Blanchfield form

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1 Introduction

After Poincar\'{e} and Lefschetz, a closed oriented manifold $ {{Stub}} == Introduction == <wikitex>; 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 $\langle\, p \, , \, q\, \rangle \in \mathbb{Z}$. 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: $$L_N \colon TH_{\ell}(N;\mathbb{Z}) \times TH_{n-\ell-1}(N;\mathbb{Z}) \to \Q/\mathbb{Z}$$ such that $$L_N(x,y) = (-)^{\ell(n-\ell-1)}L_N(y,x).$$ Given $[x] \in TH_\ell(N;\mathbb{Z})$ and $[y] \in TH_{n-\ell-1}(N;\mathbb{Z})$ represented by cycles $x \in C_\ell(N;\mathbb{Z})$, let $w \in C_{\ell+1}(N;\mathbb{Z})$ be such that $\partial w = sy$, for some $s \in \mathbb{Z}$. Then we define: $$L_N([x],[y]) := \langle x, w \rangle/s \in \Q/\mathbb{Z}.$$ The resulting element is independent of the choices of $x,y,w$ and $s$. As an example, let $N = \mathbb{RP}^3$, so that $\ell=1$ and $n=3$. Now $H_1(\mathbb{RP}^3;\mathbb{Z}) \cong \mathbb{Z}_2$. Let $\theta \in H_1(\mathbb{RP}^3;\mathbb{Z})$ be the non--trivial element. To compute the linking $L_{\mathbb{RP}^3}(\theta,\theta)$, consider $\mathbb{RP}^3$ modelled as $D^3/\sim$, with antipodal points on $\partial D^2$ identified, and choose two representative N^{n}$ has a bilinear intersection form defined on its homology:

$\displaystyle I_N \colon H_k(N;\mathbb{Z}) \times H_{n-k}(N;\mathbb{Z}) \to \mathbb{Z}; ([p],[q]) \mapsto \langle p, q \rangle$ $\displaystyle I_N \colon H_k(N;\mathbb{Z}) \times H_{n-k}(N;\mathbb{Z}) \to \mathbb{Z}; ([p],[q]) \mapsto \langle p, q \rangle$

such that

$\displaystyle I_N(x,y) = (-)^{k(n-k)}I_N(y,x).$ $\displaystyle I_N(x,y) = (-)^{k(n-k)}I_N(y,x).$

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 $\langle\, p \, , \, q\, \rangle \in \mathbb{Z}$ .

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:

$\displaystyle L_N \colon TH_{\ell}(N;\mathbb{Z}) \times TH_{n-\ell-1}(N;\mathbb{Z}) \to \Q/\mathbb{Z}$ $\displaystyle L_N \colon TH_{\ell}(N;\mathbb{Z}) \times TH_{n-\ell-1}(N;\mathbb{Z}) \to \Q/\mathbb{Z}$

such that

$\displaystyle L_N(x,y) = (-)^{\ell(n-\ell-1)}L_N(y,x).$ $\displaystyle L_N(x,y) = (-)^{\ell(n-\ell-1)}L_N(y,x).$

Given $[x] \in TH_\ell(N;\mathbb{Z})$ and $[y] \in TH_{n-\ell-1}(N;\mathbb{Z})$ represented by cycles $x \in C_\ell(N;\mathbb{Z})$ , let $w \in C_{\ell+1}(N;\mathbb{Z})$ be such that $\partial w = sy$ , for some $s \in \mathbb{Z}$ . Then we define:

$\displaystyle L_N([x],[y]) := \langle x, w \rangle/s \in \Q/\mathbb{Z}.$ $\displaystyle L_N([x],[y]) := \langle x, w \rangle/s \in \Q/\mathbb{Z}.$

The resulting element is independent of the choices of $x,y,w$ and $s$ .

As an example, let $N = \mathbb{RP}^3$ , so that $\ell=1$ and $n=3$ . Now $H_1(\mathbb{RP}^3;\mathbb{Z}) \cong \mathbb{Z}_2$ . Let $\theta \in H_1(\mathbb{RP}^3;\mathbb{Z})$ be the non--trivial element. To compute the linking $L_{\mathbb{RP}^3}(\theta,\theta)$ , consider $\mathbb{RP}^3$ modelled as $D^3/\sim$ , with antipodal points on $\partial D^2$ identified, and choose two representative $1$ --chains $x$ and $y$ for $\theta$ . Let $x$ be the straight line between north and south poles and let $y$ be half of the equator. Now $2y = \partial w$ , where $w \in C_2(\mathbb{RP}^3;\mathbb{Z})$ is the 2--disk whose boundary is the equator. We see that $\langle x,w \rangle = 1$ , so that

$\displaystyle L_{\mathbb{RP}^3}(\theta,\theta) = L_{\mathbb{RP}^3}([x],[y]) = \langle x,y \rangle/2 = 1/2.$ $\displaystyle L_{\mathbb{RP}^3}(\theta,\theta) = L_{\mathbb{RP}^3}([x],[y]) = \langle x,y \rangle/2 = 1/2.$

In 1939 Reidemeister [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.

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

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.

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 form 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.

We now turn to an example. For simplicity 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 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 $K,J$ $K,J$ 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 [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 $A \subset D^4$ $A \subset D^4$ is a slice disk for $K$ $K$ , the Blanchfield form vanishes on the kernel of the 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 [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 $3$ --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 $W$ , whose existence one wishes to deny. Let $M_K$ be the closed 3--manifold obtained from zero--framed surgery on $S^3$ along $K$ . 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 $M_K$ .

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 $(M,\phi)$ .

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 $\rho^{(2)}(M_K,\phi_p)=0$ .

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 $L^2$ -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

$--chains $x$ and $y$ for $\theta$. Let $x$ be the straight line between north and south poles and let $y$ be half of the equator. Now y = \partial w$, where $w \in C_2(\mathbb{RP}^3;\mathbb{Z})$ is the 2--disk whose boundary is the equator. We see that $\langle x,w \rangle = 1$, so that $$L_{\mathbb{RP}^3}(\theta,\theta) = L_{\mathbb{RP}^3}([x],[y]) = \langle x,y \rangle/2 = 1/2.$$ 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. $$\begin{array}{rcl} I_{\widetilde{M}} \colon H_k(\widetilde{M};\mathbb{Z}) \times H_{m-k}(\widetilde{M};\mathbb{Z}) &\to & \mathbb{Z}[G]; \ ([p],[q]) & \mapsto & \sum_{g \in G} \langle g \cdot p, q \rangle g^{-1}. \end{array}$$ 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. 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 form 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. We now turn to an example. For simplicity 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 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 \cite{Fox&Milnor1959}. 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{Kearton}, which is equivalent to Levine's Seifert form obstruction \cite{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 $A \subset D^4$ is a slice disk for $K$, the Blanchfield form vanishes on the kernel of the 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{Levine, Kearton}. Levine \cite{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 \cite{RanickiHighDimKnotTheory}. For classical knots in the $--sphere, there are many non--slice knots with metabolic Blanchfield form, the first of which were found in \cite{CassonGordon}. Cochran, Orr and Teichner \cite{COT} defined an infinite filtration of the knot concordance group, each of whose associated graded groups has infinite rank \cite{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 $W$, whose existence one wishes to deny. Let $M_K$ be the closed 3--manifold obtained from zero--framed surgery on $S^3$ along $K$. 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 $M_K$. 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 $(M,\phi)$. {{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{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. == References == {{#RefList:}} [[Category:Theory]]N^{n} has a bilinear intersection form defined on its homology: