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

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${{Stub}} <wikitex>; == Background: equivariant intersection forms== In 1939 Reidemeister {{cite|Reidemeister1939}} defined an equivariant, sesquilinear intersection form on the homology of a covering space $\widetilde{M}$ of an $m$-dimensional closed manifold $M$ whose deck transformation group $G$ is abelian. $$\begin{array}{rcl} I_{\widetilde{M}} \colon H_k(\widetilde{M};\mathbb{Z}) \times H_{m-k}(\widetilde{M};\mathbb{Z}) &\to & \mathbb{Z}[G]; \ ([p],[q]) & \mapsto & \sum_{g \in G} \langle g \cdot p, q \rangle g^{-1}. \end{array}$$ where $$\langle \bullet , \bullet \rangle \colon C_k(\widetilde{M};\mathbb{Z}) \times C_{m-k}(\widetilde{M};\mathbb{Z}) \to \mathbb{Z}$$ counts transverse intersections between chains algebraically. The intersections of each possible $G$-translate of $p$ and $q$ are counted, and indexed according to the deck transformation which produced that intersection number. ==Definition of the Blanchfield form== In his 1954 Princeton PhD thesis R. C. Blanchfield {{cite|Blanchfield1957}} made the corresponding generalisation for linking forms. Let $X^{m}$ be a compact manifold, now possibly with non--empty boundary, with a surjective homomorphism $\pi_1(X) \to \Gamma$, for some free abelian group $\Gamma$. Let $\mathbb{Z}[\Gamma]$ be the group ring of $\Gamma$ and let $\Q(\Gamma)$ be its field of fractions. The $\mathbb{Z}[\Gamma]$-torsion submodule of a $\mathbb{Z}[\Gamma]$ module $P$ is the set $$TP:= \{p \in P \,|\, ap=0 \text{ for some } a \in \mathbb{Z}[\Gamma]\}.$$ The Blanchfield form is a sesquilinear $\Q(\Gamma)/\mathbb{Z}[\Gamma]$--valued pairing which is defined on the $\mathbb{Z}[\Gamma]$-torsion submodules of the homology of the $\Gamma$--cover $\widetilde{X}$ of $X$: $$\mathop{\mathrm{Bl}} \colon TH_\ell(\widetilde{X};\mathbb{Z}) \times TH_{m-\ell-1}(\widetilde{X},\partial\widetilde{X};\mathbb{Z}) \to \Q(\Gamma)/\mathbb{Z}[\Gamma].$$ Given $$[x] \in TH_\ell(\widetilde{X};\mathbb{Z}) \cong TH_\ell(X;\mathbb{Z}[\Gamma])$$ and $$[y] \in TH_{m-\ell-1}(\widetilde{X},\partial \widetilde{X};\mathbb{Z}) \cong TH_{m-\ell-1}(X,\partial X;\mathbb{Z}[\Gamma])$$ represented by cycles $x \in C_\ell(\widetilde{X};\mathbb{Z})$ and $y \in C_{m-\ell-1}(\widetilde{X},\partial \widetilde{X};\mathbb{Z})$, let $w \in C_{m-\ell}(\widetilde{X},\partial\widetilde{X};\mathbb{Z})$ be such that $\partial w = \Delta y$, for some $\Delta \in \mathbb{Z}[\Gamma]$. Then we define: $$\mathop{\mathrm{Bl}}([x],[y]) := \sum_{g \in \Gamma} \langle g \cdot x, w \rangle g^{-1}/\Delta \in \Q(\Gamma)/\mathbb{Z}[\Gamma],$$ where $\Gamma$ acts on $C_\ell(\widetilde{X};\mathbb{Z})$ by the action induced from the deck transformation. ==Example of knot exteriors== We now turn to an example. We will focus on the case of knots in $S^3$. For a knot $K \subset S^3$, let $X_K$ denote its exterior, which is the complement of a regular neighbourhood of $K$: $X_K:= S^3 \setminus \nu K$. Now $\ell=1$, $m=3$ and the abelianisation gives a homomorphism $\pi_1(X_K) \to H_1(X_K;\mathbb{Z}) \xrightarrow{\simeq} \mathbb{Z}$. The Blanchfield form can in this case be defined without relative homology, on $H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \cong H_1(\widetilde{X}_K;\mathbb{Z})$. The form $$\mathop{\mathrm{Bl}} \colon H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \times H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \to \Q(\mathbb{Z})/\mathbb{Z}[\mathbb{Z}]$$ is non--singular, sesquilinear and Hermitian. Note that $H_1(X_K;\mathbb{Z}[\mathbb{Z}])$ is entirely $\mathbb{Z}[\mathbb{Z}]$--torsion, so $H_1(X_K;\mathbb{Z}[\mathbb{Z}]) = TH_1(X_K;\mathbb{Z}[\mathbb{Z}])$. The adjoint of this form is given by the following sequence of homomorphisms: $$\begin{aligned} &H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \to H_1(X_K,\partial