Blanchfield form
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[edit] 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)

The intersections of each possible -translate of
and
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 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)

![\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)




![\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 the situation with the linking form. Just change to
and
to
. 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 . 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/c/5/f/c5f646cf3978122201f5ffd09c06f0d7.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.
[edit] 4 Applications to knot concordance




![P \subset H_1(X_K;\mathbb{Z}[\mathbb{Z}])](/images/math/9/3/8/938003a52a52d5c1016d2ad7d6deaa60.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)


![\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 4.1 (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 form page, for the definition of a presentation of a linking form.
[edit] 5 References
- [Blanchfield1957] R. C. Blanchfield, Intersection theory of manifolds with operators with applications to knot theory, Ann. of Math. (2) 65 (1957), 340–356. MR0085512 (19,53a) Zbl 0080.16601
- [Borodzik&Friedl2012] M. Borodzik and S. Friedl, The unknotting number and classical invariants I, (2012). Available at the arXiv:1203.3225.
- [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
- [Cochran&Harvey&Leidy2009] T. D. Cochran, S. Harvey and C. Leidy, Knot concordance and higher-order Blanchfield duality, Geom. Topol. 13 (2009), no.3, 1419–1482. MR2496049 (2009m:57006) Zbl 1175.57004
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