Linking form

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==Background: intersection forms==
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== Definition==
<wikitex>;
<wikitex>;
After Poincaré and Lefschetz, a closed oriented manifold $N^{n}$ has a bilinear intersection form defined on its homology. 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 $\langle\, p \, , \, q\, \rangle \in \mathbb{Z}$.
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After Seifert \cite{Seifert1933}, a closed oriented $n$-manifold $N$ has a bilinear ''linking form''
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$$L_N \colon TH_{\ell}(N;\Zz) \times TH_{n-\ell-1}(N;\Zz) \to \mathbb{Q}/\Zz.$$
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Here the torsion part of an abelian group $P$ is the subgroup
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$$TP:= \{p \in P \,|\, ap=0 \text{ for some } a \in \Z\}.$$
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The linking form is the analogue for the torsion part of the homology of the [[Intersection_form|intersection product]] which vanishes on the torsion part.
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The linking form is defined as follows. Take $[x] \in TH_\ell(N;\Zz)$ and $[y] \in TH_{n-\ell-1}(N;\Zz)$ represented by cycles $x \in C_\ell(N;\Zz)$ and $y \in C_{n-\ell-1}(N,\Zz)$.
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There is $Y\in C_{n-\ell}(N;\Z)$ such that $\partial Y = sy$ for some $s \in \Zz$.
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Define
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$$L_N([x],[y]):= \langle x, Y \rangle/s \in \mathbb{Q}/\Zz$$ to be the [[Intersection_form#Definition| intersection number]] of $x$ and $Y$ divided by $s$ and taken modulo 1.
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For fixed $N,[x],[y]$ the resulting `residie modulo 1' is independent of the choices of $x,y,Y$ and $s$.
The intersection form is defined by
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We have
$$I_N \colon H_k(N;\mathbb{Z}) \times H_{n-k}(N;\mathbb{Z}) \to \mathbb{Z}; ([p],[q]) \mapsto \langle p, q \rangle$$
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$$L_N(x,y) = (-)^{\ell(n-\ell)}L_N(y,x).$$
and is such that
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$$I_N(x,y) = (-)^{k(n-k)}I_N(y,x).$$
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</wikitex>
</wikitex>
== Definition of the linking form==
<wikitex>;
By bilinearity, the intersection form vanishes on the torsion part of the homology. The torsion part of an abelian group $P$ is the set $$TP:= \{p \in P \,|\, ap=0 \text{ for some } a \in \mathbb{Z}\}.$$
The analogue of the intersection pairing for the torsion part of the homology of a closed oriented manifold $N^n$ is the bilinear $\mathbb{Q}/\mathbb{Z}$--valued linking form, which is due to Seifert:
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==Definition via cohomology==
$$L_N \colon TH_{\ell}(N;\mathbb{Z}) \times TH_{n-\ell-1}(N;\mathbb{Z}) \to \mathbb{Q}/\mathbb{Z}$$
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such that
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$$L_N(x,y) = (-)^{\ell(n-\ell-1)}L_N(y,x)$$
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and computed as follows. Given $[x] \in TH_\ell(N;\mathbb{Z})$ and $[y] \in TH_{n-\ell-1}(N;\mathbb{Z})$ represented by cycles $x \in C_\ell(N;\mathbb{Z})$ and $y \in C_{n-\ell-1}(N,\mathbb{Z})$, let $w \in C_{\ell+1}(N;\mathbb{Z})$ be such that $\partial w = sy$, for some $s \in \mathbb{Z}$. Then we define:
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$$L_N([x],[y]) := \langle x, w \rangle/s \in \mathbb{Q}/\mathbb{Z}.$$
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The resulting element is independent of the choices of $x,y,w$ and $s$.
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</wikitex>
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==Definition via homology==
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<wikitex>;
<wikitex>;
Let $x \in TH_{\ell}(N;\mathbb{Z})$ and let $y \in TH_{n-\ell-1}(N;\mathbb{Z})$. Note that we have Poincaré duality isomorphisms
Let $x \in TH_{\ell}(N;\mathbb{Z})$ and let $y \in TH_{n-\ell-1}(N;\mathbb{Z})$. Note that we have Poincaré duality isomorphisms
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Associated to the short exact sequence of coefficients
Associated to the short exact sequence of coefficients
$$0 \to \mathbb{Z} \to \mathbb{Q} \to \mathbb{Q}/\mathbb{Z} \to 0$$
$$0 \to \mathbb{Z} \to \mathbb{Q} \to \mathbb{Q}/\mathbb{Z} \to 0$$
is the Bockstein long exact sequence in cohomology.
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is the Bockstein long exact sequence in cohomology:
$$H^{n-\ell-1}(N;\mathbb{Q}) \to H^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z}) \xrightarrow{\beta} H^{n-\ell}(N;\mathbb{Z}) \to H^{n-\ell-1}(N;\mathbb{Q}).$$
