̃_{q+1}) is a monomorphism with cokernel B, and that p.d._{Λ}B ≤ 1. Then there is a finite (q+2)- complex X_{κ} such that π_{1}(X_{κ}) ≅ Z, π_{q}(X_{κ}) ≅ A, π_{q+1}(X_{κ}) ≅ π̃_{q+1}, H̃_{j}(X_{κ}; Λ) = 0 if j ≠ q or q + 1, and κ(X) = κ. The homotopy type of X is uniquely determined by A, π̃_{q+1} and κ.

Proof. These invariants determine a Postnikov (q + 1)-stage P = P_{q+1}(κ). We may assume the k-skeleton P^{[k]} is finite for all k ≥ 0, since A and π̃_{q+1} are finitely generated Λ-modules and Λ is noetherian. The hypothesis on κ implies that H_{q+1}(P^{[q+2]}; Λ) = H_{q+1}(P; Λ) ≅ B and that composition with η_{q} determines a monomorphism from A/2A to π̃_{q+1}. Hence the Hurewicz homomorphism for P^{[q+2]} in degree q + 2 is onto. Since p.d._{Λ}B ≤ 1 and P^{[q+2]} is a finite (q + 2)- complex H_{q+2}(P^{[q+2]}; Λ) is a finitely generated free module. Attaching (q + 3)-cells to P^{[q+2]} along representatives for a basis of H_{q+2}(P^{[q+2]}; Λ) gives a finite (q + 3)- complex with trivial homology in degrees ≥ q + 2, and which is therefore homotopy equivalent to a finite (q + 2)-complex X_{κ}, by [24].

Let Y be a (q+2)-complex with Postnikov (q+1)-stage f : Y → P and such that H_{q+2}(Y; Λ) = 0. Then f factors through a map g : Y → X_{κ}, since we may construct P by adjoining cells of dimension ≥ q + 3 to X_{κ}, to kill the higher homotopy groups. The lift of g to the universal covers induces isomorphisms on homology and so is a homotopy equivalence, by the Hurewicz and Whitehead Theorems. Therefore g is a homotopy equivalence. □

It is easy to construct a finite (q + 2)-complex with equivariant chain complex of the required chain homotopy type, using only the surjectivity of the Hurewicz homomorphism in degree q + 1 (and not appealing to [24]).

However not all such complexes admit Poincaré duality isomorphisms. Let D_{∗} = D(C^{rel})[−2q − 2]_{∗}. It is clear from Theorem 1 that there is a chain homotopy equivalence D_{∗} ≃ C_{∗}^{rel} if and only if k(A) and κ(C_{∗}) correspond.

An analogous but simpler argument shows that if K is a simple (2q −1)-knot the homotopy type of X = X(K) is determined by the module A = H_{q}(X; Λ) alone.

5. Farber quintuples and Kearton F -forms

The Farber quintuple of K is q_{F} (K) = (A, Π, α, ℓ, ψ), where A = H_{q}(X; Λ), Π = π_{q+2}^{st}(X′), α : A/2A → Π is the monomorphism determined by composition with η_{q}η_{q+1}, ℓ : T (A) × T (A) → ℚ/ℤ is the torsion linking pairing and ψ : Π × Π → Z/4Z < π_{3}^{st} is a bilinear pairing derived from Spanier-Whitehead duality for a Seifert hypersurface for K. The pairings ℓ and ψ are (−1)^{q+1}-symmetric, nonsingular and t acts isometrically. In other words the adjoint functions Ad(ℓ) and Ad(ψ) are Λ-isomorphisms, where Ad(ℓ) : T (A) → Hom_{ℤ}(T (A), ℚ/ℤ) is given by Ad(ℓ)(b)(a) = ℓ(a, b) for all a, b ∈ T (A), and Ad(ψ) is defined similarly. Let β = α^{∗}Ad(ψ) : Π → Hom_{𝔽_{2}}(A/2A, 𝔽_{2}), so that β(p)(a) = ψ(p, α(a)) for all a ∈ A/2A and p ∈ Π. Then β is an epimorphism and Ker(β) = Im(α).

