Blanchfield form

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which arise from the long exact sequence of a pair, equivariant Poincaré-Lefschetz duality, a Bockstein homomorphism, and universal coefficients. Showing that these maps are isomorphisms proves that $\mathop{\mathrm{Bl}}$ is non-singular. A good exercise is to trace through this sequence of isomorphisms to check that it really does coincide with the definition of the Blanchfield form given above.
which arise from the long exact sequence of a pair, equivariant Poincaré-Lefschetz duality, a Bockstein homomorphism, and universal coefficients. Showing that these maps are isomorphisms proves that $\mathop{\mathrm{Bl}}$ is non-singular. A good exercise is to trace through this sequence of isomorphisms to check that it really does coincide with the definition of the Blanchfield form given above.
A formula for the Blanchfield pairing of a knot in terms of a Seifert matrix for it is given in \cite{Kearton1975}. The algebraic relationship between Seifert matrices and the Blanchfield pairing was studied in \cite{Trotter1978} and \cite{Ranicki2003}.
+
A formula for the Blanchfield pairing of a knot in terms of a Seifert matrix for it is given in \cite{Kearton1975}. The algebraic relationship between Seifert matrices and the Blanchfield pairing was studied in \cite{Trotter1973},\cite{Trotter1978} and \cite{Ranicki2003}. In particular, every Blanchfield form is determined by a Seifert matrix, and two Seifert matrices are $S$-equivalent (resp. cobordant) if and only if they determine isomorphic (resp. cobordant) Blanchfield forms.

Revision as of 09:22, 30 March 2013

This page has not been refereed. The information given here might be incomplete or provisional.

1 Background: equivariant intersection forms

In 1939 Reidemeister [Reidemeister1939] defined an equivariant, sesquilinear intersection form on the homology of a covering space \widetilde{M} of an m-dimensional closed manifold M whose deck transformation group G is abelian.

\displaystyle \begin{array}{rcl} I_{\widetilde{M}} \colon H_k(\widetilde{M};\mathbb{Z}) \times H_{m-k}(\widetilde{M};\mathbb{Z}) &\to & \mathbb{Z}[G]; \\ ([p],[q]) & \mapsto &  \sum_{g \in G} \langle g \cdot p, q \rangle g^{-1}, \end{array}
where
\displaystyle \langle \cdot , \cdot \rangle \colon C_k(\widetilde{M};\mathbb{Z}) \times C_{m-k}(\widetilde{M};\mathbb{Z}) \to \mathbb{Z}
counts transverse intersections between chains algebraically.

The intersections of each possible G-translate of p and q are counted, and indexed according to the deck transformation which produced that intersection number.


2 Definition of the Blanchfield form

In his 1954 Princeton PhD thesis R. C. Blanchfield [Blanchfield1957] made the corresponding generalisation for linking forms. Let X^{m} be a compact manifold, now possibly with non-empty boundary, with a surjective homomorphism \pi_1(X) \to \Gamma, for some free abelian group \Gamma. Let \mathbb{Z}[\Gamma] be the group ring of \Gamma and let \Q(\Gamma) be its field of fractions.

The \mathbb{Z}[\Gamma]-torsion submodule of a \mathbb{Z}[\Gamma] module P is the submodule
\displaystyle TP:= \{p \in P \,|\, ap=0 \text{ for some } a \in \mathbb{Z}[\Gamma]\}.

The Blanchfield form is a sesquilinear \Q(\Gamma)/\mathbb{Z}[\Gamma]--valued pairing which is defined on the \mathbb{Z}[\Gamma]-torsion submodules of the homology of the \Gamma--cover \widetilde{X} of X:

