Blanchfield form
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1 Background: equivariant intersection forms
In 1939 Reidemeister [Reidemeister1939] defined an equivariant, sesquilinear intersection form on the homology of a covering space of an
-dimensional closed manifold
whose deck transformation group
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}](/images/math/9/5/f/95ff1aef5a0aeceaa0a316f97a96c1d3.png)
![\displaystyle \langle \cdot , \cdot \rangle \colon C_k(\widetilde{M};\mathbb{Z}) \times C_{m-k}(\widetilde{M};\mathbb{Z}) \to \mathbb{Z}](/images/math/6/8/4/684e5fd2ebba69477048dee7ecf2fae6.png)
The intersections of each possible -translate of
and
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 be a compact manifold, now possibly with non-empty boundary, with a surjective homomorphism
, for some free abelian group
. Let
be the group ring of
and let
be its field of fractions.
![\mathbb{Z}[\Gamma]](/images/math/a/3/1/a3125186741639dcc06e77c618736635.png)
![\mathbb{Z}[\Gamma]](/images/math/a/3/1/a3125186741639dcc06e77c618736635.png)
![P](/images/math/6/b/5/6b52835f794dc38160c3157e48761ad3.png)
![\displaystyle TP:= \{p \in P \,|\, ap=0 \text{ for some } a \in \mathbb{Z}[\Gamma]\}.](/images/math/2/3/6/236be0676d35583ae0d92faf468aa3cc.png)
The Blanchfield form is a sesquilinear --valued pairing which is defined on the
-torsion submodules of the homology of the
--cover
of
:
![\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].](/images/math/6/8/e/68e7badc994282925b3d2464324f747e.png)
![\displaystyle [x] \in TH_\ell(\widetilde{X};\mathbb{Z}) \cong TH_\ell(X;\mathbb{Z}[\Gamma])](/images/math/d/5/5/d55dd349382e40c3e000068222d15ffe.png)
![\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])](/images/math/a/1/2/a1287db438b45a10e05cb69d69c33415.png)
![x \in C_\ell(\widetilde{X};\mathbb{Z})](/images/math/3/c/5/3c5081dc66d23a011d6e189319f5a58d.png)
![y \in C_{m-\ell-1}(\widetilde{X},\partial \widetilde{X};\mathbb{Z})](/images/math/c/e/4/ce475ea5436509211d4e1fb136010b36.png)
![w \in C_{m-\ell}(\widetilde{X},\partial\widetilde{X};\mathbb{Z})](/images/math/e/f/4/ef420d17dc2c295971dbdd458c098afc.png)
![\partial w = \Delta y](/images/math/9/6/d/96d319a4d370b2807417c4e1e291a7b1.png)
![\Delta \in \mathbb{Z}[\Gamma]](/images/math/0/3/a/03a07519b417c559188d4f50e3899ac2.png)
![\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],](/images/math/a/f/3/af39d01e2093f30d56fb279124d600f7.png)
where acts on
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 to
and
to
. 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 . For a knot
, let
denote its exterior, which is the complement of a regular neighbourhood of
:
. Now
,
and the abelianisation gives a homomorphism
. The Blanchfield form can in this case be defined without relative homology, on
. 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}]](/images/math/8/3/b/83bf910881a456853f02bd9e86db3391.png)
is non--singular, sesquilinear and Hermitian. Note that is entirely
-torsion, so
. 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}](/images/math/f/f/3/ff3d3ed36b27c13baf9f59dd8c075d0e.png)
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 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 -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](/images/math/9/3/b/93b70fef15c2a238240720ea02b37d3c.png)
![S^3 \times I](/images/math/b/0/4/b046968e2ff58e7dbce3f942ab46bd32.png)
![K \times \{0\} \cup -J \times \{1\}](/images/math/d/3/4/d34f851b0f34f0cbc73243564d3b0752.png)
![D^4](/images/math/0/5/7/05756a0b7691f04a889e51706e300415.png)
![P \subset H_1(X_K;\mathbb{Z}[\mathbb{Z}])](/images/math/9/3/8/938003a52a52d5c1016d2ad7d6deaa60.png)
![\mathop{\mathrm{Bl}}](/images/math/0/5/0/05030d9fd7c65c3c462a5e701a86a2c8.png)
![A \subset D^4](/images/math/d/f/3/df34f85abeb8110ea51e972f90d46284.png)
![K](/images/math/8/6/7/8673dabfddfa6347e822f593dfa1d916.png)
![\displaystyle H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \to H_1(D^4\setminus \nu A;\mathbb{Z}[\mathbb{Z}]).](/images/math/6/0/b/60b126ae6a3f36837e2e3b54015454e9.png)
For high-dimensional knots, , where
, the Blanchfield form is metabolic if and only if
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 -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
, whose existence one wishes to deny. Let
be the closed 3-manifold obtained from zero-framed surgery on
along
. Note that
. Then the kernel
![\displaystyle P:= \ker(H_1(M_K;\mathbb{Q}[\mathbb{Z}]) \to H_1(W;\mathbb{Q}[\mathbb{Z}]))](/images/math/0/2/c/02ccdfcc84890179b9f6e1a8562afd3d.png)
![M_K](/images/math/8/2/4/82445669aaddbb28e620b10ae7d3e1f3.png)
![p \in P](/images/math/f/2/9/f29a1fee548460c6ec7ff288a67e9b5b.png)
![\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)](/images/math/e/d/a/eda3ac9ad1e52f718ad1762a9101c66b.png)
extends over .
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 and a representation
there is defined a real number
called the Cheeger-Gromov-Von Neumann
-invariant of
, which can be computed using
-signatures of 4-manifolds with boundary
.
Theorem 8.1 (Cochran-Orr-Teichner).
Let be a slice knot. Then there exists a metaboliser
for the rational Blanchfield form of
such that for each
there is a representation
for which
.
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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- [Casson&Gordon1986] A. J. Casson and C. M. Gordon, Cobordism of classical knots, with an appendix by P. M. Gilmer. Progr. Math., 62, À la recherche de la topologie perdue, 181–199, Birkhäuser Boston, Boston, MA, 1986. MR900252
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