These pairings interact in the following way. If M is a Λ-module and m ∈ M let [m]_{2} be the image of m in M/2M. Let γ : Π → T (A) be the homomorphism determined by ψ(p, α([x]_{2})) = ℓ(γ(p), x) for all x ∈ T (A) and p ∈ Π and nonsingularity of ℓ. Then α([γ(p)]_{2}) = 2p, for all p ∈ Π. (In particular, γ = 0 if and only if T (A) has odd order, and then Π has exponent 2, ℓ and ψ are independent and ψ is symmetric.) There is an obvious notion of sum of Farber quintuples, and q_{F} (K#K̃) = q_{F} (K) ⊕ q_{F} (K̃).

In the torsion free case (T (A) = 0) there is an alternative system of invariants, due to Kearton [14]. Reducing sequence (2) above modulo (2) gives a sequence

E : 0 → A/2A → π̃_{q+1}/2π̃_{q+1} → B/2B → 0

which is exact since B is ℤ-torsion free. The Kearton F -form of K involves the exact sequence E and an hermitean pairing [−, −] on π̃_{q+1}/2π̃_{q+1} with values in 𝔽_{2}(t)/𝔽_{2}Λ (and some auxiliary data). The pairing is defined geometrically, in terms of intersection numbers. It may be identified with an isometric structure on a 𝔽_{2}- valued pairing, via the natural isomorphism τ_{M} : M^{∗} → F (M) defined in §2 above, and the F -form may be reformulated as a pair (E, b), where b : E → E^{∗} is an isomorphism such that b^{∗} = (−1)^{q}b [26]. It is a complete invariant for torsion free simple 2q-knots if q ≥ 3 [9]. (The pairing [−, −] is defined for any simple 2q-knot, and is nonsingular if T (A) is odd. The latter hypothesis is needed only for Lemmas 1.10 and 1.13 of §1 of [14].)

These invariants are defined for q ≥ 1, but simple 2-knots are topologically trivial, and the natural invariants for simple 4-knots are not stable, so it is unlikely that the Farber quintuple is a complete invariant for these. (See instead [11].) In the torsion free case Π ≅ π̃_{q+1}/2π̃_{q+1} and the Farber quintuple is algebraically equivalent to a Kearton F -form (although it is not clear how the pairings ψ and [−, −] correspond). This suggests that the Farber quintuple may suffice when q = 3. Can the Farber invariant be extended to all even-dimensional knots as an additive invariant?

6. Relating the invariants

The first three ingredients of q_{F} (K) are determined by the homotopy type of X alone, while ℓ and ψ are manifestations of Poincaré duality for the pair (X, ∂X). We shall show that, conversely, q_{F} (K) determines π̃_{q+1} and κ(X) and hence the homotopy type of X, at least if T (A) has odd order. We shall consider how the duality pairings and the inclusion of ∂X interact in the next section.

Let pd_{Z} : B = H_{q+1}(X; Λ) → e^{1}A and pd_{2} : H_{q+1}(X; 𝔽_{2}Λ) → E(A/2A) = \overline{Ext_{𝔽_{2}Λ}^{1}(A/2A, 𝔽_{2}Λ)} be the isomorphisms determined by Poincaré duality and the Universal Coefficient spectral sequence, and let τ : E(A/2A) ≅ Hom_{𝔽_{2}}(A/2A, 𝔽_{2}) be the isomorphism defined in §2. Let ρ_{2} : B = H_{q+1}(X; Λ) → H_{q+1}(X; 𝔽_{2}Λ) be reduction mod (2). Reduction mod (2) also induces a natural isomorphism 𝔽_{2} ⊗_{ℤ} e^{1}(A/T (A)) ≅ E(A/(T (A), 2A)) and hence a natural transformation e : e^{1}A ≅ e^{1}(A/T (A)) → E(A/(T (A), 2A)) → E(A/2A), and the diagram

(4)  \begin{matrix}B & ⟶ & H^{q+1}(X; Λ) & ⟶ & e^{1}A \\ ↓^{ρ_{2}} & & ↓ & & ↓^{e} \\ H_{q+1}(X; 𝔽_{2}Λ) & ⟶ & H^{q+1}(X; 𝔽_{2}Λ) & ⟶ & E(A/2A) \end{matrix}

is commutative.

