Blanchfield form
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− | After Poincar\'{e} and Lefschetz, a closed manifold $N^{n}$ has a bilinear intersection form defined on its homology: | + | After Poincar\'{e} and Lefschetz, a closed oriented manifold $N^{n}$ has a bilinear intersection form defined on its homology: |
− | $$ | + | $$I_N \colon H_k(N;\mathbb{Z}) \times H_{n-k}(N;\mathbb{Z}) \to \mathbb{Z}; ([p],[q]) \mapsto \langle p, q \rangle$$ |
+ | such that | ||
+ | $$I_N(x,y) = (-)^{k(n-k)}I_N(y,x).$$ | ||
Given a ${k}$--chain $p \in C_{k}(N;\mathbb{Z})$ and an $(n-k)$--chain $q \in C_{n-k}(N;\mathbb{Z})$ which is transverse to $q$, the signed count of the intersections between $p$ and $q$ gives an intersection number $\langle\, p \, , \, q\, \rangle \in \mathbb{Z}$. | Given a ${k}$--chain $p \in C_{k}(N;\mathbb{Z})$ and an $(n-k)$--chain $q \in C_{n-k}(N;\mathbb{Z})$ which is transverse to $q$, the signed count of the intersections between $p$ and $q$ gives an intersection number $\langle\, p \, , \, q\, \rangle \in \mathbb{Z}$. | ||
By bilinearity, the intersection form vanishes on the torsion part of the homology. The analogue of the intersection pairing for the torsion part of the homology of a closed manifold $N^n$ is the bilinear $\Q/\mathbb{Z}$--valued linking form, which is due to Seifert: | By bilinearity, the intersection form vanishes on the torsion part of the homology. The analogue of the intersection pairing for the torsion part of the homology of a closed manifold $N^n$ is the bilinear $\Q/\mathbb{Z}$--valued linking form, which is due to Seifert: | ||
− | $$L_N \colon TH_{\ell}(N;\mathbb{Z}) \times TH_{n-\ell-1}(N;\mathbb{Z}) \to \Q/\mathbb{Z}.$$ | + | $$L_N \colon TH_{\ell}(N;\mathbb{Z}) \times TH_{n-\ell-1}(N;\mathbb{Z}) \to \Q/\mathbb{Z}$$ |
+ | such that | ||
+ | $$L_N(x,y) = (-)^{\ell(n-\ell-1)}L_N(y,x).$$ | ||
Revision as of 17:23, 6 January 2013
This page has not been refereed. The information given here might be incomplete or provisional. |
1 Introduction
After Poincar\'{e} and Lefschetz, a closed oriented manifold has a bilinear intersection form defined on its homology:
such that
Given a --chain and an --chain which is transverse to , the signed count of the intersections between and gives an intersection number .
By bilinearity, the intersection form vanishes on the torsion part of the homology. The analogue of the intersection pairing for the torsion part of the homology of a closed manifold is the bilinear --valued linking form, which is due to Seifert:
such that
Given and represented by cycles , let be such that , for some . Then we define:
The resulting element is independent of the choices of and .
As an example, let , so that and . Now . Let be the non--trivial element. To compute the linking , consider modelled as , with antipodal points on identified, and choose two representative --chains and for . Let be the straight line between north and south poles and let be half of the equator. Now , where is the 2--disk whose boundary is the equator. We see that , so that
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.
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 setThe Blanchfield form is a sesquilinear --valued form 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.
We now turn to an example. For simplicity 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 Poincar\'{e}--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.
One of the Blanchfield form's main applications is in knot concordance, a notion first defined in [Fox&Milnor1959]. 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 [Kearton], which is equivalent to Levine's Seifert form obstruction [Levine], but which is more intrinsic, since for a given knot there are many Seifert surfaces but only one knot exterior. The proof that 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 mapFor high--dimensional knots, , where , the Blanchfield form is metabolic if and only if is slice [Levine, Kearton]. Levine [Levine2] 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 [RanickiHighDimKnotTheory].
For classical knots in the --sphere, there are many non--slice knots with metabolic Blanchfield form, the first of which were found in [CassonGordon]. Cochran, Orr and Teichner [COT] defined an infinite filtration of the knot concordance group, each of whose associated graded groups has infinite rank [COT2, 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 r\^{o}le 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 . Then the kernel \[P:= \ker(H_1(M_K;\Q[\mathbb{Z}]) \to H_1(W;\Q[\mathbb{Z}]))\] is a metaboliser for the rational Blanchfield form of .
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 .
Theorem 1.1 Cochran-Orr-Teichner. Let be a slice knot. Then there exists a metaboliser for the Blanchfield form of such that for each there is a representation for which .
