Blanchfield form

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


[edit] 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}
\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.

[edit] 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 \mathbb{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 \mathbb{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 \mathbb{Q}(\Gamma)/\mathbb{Z}[\Gamma].
\displaystyle [x] \in TH_\ell(\widetilde{X};\mathbb{Z}) \cong TH_\ell(X;\mathbb{Z}[\Gamma])
\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 \mathbb{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 the situation with the linking form. 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.

[edit] 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 \mathbb{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;\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}} 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.

[edit] 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 4.1 (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 form page, for the definition of a presentation of a linking form.

[edit] 5 References

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