X_K;\mathbb{Z}[\mathbb{Z}]) \to H^2(X_K;\mathbb{Z}[\mathbb{Z}]) \ & \to H^1(X_K;\Q(\mathbb{Z})/\Q[\mathbb{Z}]) \to \Hom_{\mathbb{Z}[\mathbb{Z}]}(H_1(X_K;\mathbb{Z}[\mathbb{Z}]),\Q(\mathbb{Z})/\Q[\mathbb{Z}]), \end{aligned} $$ 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 $\mathop{\mathrm{Bl}}$ 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. ==Applications to knot concordance== One of the Blanchfield form's main applications is in knot concordance, a notion first defined in \cite{Fox&Milnor1966}. Two knots $K,J$ are concordant if there is an annulus embedded in $S^3 \times I$ whose boundary is $K \times \{0\} \cup -J \times \{1\}$. A knot which is concordant to the unknot is called a slice knot; equivalently a slice knot bounds an embedded disk in $D^4$. We say that a Blanchfield form is metabolic if there is a submodule $P \subset H_1(X_K;\mathbb{Z}[\mathbb{Z}])$ which is self--orthogonal with respect to $\mathop{\mathrm{Bl}}$, called a metaboliser. The Blanchfield form of a slice knot is metabolic, so that the Blanchfield form provides an obstruction to concordance \cite{Kearton1975}, which is equivalent to Levine's Seifert form obstruction \cite{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 $A \subset D^4$ is a slice disk for $K$, the Blanchfield form vanishes on the kernel of the inclusion induced map $$H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \to H_1(D^4\setminus \nu A;\mathbb{Z}[\mathbb{Z}]).$$ For high-dimensional knots, $K \colon S^{2k-1} \subset S^{2k+1}$, where $k \geq 3$, the Blanchfield form is metabolic if and only if $K$ is slice \cite{Levine1969a}, \cite{Kearton1975}. Levine \cite{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 \cite{Ranicki1998}. For classical knots in the $-sphere, there are many non-slice knots with metabolic Blanchfield form, the first of which were found in \cite{Casson&Gordon1986}. Cochran, Orr and Teichner \cite{Cochran&Orr&Teichner2003} defined an infinite filtration of the knot concordance group, each of whose associated graded groups has infinite rank \cite{Cochran&Orr&Teichner2004}, \cite{Cochran&Teichner2007}, \cite{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 $W$, whose existence one wishes to deny. Let $M_K$ be the closed 3-manifold obtained from zero-framed surgery on $S^3$ along $K$. Note that $\partial W = M_K$. Then the kernel $$P:= \ker(H_1(M_K;\mathbb{Q}[\mathbb{Z}]) \to H_1(W;\mathbb{Q}[\mathbb{Z}]))$$ is a metaboliser for the rational Blanchfield form of $M_K$. Therefore, given $p \in P$, the representation $$H_1(M_K;\mathbb{Q}[\mathbb{Z}]) \to \mathbb{Q}(\mathbb{Z})/\mathbb{Q}[\mathbb{Z}];\; x \mapsto \mathop{\mathrm{Bl}}(p,x)$$ extends over $W$. We give a special case of the results of \cite{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 $M$ and a representation $\phi \colon \pi_1(M) \to G$ there is defined a real number $\rho^{(2)}(M,\phi)$ called the Cheeger-Gromov-Von Neumann $\rho$-invariant of $(M,\phi)$, which can be computed using $L^{(2)}$-signatures of 4-manifolds with boundary $M_K$. {{beginthm |Theorem |(Cochran-Orr-Teichner)}} Let $K$ be a slice knot. Then there exists a metaboliser $P = P^{\bot}$ for the Blanchfield form of $K$ such that for each $p \in P$ there is a representation $\phi_p \colon \pi_1(M_K) \to G$ for which $\rho^{(2)}(M_K,\phi_p)=0$. {{endthm}} It is somewhat remarkable that the classical Blanchfield form continues to have new and interesting applications. For example, Borodzik and Friedl \cite{Borodzik&Friedl2012} recently used the minimal size of a square matrix which represents the Blanchfield form of a given knot to compute many previously unknown unknotting numbers of low crossing number knots. </wikitex> == References == {{#RefList:}} [[Category:Theory]] [[Category:Definitions]]\widetilde{M}$ of an $m$ $m$ -dimensional closed manifold $M$ $M$ whose deck transformation group $G$ $G$ is abelian.