Theorem 4. The module π̃_{q+1} is the pullback of τe and β over Hom_{𝔽_{2}}(A/2A, 𝔽_{2}), i.e., it is the fibre sum π̃_{q+1} ≅ {(b, p) ∈ e^{1}A × Π | τe(b) = β(p)}.

Proof. The Atiyah-Hirzebruch spectral sequence for the stable homotopy of X′ gives the exact sequence (2) (if q > 2) and also an exact sequence

(5)  0 → H_{q}(X′; π_{2}^{st}) → Π ⟶^{θ} H_{q+1}(X′; π_{1}^{st}) → 0.

We may identify H_{q}(X′; π_{1}^{st}) and H_{q}(X′; π_{2}^{st}) with A/2A, since π_{1}^{st} = π_{2}^{st} = Z/2Z. This sequence is an exact sequence of Λ-modules, by the naturality of the Atiyah-Hirzebruch spectral sequence. Moreover β = τpd_{2}θ. (See Theorem 3.3(b) of [6].) Composition (on the right) with η_{q+1} induces a homomorphism η_{q+1}^{∗} : π̃_{q+1} → Π, which factors through π̃_{q+1}/2π̃_{q+1}, since η_{q+1} has order 2. The sequences (2) and (5) form the rows of a commutative diagram

(6) \begin{matrix}0 & ⟶ & A/2A & ⟶^{η_{q}^{∗}} & π̃_{q+1} & ⟶ & B = H_{q+1}(X′; ℤ) & ⟶ & 0 \\ ↓ & & ↓^{=} & & ↓^{η_{q+1}^{∗}} & ↓^{ρ_{2}} & & ↓ \\ 0 & ⟶ & A/2A & ⟶^{α} & Π & ⟶^{θ} & H_{q+1}(X′; 𝔽_{2}) & ⟶ & 0. \end{matrix}

The theorem now follows easily. □

Reduction mod (2) induces a monomorphism from B/2B to H_{q+1}(X′; 𝔽_{2}), with cokernel T or(A, Z/2Z), while η_{q+1}^{∗} factors through π̃_{q+1}/2π̃_{q+1}, since 2η_{q+1} = 0.

Corollary 5. If T (A) has odd order the homotopy type of X is determined directly by q_{F} (K).

Proof. Since T (A) has odd order T or(A, Z/2Z) = 0, so η^{∗} is an epimorphism, and Ext_{Λ}^{2}(A, π̃_{q+1}) ≅ Ext_{Λ}^{2}(T (A), e^{1}(A/T (A))), and is the odd-order summand of H^{q+2}(L_{Z}(A, q); π̃_{q+1}).

Since the image of κ(X) in the 2-primary summand is η_{q}^{∗} it follows that [κ(X)] is determined by [κ(C)] and α, and hence by A and α. Thus the homotopy type of X is determined directly by q_{F} (K), as in Theorems 3 and 4. □

If A = 2A then Π = 0 and so q_{F} (K) reduces to the pair (A, ℓ). In this case π̃_{q+1} ≅ e^{1}A and κ(X) = κ(C) ∈ Ext_{Λ}^{2}(A, e^{1}A). If instead A is torsion free Ext_{Λ}^{2}(A, π̃_{q+1}) = 0 and so κ(C) = 0. Hence κ(X) is determined by α, which lifts to a monomorphism A/2A ≅ {(0, p) ∈ B × Π | β(p) = 0} ≤ π̃_{q+1} in the above fibre sum. (In this case κ(C) = 0.)