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$$H^{n-\ell-1}(N;\mathbb{Q}) \to H^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z}) \xrightarrow{\beta} H^{n-\ell}(N;\mathbb{Z}) \to H^{n-\ell}(N;\mathbb{Q}).$$
Choose $z \in TH^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z})$ such that $\beta(z) = PD(x)$. This is always possible since torsion elements in $H^{n-\ell}(N;\mathbb{Z})$ map to zero in $H^{n-\ell}(N;\mathbb{Q})$. There is a cup product:
Choose $z \in TH^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z})$ such that $\beta(z) = PD(x)$. This is always possible since torsion elements in $H^{n-\ell}(N;\mathbb{Z})$ map to zero in $H^{n-\ell}(N;\mathbb{Q})$. There is a cup product:
$$\cup \colon H^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z}) \otimes H^{\ell+1}(N;\mathbb{Z}) \to H^{n}(N;\mathbb{Q}/\mathbb{Z}).$$
$$\cup \colon H^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z}) \otimes H^{\ell+1}(N;\mathbb{Z}) \to H^{n}(N;\mathbb{Q}/\mathbb{Z}).$$
Compute $a:= z \cup PD(y)$. Then the Kronecker pairing:
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Then the Kronecker pairing,
$$\langle a,[N] \rangle \in \mathbb{Q}/\mathbb{Z}$$
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$$\langle z \cup PD(y),[N] \rangle \in \mathbb{Q}/\mathbb{Z},$$
yields $L_N(x,y)$.
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of $z \cup PD(y)$ with the fundamental class of $N$ yields $L_N(x,y)$.
</wikitex>
</wikitex>
==Example of 3-dimensional projective space==
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==Examples of 3-dimensional projective and lens spaces==
<wikitex>;
<wikitex>;
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
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
$$L_{\mathbb{RP}^3}(\theta,\theta) = L_{\mathbb{RP}^3}([x],[y]) = \langle x,y \rangle/2 = 1/2.$$
$$L_{\mathbb{RP}^3}(\theta,\theta) = L_{\mathbb{RP}^3}([x],[y]) = \langle x,y \rangle/2 = 1/2.$$
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Generalising the above example, the 3-dimensional [[Lens spaces|lens space]] $N_{p,q} := L(p; q, 1)$ has $H_1(N_{p,q};\mathbb{Z}) \cong \mathbb{Z}_p$. The linking form is given on a generator $\theta \in H_1(N_{p,q};\mathbb{Z})$ by $L_{N_{p,q}}(\theta,\theta) = q/p$. Note that $N_{2,1} \cong \mathbb{RP}^3$, so this is consistent with the above example.
</wikitex>
</wikitex>
==Example of lens spaces==
<wikitex>;
Generalising the above example, the 3-dimensional lens space $N_{p,q}$ has $H_1(N_{p,q};\mathbb{Z}) \cong \mathbb{Z}_p$. The linking form is given on a generator $\theta \in H_1(N_{p,q};\mathbb{Z})$ by $L_{N_{p,q}}(\theta,\theta) = p/q$. Note that $L_{2,1} \cong \mathbb{RP}^3$, so this is consistent with the above example.
</wikitex>
==Presentations of linking forms==
==Presentations of linking forms==
<wikitex>;
<wikitex>;
A presentation for a linking form is an exact sequence:
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A presentation for a middle dimensional linking form on $N^{2\ell +1}$
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$$L_N \colon TH_{\ell}(N;\mathbb{Z}) \times TH_{\ell}(N;\mathbb{Z}) \to \mathbb{Q}/\mathbb{Z}$$
$$0 \to F \xrightarrow{\Phi} F^* \to TH_1(N;\mathbb{Z}) \xrightarrow{\partial} 0,$$
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is an exact sequence:
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$$0 \to F \xrightarrow{\Phi} F^* \to TH_{\ell}(N;\mathbb{Z}) \xrightarrow{\partial} 0,$$
where $F$ is a free abelain group and the linking $L_N(x,y)$ can be computed as follows. Let $x',y' \in F^*$ be such that $\partial(x')=x$ and $\partial(y')=y$. Then we can tensor with $\mathbb{Q}$ to obtain an isomorphism
where $F$ is a free abelain group and the linking $L_N(x,y)$ can be computed as follows. Let $x',y' \in F^*$ be such that $\partial(x')=x$ and $\partial(y')=y$. Then we can tensor with $\mathbb{Q}$ to obtain an isomorphism
$$\Phi \otimes \mathop{\mathrm{Id}} \colon F \otimes_{\mathbb{Z}} \mathbb{Q} \xrightarrow{\cong} F^* \otimes_{\mathbb{Z}} \mathbb{Q}.$$
$$\Phi \otimes \mathop{\mathrm{Id}} \colon F \otimes_{\mathbb{Z}} \mathbb{Q} \xrightarrow{\cong} F^* \otimes_{\mathbb{Z}} \mathbb{Q}.$$
The linking form is given by:
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The linking form of $N$ is then given by:
$$L_N(x,y) = (x' \otimes 1)((\Phi\otimes \mathop{\mathrm{Id}})^{-1}(y')).$$
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$$L_N(x,y) = -(x' \otimes 1)((\Phi\otimes \mathop{\mathrm{Id}})^{-1}(y'\otimes 1)).$$
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This formula, in particular the appearance of the sign, is explained in \cite[Section 3]{Gordon&Litherland1978} and \cite[Proof of Theorem 2.1]{Alexander&Hamrick&Vick1976}.