$\displaystyle \begin{array}{rcl} I_{\widetilde{M}} \colon H_k(\widetilde{M};\mathbb{Z}) \times H_{m-k}(\widetilde{M};\mathbb{Z}) &\to & \mathbb{Z}[G]; \\ ([p],[q]) & \mapsto & \sum_{g \in G} \langle g \cdot p, q \rangle g^{-1}. \end{array}$ $\displaystyle \begin{array}{rcl} I_{\widetilde{M}} \colon H_k(\widetilde{M};\mathbb{Z}) \times H_{m-k}(\widetilde{M};\mathbb{Z}) &\to & \mathbb{Z}[G]; \\ ([p],[q]) & \mapsto & \sum_{g \in G} \langle g \cdot p, q \rangle g^{-1}. \end{array}$

where

$\displaystyle \langle \bullet , \bullet \rangle \colon C_k(\widetilde{M};\mathbb{Z}) \times C_{m-k}(\widetilde{M};\mathbb{Z}) \to \mathbb{Z}$ $\displaystyle \langle \bullet , \bullet \rangle \colon C_k(\widetilde{M};\mathbb{Z}) \times C_{m-k}(\widetilde{M};\mathbb{Z}) \to \mathbb{Z}$

counts transverse intersections between chains algebraically.

The intersections of each possible $G$ -translate of $p$ and $q$ 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 $X^{m}$ be a compact manifold, now possibly with non-empty boundary, with a surjective homomorphism $\pi_1(X) \to \Gamma$ , for some free abelian group $\Gamma$ . Let $\mathbb{Z}[\Gamma]$ be the group ring of $\Gamma$ and let $\Q(\Gamma)$ be its field of fractions.

The $\mathbb{Z}[\Gamma]$ $\mathbb{Z}[\Gamma]$ -torsion submodule of a $\mathbb{Z}[\Gamma]$ $\mathbb{Z}[\Gamma]$ module $P$ $P$ is the set

$\displaystyle TP:= \{p \in P \,|\, ap=0 \text{ for some } a \in \mathbb{Z}[\Gamma]\}.$ $\displaystyle TP:= \{p \in P \,|\, ap=0 \text{ for some } a \in \mathbb{Z}[\Gamma]\}.$

The Blanchfield form is a sesquilinear $\Q(\Gamma)/\mathbb{Z}[\Gamma]$ --valued pairing which is defined on the $\mathbb{Z}[\Gamma]$ -torsion submodules of the homology of the $\Gamma$ --cover $\widetilde{X}$ of $X$ :