We may also ask how we may derive Π and α directly from A, π̃_{q+1} and κ(X). If T (A) is odd Π ≅ π̃_{q+1}/2π̃_{q+1} and α is the composite of the monomorphism η_{q}^{∗} with reduction mod (2). In this case q_{F} (K) is algebraically equivalent to the torsion linking pairing ℓ together with an F -form (with associated module A/T (A)).

In [6] the pairing ψ is defined in terms of Spanier-Whitehead duality for a Seifert hypersurface V for the knot K. Let u : V ∧ V → S^{n+1} and z : V → V be the duality pairing and carving map of [6], respectively. Let ψ_{V} (α, β) = u(α ∧ β) ∈ π_{2q+4}^{st}(S^{2q+1}) = π_{3}^{st} ≅ Z/24Z for all α, β ∈ π_{q+2}^{st}(V). Let P = ℤ[z], z̄ = 1 − z and L = P [(zz̄)^{−1}]. Then π_{q+2}^{st}(X′) ≅ L ⊗_{P} π_{q+2}^{st}(V) and ψ_{V} extends to a pairing ψ on π_{q+2}^{st}(X′) which is independent of the choice of Seifert hypersurface V. Is there a direct definition of ψ which does not involve the choice of a Seifert hypersurface?

For instance, can it be defined directly in terms of a Z-equivariant SW duality? (See also [5] for the case of fibred simple 2q-knots.)

We may define similar pairings Ψ_{V} and Ψ on π_{q+1}^{st}(V) and π_{q+1}^{st}(X′), with values in π_{1}^{st} = Z/2Z. Compostion on the right with η^{2} induces a monomorphism from π_{1}^{st} to π_{3}^{st}, and η ∧ η = η^{2} ∈ π_{2}^{st}, by X.8.12 of [25]. Hence Ψ(a, b) = ψ(aη_{q+1}, bη_{q+1}) for all a, b ∈ π_{q+1}^{st}(X′). If T (A) is odd η_{q+1}^{∗} induces an isomorphism π_{q+1}^{st}(X′) ≅ π_{q+2}^{st}(X′) and so ψ and Ψ are equivalent. We expect that the pairings [−, −] and Ψ may then be related as follows:

Lemma? If T (A) is odd Ψ(u, v) = R([u, v]) for all u, v ∈ π̃_{q+1}/2π̃_{q+1}.

7. Inclusion of the boundary

Let Y = S^{2q} ×S^{1} have the product orientation and let J_{K} : Y ≅ ∂X ⊂ X be the inclusion of the boundary. The oriented homotopy type of X and the class [J(K)] of J_{K} modulo homotopy and composition with orientation preserving self homotopy equivalences of X and Y together determine the oriented homotopy type of the pair (X, ∂X). The difficulty is in giving a practical characterization of [J(K)]. In this section we shall mention two of the known constraints.

Let J : Y → X be a map which induces an isomorphism on fundamental groups. Let M(J) be the mapping cylinder of J. There is a canonical homotopy equivalence M(J) ≃ X. The orientation for Y determines a generator μ_{J} for H_{2q+2}(M(J), Y; ℤ) ≅ H_{2q+1}(Y; ℤ). Since the universal covers of M(J) ≃ X and Y are highly connected cap product with μ_{J} determines homomorphisms \overline{H^{q+1}(X; Λ)} → H_{q+1}(X; Λ) and \overline{H^{q+2}(X; Λ)} → H_{q}(X; Λ). The pair (M(J), Y) is a P D_{2q+2}-pair if and only if these homomorphisms are isomorphisms. (The corresponding homomorphisms in other degrees are isomorphisms, in all cases.)

