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.
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.
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 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
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 [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
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
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
- [Cochran&Orr&Teichner2003] T. D. Cochran, K. E. Orr and P. Teichner, Knot concordance, Whitney towers and -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
- [Cochran&Teichner2007] T. D. Cochran and P. Teichner, Knot concordance and von Neumann -invariants, Duke Math. J. 137 (2007), no.2, 337–379. MR2309149 (2008f:57005) Zbl 1186.57006
- [Fox&Milnor1966] R. H. Fox and J. W. Milnor, Singularities of -spheres in -space and cobordism of knots, Osaka J. Math. 3 (1966), 257–267. MR0211392 (35 #2273) Zbl 0146.45501
- [Kearton1975] C. Kearton, Cobordism of knots and Blanchfield duality, J. London Math. Soc. (2) 10 (1975), no.4, 406–408. MR0385873 (52 #6732) Zbl 0305.57015
- [Levine1969a] J. Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969), 229–244. MR0246314 (39 #7618) Zbl 0176.22101
- [Levine1977] J. Levine, Knot modules. I, Trans. Amer. Math. Soc. 229 (1977), 1–50. MR0461518 (57 #1503) Zbl 0422.57007
- [Ranicki1998] A. Ranicki, High-dimensional knot theory, Springer-Verlag, 1998. MR1713074 (2000i:57044) Zbl 1059.19003
- [Ranicki2003] A. Ranicki, Blanchfield and Seifert algebra in high-dimensional knot theory, Mosc. Math. J. 3 (2003), no.4, 1333–1367. MR2058802 (2005b:19008) Zbl 1059.19003
- [Reidemeister1939] K. Reidemeister, Durchschnitt und Schnitt von Homotopieketten, Monatsh. Math. Phys. 48 (1939), 226–239. MR0000634 (1,105h) Zbl 0021.43104
- [Trotter1973] H. F. Trotter, On -equivalence of Seifert matrices, Invent. Math. 20 (1973), 173–207. MR0645546 (58 #31100) Zbl 0269.15009
- [Trotter1978] H. F. Trotter, Knot module and Seifert matrices, Knot theory (Proc. Sem., Plans-sur-Bex, 1977), Springer (1978), 291–299. MR521739 (81c:57019) Zbl 407.57015