$\displaystyle \mathop{\mathrm{Bl}} \colon TH_\ell(\widetilde{X};\mathbb{Z}) \times TH_{m-\ell-1}(\widetilde{X},\partial\widetilde{X};\mathbb{Z}) \to \Q(\Gamma)/\mathbb{Z}[\Gamma].$ $\displaystyle \mathop{\mathrm{Bl}} \colon TH_\ell(\widetilde{X};\mathbb{Z}) \times TH_{m-\ell-1}(\widetilde{X},\partial\widetilde{X};\mathbb{Z}) \to \Q(\Gamma)/\mathbb{Z}[\Gamma].$

Given

$\displaystyle [x] \in TH_\ell(\widetilde{X};\mathbb{Z}) \cong TH_\ell(X;\mathbb{Z}[\Gamma])$ $\displaystyle [x] \in TH_\ell(\widetilde{X};\mathbb{Z}) \cong TH_\ell(X;\mathbb{Z}[\Gamma])$

and

$\displaystyle [y] \in TH_{m-\ell-1}(\widetilde{X},\partial \widetilde{X};\mathbb{Z}) \cong TH_{m-\ell-1}(X,\partial X;\mathbb{Z}[\Gamma])$ $\displaystyle [y] \in TH_{m-\ell-1}(\widetilde{X},\partial \widetilde{X};\mathbb{Z}) \cong TH_{m-\ell-1}(X,\partial X;\mathbb{Z}[\Gamma])$

represented by cycles $x \in C_\ell(\widetilde{X};\mathbb{Z})$ $x \in C_\ell(\widetilde{X};\mathbb{Z})$ and $y \in C_{m-\ell-1}(\widetilde{X},\partial \widetilde{X};\mathbb{Z})$ $y \in C_{m-\ell-1}(\widetilde{X},\partial \widetilde{X};\mathbb{Z})$ , let $w \in C_{m-\ell}(\widetilde{X},\partial\widetilde{X};\mathbb{Z})$ $w \in C_{m-\ell}(\widetilde{X},\partial\widetilde{X};\mathbb{Z})$ be such that $\partial w = \Delta y$ $\partial w = \Delta y$ , for some $\Delta \in \mathbb{Z}[\Gamma]$ $\Delta \in \mathbb{Z}[\Gamma]$ . Then we define:

$\displaystyle \mathop{\mathrm{Bl}}([x],[y]) := \sum_{g \in \Gamma} \langle g \cdot x, w \rangle g^{-1}/\Delta \in \Q(\Gamma)/\mathbb{Z}[\Gamma],$ $\displaystyle \mathop{\mathrm{Bl}}([x],[y]) := \sum_{g \in \Gamma} \langle g \cdot x, w \rangle g^{-1}/\Delta \in \Q(\Gamma)/\mathbb{Z}[\Gamma],$

where $\Gamma$ acts on $C_\ell(\widetilde{X};\mathbb{Z})$ by the action induced from the deck transformation.

3 Example of knot exteriors

We now turn to an example. We will focus on the case of knots in $S^3$ . For a knot $K \subset S^3$ , let $X_K$ denote its exterior, which is the complement of a regular neighbourhood of $K$ : $X_K:= S^3 \setminus \nu K$ . Now $\ell=1$ , $m=3$ and the abelianisation gives a homomorphism $\pi_1(X_K) \to H_1(X_K;\mathbb{Z}) \xrightarrow{\simeq} \mathbb{Z}$ . The Blanchfield form can in this case be defined without relative homology, on $H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \cong H_1(\widetilde{X}_K;\mathbb{Z})$ . The form

$\displaystyle \mathop{\mathrm{Bl}} \colon H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \times H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \to \Q(\mathbb{Z})/\mathbb{Z}[\mathbb{Z}]$ $\displaystyle \mathop{\mathrm{Bl}} \colon H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \times H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \to \Q(\mathbb{Z})/\mathbb{Z}[\mathbb{Z}]$

is non--singular, sesquilinear and Hermitian. Note that $H_1(X_K;\mathbb{Z}[\mathbb{Z}])$ is entirely $\mathbb{Z}[\mathbb{Z}]$ --torsion, so $H_1(X_K;\mathbb{Z}[\mathbb{Z}]) = TH_1(X_K;\mathbb{Z}[\mathbb{Z}])$ . The adjoint of this form is given by the following sequence of homomorphisms:

