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 spaceTex syntax errorof an -dimensional closed manifold whose deck transformation group is abelian.
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.
The -torsion submodule of a module is the submoduleThe Blanchfield form is a sesquilinear --valued pairing which is defined on the -torsion submodules of the homology of the --cover of :
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
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:
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 [Trotter1978] and [Ranicki2003].
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 are concordant if there is an annulus embedded in whose boundary is . A knot which is concordant to the unknot is called a slice knot; equivalently a slice knot bounds an embedded disk in . We say that a Blanchfield form is metabolic if there is a submodule which is self-orthogonal with respect to , 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 is a slice disk for , the Blanchfield form vanishes on the kernel of the inclusion induced mapFor 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 exteriorTex syntax error, whose existence one wishes to deny. Let be the closed 3-manifold obtained from zero-framed surgery on along . Note that . Then the kernel
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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
- [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
- [Cochran&Orr&Teichner2003] T. D. Cochran, K. E. Orr and P. Teichner, Knot concordance, Whitney towers and
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-signatures, Ann. of Math. (2) 157 (2003), no.2, 433–519. MR1973052 (2004i:57003) Zbl 1044.57001 - [Cochran&Orr&Teichner2004] T. D. Cochran, K. E. Orr and P. Teichner, Structure in the classical knot concordance group, Comment. Math. Helv. 79 (2004), no.1, 105–123. MR2031301 (2004k:57005) Zbl 1061.57008
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