The map J′ : Y′ = S^{2q} × R → X′ covering J determines an element λ_{J} ∈ π_{2q}(X′) = π_{2q}(X), such that (t − 1)λ_{J} = 0. In the fibred case the Poincaré duality constraints depend only on λ_{J}. The longitude λ_{K} corresponds to J_{K}. Lemma 3 of [11] asserted that if X is an n-dimensional homology circle with H_{n}(X; Λ) = 0 and two maps j_{1}, j_{2} : S^{n} × S^{1} → X induce isomorphisms on π_{1} and agree on S^{n} ∨ S^{1} then there is a self-homotopy equivalent f : X → X such that j_{2} ∼ fj_{1}. If this were correct it would follow not only that λ_{K} determines J_{K} for 1-simple knots but also that such knots are determined by their exteriors. The proof of Lemma 3 in [11] is wrong, and thus the later assumptions in that paper that the homotopy quadruple discussed there is a complete invariant for simple 4-knots are unjustified. (We thank W.Richter for pointing out our error). Nevertheless it remains possible that the result may be true, at least if X′ is sufficiently highly connected.

The inclusion of the boundary of a Seifert hypersurface V for K has trivial suspension in π_{2q+1}(SV) [22]. Therefore λ_{K} is in the kernel of the suspension E : π_{2q}(X′) → π_{2q+1}(SX′). The group π_{2q}(X′) is just outside the stable range, since X′ is (q − 1)-connected, and sits near one extremity of the EHP sequence (Theorem XII.2.2 of [25]):

π_{3q−2}(X′) → ⋯ → π_{2q+2}(SX) → π_{2q+2}(X′ ∗ X′) → π_{2q}(X′) → π_{2q+1}(SX′) → 0.

Moreover X′ ≃ M_{A} ∨ M_{B} is a wedge of Moore spaces M_{A} = M(A, q) and M_{B} = M(B, q + 1), since X′ is (q − 1)-connected, H_{j}(X′; ℤ) = 0 for j > q + 1 and q > 2. Hence π_{2q}(X′) ≅ π_{2q}(M_{A}) ⊕ π_{2q}(M_{B}) ⊕ [A, B], where [A, B] is the subgroup generated by all Whitehead products [x, y] with x ∈ π_{q}(M_{A}) ≅ A and y ∈ π_{q+1}(M_{B}) ≅ B. The summand π_{2q}(M_{B}) is stable, while all Whitehead products have trivial suspension.

Suppose now that K is fibred and A ≅ Z^{β} is torsion-free. Then M_{A} ≃ ∨^{β}S^{q} and M_{B} ≃ ∨^{β}S^{q+1}. Let {i_{q}^{m}}_{m≤β} and {i_{q+1}^{n}}_{n≤β} be bases for A and the image of B ≅ π_{q+1}(∨^{β}S^{q+1}) in π_{q+1}(X′), respectively. In this case Ker(E) is generated by all Whitehead products i_{mm′} = [i_{q}^{m}, i_{q}^{m′}η_{q+1}] and j_{mn} = [i_{q}^{m}, i_{q+1}^{n}] for all m, m′, n ≤ β. Thus λ_{K} = Σc_{mm′}i_{mm′} + Σd_{mn}j_{mn} for some coefficients c_{mm′} ∈ 𝔽_{2} and d_{mn} ∈ ℤ. The integral coefficients d_{mn} may be detected by computing cup products in (S^{q} ∨ S^{q+1}) ∪_{pλ} e^{2q+1} for suitable projections p : X′ → S^{q} ∨ S^{q+1}. (See [12], and note that S^{q} ∨ S^{q+1} ∪_{[i,j]} e^{2q+1} ≃ S^{q} × S^{q+1}.) Hence the matrix D = [d_{mn}] is invertible, since (X′, λ_{K}) is a P D_{2q+1}-complex. and we may choose the basis for B so that D = I_{β}.

Question. How are the coefficients c_{mm′} determined by the pairing Ψ?