$\displaystyle \begin{aligned} &H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \to H_1(X_K,\partial X_K;\mathbb{Z}[\mathbb{Z}]) \to H^2(X_K;\mathbb{Z}[\mathbb{Z}]) \\ & \to H^1(X_K;\Q(\mathbb{Z})/\Q[\mathbb{Z}]) \to \Hom_{\mathbb{Z}[\mathbb{Z}]}(H_1(X_K;\mathbb{Z}[\mathbb{Z}]),\Q(\mathbb{Z})/\Q[\mathbb{Z}]), \end{aligned}$ $\displaystyle \begin{aligned} &H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \to H_1(X_K,\partial X_K;\mathbb{Z}[\mathbb{Z}]) \to H^2(X_K;\mathbb{Z}[\mathbb{Z}]) \\ & \to H^1(X_K;\Q(\mathbb{Z})/\Q[\mathbb{Z}]) \to \Hom_{\mathbb{Z}[\mathbb{Z}]}(H_1(X_K;\mathbb{Z}[\mathbb{Z}]),\Q(\mathbb{Z})/\Q[\mathbb{Z}]), \end{aligned}$

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 $\mathop{\mathrm{Bl}}$ 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.

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 $K,J$ $K,J$ are concordant if there is an annulus embedded in $S^3 \times I$ $S^3 \times I$ whose boundary is $K \times \{0\} \cup -J \times \{1\}$ $K \times \{0\} \cup -J \times \{1\}$ . A knot which is concordant to the unknot is called a slice knot; equivalently a slice knot bounds an embedded disk in $D^4$ $D^4$ . We say that a Blanchfield form is metabolic if there is a submodule $P \subset H_1(X_K;\mathbb{Z}[\mathbb{Z}])$ $P \subset H_1(X_K;\mathbb{Z}[\mathbb{Z}])$ which is self-orthogonal with respect to $\mathop{\mathrm{Bl}}$ $\mathop{\mathrm{Bl}}$ , 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 $A \subset D^4$ $A \subset D^4$ is a slice disk for $K$ $K$ , the Blanchfield form vanishes on the kernel of the inclusion induced map

$\displaystyle H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \to H_1(D^4\setminus \nu A;\mathbb{Z}[\mathbb{Z}]).$ $\displaystyle H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \to H_1(D^4\setminus \nu A;\mathbb{Z}[\mathbb{Z}]).$

For high-dimensional knots, $K \colon S^{2k-1} \subset S^{2k+1}$ , where $k \geq 3$ , the Blanchfield form is metabolic if and only if $K$ 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 $3$ -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 $W$ , whose existence one wishes to deny. Let $M_K$ be the closed 3-manifold obtained from zero-framed surgery on $S^3$ along $K$ . Note that $\partial W = M_K$ . Then the kernel

$\displaystyle P:= \ker(H_1(M_K;\mathbb{Q}[\mathbb{Z}]) \to H_1(W;\mathbb{Q}[\mathbb{Z}]))$ $\displaystyle P:= \ker(H_1(M_K;\mathbb{Q}[\mathbb{Z}]) \to H_1(W;\mathbb{Q}[\mathbb{Z}]))$

is a metaboliser for the rational Blanchfield form of $M_K$ $M_K$ . Therefore, given $p \in P$ $p \in P$ , the representation

$\displaystyle H_1(M_K;\mathbb{Q}[\mathbb{Z}]) \to \mathbb{Q}(\mathbb{Z})/\mathbb{Q}[\mathbb{Z}];\; x \mapsto \mathop{\mathrm{Bl}}(p,x)$ $\displaystyle H_1(M_K;\mathbb{Q}[\mathbb{Z}]) \to \mathbb{Q}(\mathbb{Z})/\mathbb{Q}[\mathbb{Z}];\; x \mapsto \mathop{\mathrm{Bl}}(p,x)$

extends over $W$ .