It is somewhat remarkable that the classical Blanchfield form continues to have new and interesting applications. For example, Borodzik and Friedl [BorodzikFriedl12I, BorodzikFriedl12II] recently used the minimal size of a matrix which represents the Blanchfield form of a given knot to compute many previously unknown unknotting numbers of low crossing number knots.
2 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
- [BorodzikFriedl12I, BorodzikFriedl12II] Template:BorodzikFriedl12I, BorodzikFriedl12II
- [COT] Template:COT
- [COT2, Cochran&Teichner2007, Cochran&Harvey&Leidy2009] Template:COT2, Cochran&Teichner2007, Cochran&Harvey&Leidy2009
- [CassonGordon] Template:CassonGordon
- [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
- [Fox&Milnor1959] Template:Fox&Milnor1959
- [Kearton] Template:Kearton
- [Levine] Template:Levine
- [Levine, Kearton] Template:Levine, Kearton
- [Levine2] Template:Levine2
- [RanickiHighDimKnotTheory] Template:RanickiHighDimKnotTheory
- [Reidemeister1939] K. Reidemeister, Durchschnitt und Schnitt von Homotopieketten, Monatsh. Math. Phys. 48 (1939), 226–239. MR0000634 (1,105h) Zbl 0021.43104
such that
Given a --chain and an --chain which is transverse to , the signed count of the intersections between and gives an intersection number .
By bilinearity, the intersection form vanishes on the torsion part of the homology. The analogue of the intersection pairing for the torsion part of the homology of a closed manifold is the bilinear --valued linking form, which is due to Seifert:
such that
Given and represented by cycles , let be such that , for some . Then we define:
The resulting element is independent of the choices of and .
As an example, let , so that and . Now . Let be the non--trivial element. To compute the linking , consider modelled as , with antipodal points on identified, and choose two representative --chains and for . Let be the straight line between north and south poles and let be half of the equator. Now , where is the 2--disk whose boundary is the equator. We see that , so that
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.
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 setThe Blanchfield form is a sesquilinear --valued form 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.
We now turn to an example. For simplicity 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 Poincar\'{e}--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.
One of the Blanchfield form's main applications is in knot concordance, a notion first defined in [Fox&Milnor1959]. 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 [Kearton], which is equivalent to Levine's Seifert form obstruction [Levine], but which is more intrinsic, since for a given knot there are many Seifert surfaces but only one knot exterior. The proof that 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 mapFor high--dimensional knots, , where , the Blanchfield form is metabolic if and only if is slice [Levine, Kearton]. Levine [Levine2] 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 [RanickiHighDimKnotTheory].
For classical knots in the --sphere, there are many non--slice knots with metabolic Blanchfield form, the first of which were found in [CassonGordon]. Cochran, Orr and Teichner [COT] defined an infinite filtration of the knot concordance group, each of whose associated graded groups has infinite rank [COT2, 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 r\^{o}le 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 . Then the kernel \[P:= \ker(H_1(M_K;\Q[\mathbb{Z}]) \to H_1(W;\Q[\mathbb{Z}]))\] is a metaboliser for the rational Blanchfield form of .
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 .
Theorem 1.1 Cochran-Orr-Teichner. Let be a slice knot. Then there exists a metaboliser for the Blanchfield form of such that for each there is a representation for which .
It is somewhat remarkable that the classical Blanchfield form continues to have new and interesting applications. For example, Borodzik and Friedl [BorodzikFriedl12I, BorodzikFriedl12II] recently used the minimal size of a matrix which represents the Blanchfield form of a given knot to compute many previously unknown unknotting numbers of low crossing number knots.
2 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
- [BorodzikFriedl12I, BorodzikFriedl12II] Template:BorodzikFriedl12I, BorodzikFriedl12II
- [COT] Template:COT
- [COT2, Cochran&Teichner2007, Cochran&Harvey&Leidy2009] Template:COT2, Cochran&Teichner2007, Cochran&Harvey&Leidy2009
- [CassonGordon] Template:CassonGordon
- [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
- [Fox&Milnor1959] Template:Fox&Milnor1959
- [Kearton] Template:Kearton
- [Levine] Template:Levine
- [Levine, Kearton] Template:Levine, Kearton
- [Levine2] Template:Levine2
- [RanickiHighDimKnotTheory] Template:RanickiHighDimKnotTheory
- [Reidemeister1939] K. Reidemeister, Durchschnitt und Schnitt von Homotopieketten, Monatsh. Math. Phys. 48 (1939), 226–239. MR0000634 (1,105h) Zbl 0021.43104