An analogous but simpler argument applies if K is a fibred simple (4k − 1)-knot. For then X′ ≃ ∨^{β}S^{2k} and Ker(E) ≤ π_{4k−1}(X′) is generated by the Whitehead products [i_{2k}^{m}, i_{2k}^{n}] for m, n ≤ β. The coefficients e_{mn} of λ_{K} = Σe_{mn}[i_{2k}^{m}, i_{2k}^{n}] may be detected by cup products in S^{2k} ∪_{pλ} e^{4k} or (S^{2k} ∨ S^{2k}) ∪_{pλ} e^{4k} for suitable projections p : X′ → S^{2k} (and using the Hopf invariant, as on page 211 of [18]) or p : X′ → S^{2k} ∨ S^{2k} (and using[12]).

When T (A) ≠ 0 there may be elements of Ker(E) which are not Whitehead products, as can be seen already when A = Z/pZ. This is the situation that arises for odd finite simple 2q-knots, for which A = T (A) is finite of odd order and the Farber quintuple reduces to the torsion linking pairing ℓ. (See [15].)

8. Hermitian self-dualities

Let C be an additive category with an involution ∗ and let ε = ±1. An εhermitean pairing on an object C of C is defined to be an isomorphism ϕ : C → C^{∗} such that ϕ^{∗} = εϕ. This categorical point of view is particularly useful if the Krull-Schmidt Theorem holds for the objects of C and the endomorphism rings of objects are radically complete. For then the analysis of the decompositions of hermitean pairings into orthogonal direct sums may be reduced to corresponding questions for pairings over (skew) fields [19].

Consider quadruples Q = (A, C, θ, E) where A and C are finitely generated Λmodules on which t − 1 acts invertibly, θ : T (A) → e^{2}C is an isomorphism and E is an exact sequence of Λ-modules

(7)  0 → A/2A ⟶^{α} Π ⟶^{ω} e^{2}(C/2C) → 0.

A morphism between two such quadruples is a triple ϕ = (f, g, h) where f : A → A′, g : C′ → C and h : Π → Π′ such that θ′f|_{T (A)} = e^{2}gθ, hα = α′[f]_{2} and ω′h = e^{2}[g]_{2}ω. The category Q of such quadruples is additive.

Define a duality functor by Q^{∗} = (A, C, θ, E)^{∗} = (C, A, e^{2}θ, e^{2}E) and (f, g, h)^{∗} = (g, f, e^{2}h). An isomorphism ϕ = (f, g, h) from Q to Q^{∗} is an ε-symmetric selfduality of Q if ϕ^{∗} = εϕ, in which case f and g are isomorphisms from A to C, f = εg and h = εe^{2}h. A pair (Q, ϕ) with ϕ an ε-symmetric self-duality of q is hyperbolic if there is an object N in Q such that Q ≅ N ⊕ N^{∗} and ϕ has matrix ( \begin{matrix}0 & ε \\ 1 & 0 \end{matrix} ) with respect to this decomposition.

Theorem 6. Farber quintuples for 2q-knots correspond bijectively to pairs (Q, ϕ) where Q = (A, C, θ, E) is an object in Q and ϕ = (f, g, h) is a (−1)^{q+1}-symmetric self-duality of Q such that α[γ(p)]_{2} = 2p for all p ∈ Π, where γ(p) is determined by e^{2}gθ(γ(p)(x) = σ_{Π}h(α([x]_{2})(p) for all x ∈ T (A) and p ∈ Π.

Proof. Let (A, Π, α, ℓ, ψ) be a Farber quintuple with (−1)^{q+1}-symmetric pairings ℓ and ψ and let β be the associated epimorphism. Let C = A, θ = σ_{A}^{−1}Ad(ℓ), ω = σ_{A}^{−1}β and E be the exact sequence determined by α and ω. Then Q = (A, A, θ, E) is in Q and (1_{A}, 1_{A}, σ_{Π}^{−1}Ad(ψ)) is a (−1)^{q+1}-symmetric self-duality of Q.