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 $M$ and a representation $\phi \colon \pi_1(M) \to G$ there is defined a real number $\rho^{(2)}(M,\phi)$ called the Cheeger-Gromov-Von Neumann $\rho$ -invariant of $(M,\phi)$ , which can be computed using $L^{(2)}$ -signatures of 4-manifolds with boundary $M_K$ .

Let $K$ be a slice knot. Then there exists a metaboliser $P = P^{\bot}$ for the rational Blanchfield form of $K$ such that for each $p \in P$ there is a representation $\phi_p \colon \pi_1(M_K) \to G$ for which $\rho^{(2)}(M_K,\phi_p)=0$ .

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 represents the Blanchfield form of a given knot to compute many previously unknown unknotting numbers of low crossing number knots.

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 $L^2$ -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 $\rho$ -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 $2$ -spheres in $4$ -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
[Reidemeister1939] K. Reidemeister, Durchschnitt und Schnitt von Homotopieketten, Monatsh. Math. Phys. 48 (1939), 226–239. MR0000634 (1,105h) Zbl 0021.43104

Blanchfield form

Revision as of 15:53, 27 March 2013

1 Background: equivariant intersection forms

2 Definition of the Blanchfield form

3 Example of knot exteriors

4 Applications to knot concordance

References

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@@ Line 14: / Line 14: @@
 ==Definition of the Blanchfield form==
-In his 1954 Princeton PhD thesis R. C. Blanchfield {{cite|Blanchfield1957}} made the corresponding generalisation for linking forms.  Let $X^{m}$ be a compact manifold, now possibly with non--empty boundary, with a surjective homomorphism $\pi_1(X) \to \Gamma$, for some free abelian group $\Gamma$.  Let $\mathbb{Z}[\Gamma]$ be the group ring of $\Gamma$ and let $\Q(\Gamma)$ be its field of fractions.
+In his 1954 Princeton PhD thesis R. C. Blanchfield {{cite|Blanchfield1957}} made the corresponding generalisation for linking forms.  Let $X^{m}$ be a compact manifold, now possibly with non-empty boundary, with a surjective homomorphism $\pi_1(X) \to \Gamma$, for some free abelian group $\Gamma$.  Let $\mathbb{Z}[\Gamma]$ be the group ring of $\Gamma$ and let $\Q(\Gamma)$ be its field of fractions.
 The $\mathbb{Z}[\Gamma]$-torsion submodule of a $\mathbb{Z}[\Gamma]$ module $P$ is the set $$TP:= \{p \in P \,|\, ap=0 \text{ for some } a \in \mathbb{Z}[\Gamma]\}.$$
 The Blanchfield form is a sesquilinear $\Q(\Gamma)/\mathbb{Z}[\Gamma]$--valued pairing which is defined on the $\mathbb{Z}[\Gamma]$-torsion submodules of the homology of the $\Gamma$--cover $\widetilde{X}$ of $X$:
@@ Line 35: / Line 35: @@
 ==Applications to knot concordance==
-One of the Blanchfield form's main applications is in knot concordance, a notion first defined in \cite{Fox&Milnor1966}.  Two knots $K,J$ are concordant if there is an annulus embedded in $S^3 \times I$ whose boundary is $K \times \{0\} \cup -J \times \{1\}$.  A knot which is concordant to the unknot is called a slice knot; equivalently a slice knot bounds an embedded disk in $D^4$.  We say that a Blanchfield form is metabolic if there is a submodule $P \subset H_1(X_K;\mathbb{Z}[\mathbb{Z}])$ which is self--orthogonal with respect to $\mathop{\mathrm{Bl}}$, called a metaboliser.  The Blanchfield form of a slice knot is metabolic, so that the Blanchfield form provides an obstruction to concordance \cite{Kearton1975}, which is equivalent to Levine's Seifert form obstruction \cite{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 $A \subset D^4$ is a slice disk for $K$, the Blanchfield form vanishes on the kernel of the inclusion induced map $$H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \to H_1(D^4\setminus \nu A;\mathbb{Z}[\mathbb{Z}]).$$
