Blanchfield form

1 Background: equivariant intersection forms


$\displaystyle \begin{array}{rcl} I_{\widetilde{M}} \colon H_k(\widetilde{M};\mathbb{Z}) \times H_{m-k}(\widetilde{M};\mathbb{Z}) &\to & \mathbb{Z}[G]; \\ ([p],[q]) & \mapsto & \sum_{g \in G} \langle g \cdot p, q \rangle g^{-1}, \end{array}$
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$$G$-translate of $p$$p$ and $q$$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}$$X^{m}$ be a compact manifold, now possibly with non-empty boundary, with a surjective homomorphism $\pi_1(X) \to \Gamma$$\pi_1(X) \to \Gamma$, for some free abelian group $\Gamma$$\Gamma$. Let $\mathbb{Z}[\Gamma]$$\mathbb{Z}[\Gamma]$ be the group ring of $\Gamma$$\Gamma$ and let $\mathbb{Q}(\Gamma)$$\mathbb{Q}(\Gamma)$ be its field of fractions.

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

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

where $\Gamma$$\Gamma$ acts on $C_\ell(\widetilde{X};\mathbb{Z})$$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 the situation with the linking form. Just change $\mathbb{Z}$$\mathbb{Z}$ to $\mathbb{Z}[\mathbb{Z}]$$\mathbb{Z}[\mathbb{Z}]$ and $\mathbb{Q}$$\mathbb{Q}$ to $\mathbb{Q}(\mathbb{Z})$$\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$$S^3$. For a knot $K \subset S^3$$K \subset S^3$, let $X_K$$X_K$ denote its exterior, which is the complement of a regular neighbourhood of $K$$K$: $X_K:= S^3 \setminus \nu K$$X_K:= S^3 \setminus \nu K$. Now $\ell=1$$\ell=1$, $m=3$$m=3$ and the abelianisation gives a homomorphism $\pi_1(X_K) \to H_1(X_K;\mathbb{Z}) \xrightarrow{\simeq} \mathbb{Z}$$\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})$$H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \cong H_1(\widetilde{X}_K;\mathbb{Z})$. The form

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

is non--singular, sesquilinear and Hermitian. Note that $H_1(X_K;\mathbb{Z}[\mathbb{Z}])$$H_1(X_K;\mathbb{Z}[\mathbb{Z}])$ is entirely $\mathbb{Z}[\mathbb{Z}]$$\mathbb{Z}[\mathbb{Z}]$-torsion, so $H_1(X_K;\mathbb{Z}[\mathbb{Z}]) = TH_1(X_K;\mathbb{Z}[\mathbb{Z}])$$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;\mathbb{Q}(\mathbb{Z})/\mathbb{Q}[\mathbb{Z}]) \to \Hom_{\mathbb{Z}[\mathbb{Z}]}(H_1(X_K;\mathbb{Z}[\mathbb{Z}]),\mathbb{Q}(\mathbb{Z})/\mathbb{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}}$$\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$$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$$K,J$ are concordant if there is an annulus embedded in $S^3 \times I$$S^3 \times I$ whose boundary is $K \times \{0\} \cup -J \times \{1\}$$K \times \{0\} \cup -J \times \{1\}$. A knot which is concordant to the unknot is called a slice knot; equivalently a slice knot bounds an embedded disk in $D^4$$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}])$$P \subset H_1(X_K;\mathbb{Z}[\mathbb{Z}])$ which is self-orthogonal with respect to $\mathop{\mathrm{Bl}}$$\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$$A \subset D^4$ is a slice disk for $K$$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}$$K \colon S^{2k-1} \subset S^{2k+1}$, where $k \geq 3$$k \geq 3$, the Blanchfield form is metabolic if and only if $K$$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$$3$-sphere, there are many non-slice knots with metabolic Blanchfield form, the first of which were found in [Casson&Gordon1986]. Cochran, Orr and Teichner [Cochran&Orr&Teichner2003] defined an infinite filtration of the knot concordance group, each of whose associated graded groups has infinite rank [Cochran&Orr&Teichner2004], [Cochran&Teichner2007], [Cochran&Harvey&Leidy2009]. Their obstructions are obtained by defining representations into progressively more solvable groups. The Blanchfield form, and so-called higher order Blanchfield forms, play a crucial role in controlling the representations which extend from the knot exterior across a potential slice disc exterior $W$$W$, whose existence one wishes to deny. Let $M_K$$M_K$ be the closed 3-manifold obtained from zero-framed surgery on $S^3$$S^3$ along $K$$K$. Note that $\partial W = M_K$$\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$$M_K$. Therefore, given $p \in P$$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$$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$$M$ and a representation $\phi \colon \pi_1(M) \to G$$\phi \colon \pi_1(M) \to G$ there is defined a real number $\rho^{(2)}(M,\phi)$$\rho^{(2)}(M,\phi)$ called the Cheeger-Gromov-Von Neumann $\rho$$\rho$-invariant of $(M,\phi)$$(M,\phi)$, which can be computed using $L^{(2)}$$L^{(2)}$-signatures of 4-manifolds with boundary $M_K$$M_K$.

Theorem 4.1 (Cochran-Orr-Teichner). Let $K$$K$ be a slice knot. Then there exists a metaboliser $P = P^{\bot}$$P = P^{\bot}$ for the rational Blanchfield form of $K$$K$ such that for each $p \in P$$p \in P$ there is a representation $\phi_p \colon \pi_1(M_K) \to G$$\phi_p \colon \pi_1(M_K) \to G$ for which $\rho^{(2)}(M_K,\phi_p)=0$$\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 form page, for the definition of a presentation of a linking form.