Conversely, if ϕ = (f, g, h) is a (−1)^{q+1}-symmetric self-duality of (A, C, θ, E) we may define pairings ℓ on T (A) and ψ on Π by setting ℓ(a, a′) = e^{2}gθ(a′)(a) for all a, a′ ∈ T (A) and ψ(p, q) = σ_{Π}(h(q))(p) for all p, q ∈ Π. (We then have β = σ_{A}e^{2}gω.) Then (A, Π, α, ℓ, ψ) is almost a Farber quintuple; we need only the additional condition that α[γ(p)]_{2} = 2p for all p ∈ Π, where γ is defined as in §5. The equation in the statement of the theorem is a reformulation of this condition in terms of the constituents of ϕ. □

The Krull-Schmidt Theorem holds in any additive category in which (i) all idempotents split; and (ii) every object is a finite sum of objects whose endomorphism rings are local rings. (Such summands are necessarily indecomposable). It is easily verified that Q satisfies the first condition. We shall determine the objects of Q which satisfy (ii) by means of the next lemma.

Lemma 7. Let R be a local Λ-algebra which is finitely generated as a Λ-module. Then R is finite.

Proof. Let p : Λ → R be the natural homomorphism. Let J = rad(R) = R − R^{×} and m = p^{−1}(J). The skewfield R/J is finitely generated as a Λ/m-module. It follows easily that Λ/m is a field. In particular m is a maximal ideal (and R/J is a finite field). Since p(Λ)_{m} ≤ R it is an integral extension of p(Λ), and every prime ideal of p(Λ) is the restriction of a prime ideal of p(Λ)_{m}, by Theorem 5.10 of [1]. Thus p(Λ) is also a local ring, and so m is the unique maximal ideal of Λ which contains K = Ker(p). Consideration of the primary decomposition of K shows that K must be m-primary and p(Λ) = Λ/K finite. Hence R is also finite. □

An object N in an abelian category C is a torsion object if the ring End_{C}(N) = C(N, N) is finite.

Theorem 8. Let Q = (A, C, θ, E) in Q. Then the following are equivalent (1) A and C are finite Λ-modules;

(2) Q is a torsion object;

(3) Q is a finite sum of objects whose endomorphism rings are local rings. The Krull-Schmidt Theorem holds in the full subcategory of Q determined by the torsion objects, and the endomorphism rings of such objects are radically complete.

Proof. It is immediate that End_{Q}(Q) has finite index in End_{Λ}(A) × End_{Λ}(C). Thus End_{Q}(Q) is finite if and only if End_{Λ}(A) and End_{Λ}(C) are both finite. It follows easily that Q is a torsion object in Q if and only if A and C are finite. Thus (1) ⇔ (2).

In a finite ring some power of each element is idempotent. Thus a finite ring with no nontrivial idempotents is local. Since idempotents in Q split it follows that (2) ⇒ (3).

If End_{Q}(Q) is local then it is finite, by the lemma, and so Q is a torsion object. Thus (3) ⇒ (2).

The final assertion is now clear, since the radical of an Artinian ring is nilpotent. □

If (A, C, θ, E) is a torsion object in Q then θ : A ≅ e^{2}C and e^{2}(C/2C) ≅ _{2}A, and so the full subcategory of Q determined by the torsion objects is essentially the category considered in [8].

9. Double null concordance

We expect that the following assertions should be true.

Theorem? A simple 2q-knot K with q ≥ 4 is doubly null concordant if and only if q_{F} (K) is hyperbolic.

The necessity of the condition should be easy.

Corollary? A simple 2q-knot K with q ≥ 4 is stably DNC if and only if the Witt class of q_{F} (K) is trivial.

The sufficiency of the condition would follow from the theorem and the fact that every Farber quintuple is realized by some simple 2q-knot. The necessity of the condition may require extending q_{F} (−) to an invariant of all even-dimensional knots.

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14 JONATHAN A. HILLMAN

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

E-mail address: jonh@maths.usyd.edu.au