+One of the Blanchfield form's main applications is in knot concordance, a notion first defined in \cite{Fox&Milnor1966}.  Two knots $K,J$ are concordant if there is an annulus embedded in $S^3 \times I$ whose boundary is $K \times \{0\} \cup -J \times \{1\}$.  A knot which is concordant to the unknot is called a slice knot; equivalently a slice knot bounds an embedded disk in $D^4$.  We say that a Blanchfield form is metabolic if there is a submodule $P \subset H_1(X_K;\mathbb{Z}[\mathbb{Z}])$ which is self-orthogonal with respect to $\mathop{\mathrm{Bl}}$, called a metaboliser.  The Blanchfield form of a slice knot is metabolic, so that the Blanchfield form provides an obstruction to concordance \cite{Kearton1975}, which is equivalent to Levine's Seifert form obstruction \cite{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 $A \subset D^4$ is a slice disk for $K$, the Blanchfield form vanishes on the kernel of the inclusion induced map $$H_1(X_K;\mathbb{Z}[\mathbb{Z}]) \to H_1(D^4\setminus \nu A;\mathbb{Z}[\mathbb{Z}]).$$
-For high-dimensional knots, $K \colon S^{2k-1} \subset S^{2k+1}$, where $k \geq 3$, the Blanchfield form is metabolic if and only if $K$ is slice  \cite{Levine1969a}, \cite{Kearton1975}.  Levine \cite{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 \cite{Ranicki1998}.
+For high-dimensional knots, $K \colon S^{2k-1} \subset S^{2k+1}$, where $k \geq 3$, the Blanchfield form is metabolic if and only if $K$ is slice  \cite{Levine1969a}, \cite{Kearton1975}.  Levine \cite{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 \cite{Ranicki1998}.
 For classical knots in the $3$-sphere, there are many non-slice knots with metabolic Blanchfield form, the first of which were found in \cite{Casson&Gordon1986}.  Cochran, Orr and Teichner \cite{Cochran&Orr&Teichner2003} defined an infinite filtration of the knot concordance group, each of whose associated graded groups has infinite rank \cite{Cochran&Orr&Teichner2004}, \cite{Cochran&Teichner2007}, \cite{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 $W$, whose existence one wishes to deny.  Let $M_K$ be the closed 3-manifold obtained from zero-framed surgery on $S^3$ along $K$.  Note that $\partial W = M_K$.  Then the kernel
@@ Line 44: / Line 44: @@
 extends over $W$.
-We give a special case of the results of \cite{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 $M$ and a representation $\phi \colon \pi_1(M) \to G$ there is defined a real number $\rho^{(2)}(M,\phi)$ called the Cheeger-Gromov-Von Neumann $\rho$-invariant of $(M,\phi)$, which can be computed using $L^{(2)}$-signatures of 4-manifolds with boundary $M_K$.
+We give a special case of the results of \cite{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 $M$ and a representation $\phi \colon \pi_1(M) \to G$ there is defined a real number $\rho^{(2)}(M,\phi)$ called the Cheeger-Gromov-Von Neumann $\rho$-invariant of $(M,\phi)$, which can be computed using $L^{(2)}$-signatures of 4-manifolds with boundary $M_K$.
 {{beginthm |Theorem |(Cochran-Orr-Teichner)}}
-Let $K$ be a slice knot.  Then there exists a metaboliser $P = P^{\bot}$ for the Blanchfield form of $K$ such that for each $p \in P$ there is a representation $\phi_p \colon \pi_1(M_K) \to G$ for which $\rho^{(2)}(M_K,\phi_p)=0$.
+Let $K$ be a slice knot.  Then there exists a metaboliser $P = P^{\bot}$ for the rational Blanchfield form of $K$ such that for each $p \in P$ there is a representation $\phi_p \colon \pi_1(M_K) \to G$ for which $\rho^{(2)}(M_K,\phi_p)=0$.
 {{endthm}}