\displaystyle \mathop{\mathrm{Bl}} \colon TH_\ell(\widetilde{X};\mathbb{Z}) \times TH_{m-\ell-1}(\widetilde{X},\partial\widetilde{X};\mathbb{Z}) \to \Q(\Gamma)/\mathbb{Z}[\Gamma].
Given
\displaystyle [x] \in TH_\ell(\widetilde{X};\mathbb{Z}) \cong TH_\ell(X;\mathbb{Z}[\Gamma])
and
\displaystyle [y] \in TH_{m-\ell-1}(\widetilde{X},\partial \widetilde{X};\mathbb{Z}) \cong TH_{m-\ell-1}(X,\partial X;\mathbb{Z}[\Gamma])
represented by cycles x \in C_\ell(\widetilde{X};\mathbb{Z}) and y \in C_{m-\ell-1}(\widetilde{X},\partial \widetilde{X};\mathbb{Z}), let w \in C_{m-\ell}(\widetilde{X},\partial\widetilde{X};\mathbb{Z}) be such that \partial w = \Delta y, for some \Delta \in \mathbb{Z}[\Gamma]. Then we define:
\displaystyle \mathop{\mathrm{Bl}}([x],[y]) := \sum_{g \in \Gamma} \langle g \cdot x, w \rangle g^{-1}/\Delta \in \Q(\Gamma)/\mathbb{Z}[\Gamma],

where \Gamma acts on C_\ell(\widetilde{X};\mathbb{Z}) by the action induced from the deck transformation.

The Blanchfield form can also be defined via homology, using a Bockstein homomorphism and a cup product. This is entirely analogous to the treatment on the linking forms page, Section 3. Just change \mathbb{Z} to \mathbb{Z}[\mathbb{Z}] and \mathbb{Q} to \mathbb{Q}(\mathbb{Z}). In the case of knot exteriors, a homological definition is also given below.

3 Example of knot exteriors

We now turn to an example. We will focus on the case of knots in S^3. For a knot K \subset S^3, let X_K denote its exterior, which is the complement of a regular neighbourhood of K: X_K:= S^3 \setminus \nu K. Now \ell=1, m=3 and the abelianisation gives a homomorphism \pi_1(X_K) \to H_1(X_K;\mathbb{Z}) \xrightarrow{\simeq} \mathbb{Z}. The Blanchfield form can in this case be defined without relative homology, on H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \cong H_1(\widetilde{X}_K;\mathbb{Z}). The form

\displaystyle \mathop{\mathrm{Bl}} \colon H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \times H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \to \Q(\mathbb{Z})/\mathbb{Z}[\mathbb{Z}]

is non--singular, sesquilinear and Hermitian. Note that H_1(X_K;\mathbb{Z}[\mathbb{Z}]) is entirely \mathbb{Z}[\mathbb{Z}]-torsion, so H_1(X_K;\mathbb{Z}[\mathbb{Z}]) = TH_1(X_K;\mathbb{Z}[\mathbb{Z}]). The adjoint of this form is given by the following sequence of homomorphisms:

\displaystyle \begin{aligned} &H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \to H_1(X_K,\partial X_K;\mathbb{Z}[\mathbb{Z}]) \to H^2(X_K;\mathbb{Z}[\mathbb{Z}]) \\ & \to H^1(X_K;\Q(\mathbb{Z})/\Q[\mathbb{Z}]) \to \Hom_{\mathbb{Z}[\mathbb{Z}]}(H_1(X_K;\mathbb{Z}[\mathbb{Z}]),\Q(\mathbb{Z})/\Q[\mathbb{Z}]), \end{aligned}

which arise from the long exact sequence of a pair, equivariant Poincaré-Lefschetz duality, a Bockstein homomorphism, and universal coefficients. Showing that these maps are isomorphisms proves that \mathop{\mathrm{Bl}} is non-singular. A good exercise is to trace through this sequence of isomorphisms to check that it really does coincide with the definition of the Blanchfield form given above.

A formula for the Blanchfield pairing of a knot in terms of a Seifert matrix for it is given in [Kearton1975]. The algebraic relationship between Seifert matrices and the Blanchfield pairing was studied in [Trotter1973],[Trotter1978] and [Ranicki2003]. In particular, every Blanchfield form is determined by a Seifert matrix, and two Seifert matrices are S-equivalent (resp. cobordant) if and only if they determine isomorphic (resp. cobordant) Blanchfield forms.


4 Applications to knot concordance

One of the Blanchfield form's main applications is in knot concordance, a notion first defined in [Fox&Milnor1966]. Two knots K,J are concordant if there is an annulus embedded in S^3 \times I whose boundary is K \times \{0\} \cup -J \times \{1\}. A knot which is concordant to the unknot is called a slice knot; equivalently a slice knot bounds an embedded disk in D^4. We say that a Blanchfield form is metabolic if there is a submodule P \subset H_1(X_K;\mathbb{Z}[\mathbb{Z}]) which is self-orthogonal with respect to \mathop{\mathrm{Bl}}, called a metaboliser. The Blanchfield form of a slice knot is metabolic, so that the Blanchfield form provides an obstruction to concordance [Kearton1975], which is equivalent to Levine's Seifert form obstruction [Levine1969a], but which is more intrinsic, since for a given knot there are potentially many Seifert surfaces but only one knot exterior. The proof that the Blanchfield form of a slice knot is metabolic rests on the observation that, if A \subset D^4 is a slice disk for K, the Blanchfield form vanishes on the kernel of the inclusion induced map
\displaystyle H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \to H_1(D^4\setminus \nu A;\mathbb{Z}[\mathbb{Z}]).

For high-dimensional knots, K \colon S^{2k-1} \subset S^{2k+1}, where k \geq 3, the Blanchfield form is metabolic if and only if K is slice [Levine1969a], [Kearton1975]. Levine [Levine1977] classified the modules which can arise as the homology of high-dimensional knots: the key property that a knot module must satisfy is Blanchfield duality. For a comprehensive account of the algebraic theory of high-dimensional knots, such as how the Blanchfield form can be used to compute the high-dimensional knot cobordism group, see [Ranicki1998].

For classical knots in the 3-sphere, there are many non-slice knots with metabolic Blanchfield form, the first of which were found in [Casson&Gordon1986]. Cochran, Orr and Teichner [Cochran&Orr&Teichner2003] defined an infinite filtration of the knot concordance group, each of whose associated graded groups has infinite rank [Cochran&Orr&Teichner2004], [Cochran&Teichner2007], [Cochran&Harvey&Leidy2009]. Their obstructions are obtained by defining representations into progressively more solvable groups. The Blanchfield form, and so-called higher order Blanchfield forms, play a crucial role in controlling the representations which extend from the knot exterior across a potential slice disc exterior W, whose existence one wishes to deny. Let M_K be the closed 3-manifold obtained from zero-framed surgery on S^3 along K. Note that \partial W = M_K. Then the kernel

\displaystyle P:= \ker(H_1(M_K;\mathbb{Q}[\mathbb{Z}]) \to H_1(W;\mathbb{Q}[\mathbb{Z}]))
is a metaboliser for the rational Blanchfield form of M_K. Therefore, given p \in P, the representation
\displaystyle H_1(M_K;\mathbb{Q}[\mathbb{Z}]) \to \mathbb{Q}(\mathbb{Z})/\mathbb{Q}[\mathbb{Z}];\; x \mapsto \mathop{\mathrm{Bl}}(p,x)

extends over W.

We give a special case of the results of [Cochran&Orr&Teichner2003] below, which shows the use of the Blanchfield form in an archetypal obstruction theorem for knot concordance problems. Given a closed 3-manifold M and a representation \phi \colon \pi_1(M) \to G there is defined a real number \rho^{(2)}(M,\phi) called the Cheeger-Gromov-Von Neumann \rho-invariant of (M,\phi), which can be computed using L^{(2)}-signatures of 4-manifolds with boundary M_K.

Theorem 8.1 (Cochran-Orr-Teichner). Let K be a slice knot. Then there exists a metaboliser P = P^{\bot} for the rational Blanchfield form of K such that for each p \in P there is a representation \phi_p \colon \pi_1(M_K) \to G for which \rho^{(2)}(M_K,\phi_p)=0.

Finally, it is somewhat remarkable that the classical Blanchfield form continues to have new and interesting applications. For example, Borodzik and Friedl [Borodzik&Friedl2012] recently used the minimal size of a square matrix which presents the Blanchfield form of a given knot to compute many previously unknown unknotting numbers of low crossing number knots. See the linking forms page, Section 6 for the definition of a presentation of a linking form.

References

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