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Let $\ell = 1$, so $2\ell + 1 = 3$. Every 3-manifold $N$ is the boundary of a simply connected 4-manifold, which is obtained by glueing 2-handles to an integrally framed link in $S^3$ \cite{Lickorish1962}, \cite{Wallace1960}. This is sometimes called a surgery presentation for $N$. Suppose that $N$ is a rational homology 3-sphere. Let $A$ be the matrix of (self-) linking numbers of the surgery presentation link. Taking the number of link components in the surgery presentation for $N$ as the rank of $F$, the linking matrix $A$ determines a map $\Phi$ as above, which presents the linking form of $N$. The intersection form on a simply connected 4-manifold $W$ whose boundary is $N$ presents the linking form of $N$. This follows from the long exact sequence of the pair $(W,N)$ and Poincaré duality. See \cite{Boyer1986} for more details and the use of such presentations for the classification of simply connected 4-manifolds with a given boundary.
Every 3-manifold $N$ is the boundary of a simply connected 4-manifold, which is obtained by glueing 2-handles to an integrally framed link in $S^3$ \cite{Lickorish1963}, \cite{Wallace1960}. This is sometimes called a surgery presentation for $N$. Suppose that $N$ is a rational homology 3-sphere. The matrix $A$ of (self-) linking numbers of the link determines a presentation of the linking form of $N$. Taking the number of link components in the surgery presentation for $N$ as the rank of $F$, the linking matrix $A$ determines a map $\Phi$ as above. The intersection form on a simply connected 4-manifold $W$ whose boundary is $N$ presents the linking form of $N$. This follows from the long exact sequence of the pair $W,N$ and Poincar\'{e} duality.
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For example, the lens space $L(p,1)$ is the boundary of the $D^2$ bundle over $S^2$ with Euler number $-p$, so the presentation $\Phi = (-p) \colon \mathbb{Z} \to \mathbb{Z}$ presents the linking form of $L(p,1)$ \cite[Example 5.3.2]{Gompf&Stipsicz1999}.
</wikitex>
</wikitex>
==Classification of 5-manifolds==
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== Role in the classification of odd-dimensional manifolds ==
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<wikitex>;
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Linking forms play an important role in the classification of odd-dimensional manifolds.
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For closed simply connected $5$-manifolds $M$, the linking form is a complete invariant if $H_2(M; \Zz) = TH_2(M; \Zz)$. For more information in dimension $5$, see the page on [[5-manifolds: 1-connected|simply-connected 5-manifolds]].
Linking forms are central to the classification of simply connected 5-manifolds. See this [[5-manifolds: 1-connected|5-manifolds]] page. This page also described the classification of anti-symmetric linking forms.
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For the role of linking forms in the classification of smooth $(q-1)$-connected $(2q+1)$ manifolds with boundary a homotopy sphere, see \cite[Theorem 7]{Wall1967}.
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</wikitex>
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== Algebraic classification ==
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<wikitex>;
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An algebraic linking form is a non-singular bi-linear pairing
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$$ b \colon T \times T \to \Qq/\Zz$$
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on a finite abelian group $T$. It is called symmetric if $b(x, y) = b(y, x)$ and skew-symmetric
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if $b(x, y) = - b(y, x)$.
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The classification of skew-symmetric linking forms is rather simple and is due to Wall, \cite[Theorem 3]{Wall1963}.
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It is described in detail in the page on [[5-manifolds: 1-connected#Linking_forms|simply-connected 5-manifolds]].
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The classification of symmetric linking forms is rather intricate. It was begun in \cite{Wall1963} and completed by Kawauchi and Kojima: see \cite[Theorem 4.1]{Kawauchi&Kojima1980}.
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</wikitex>
== References ==
== References ==
{{#RefList:}}
{{#RefList:}}
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== External links ==
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* The Wikipedia page on [[Wikipedia:Poincare duality#Bilinear_pairings_formulation|Poincaré duality]]
[[Category:Definitions]]
[[Category:Definitions]]
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[[Category:Forgotten in Textbooks]]

Latest revision as of 12:31, 29 March 2019

This page has not been refereed. The information given here might be incomplete or provisional.

Contents

[edit] 1 Definition

After Seifert [Seifert1933], a closed oriented n-manifold N has a bilinear linking form

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

Here the torsion part of an abelian group P is the subgroup

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

The linking form is the analogue for the torsion part of the homology of the intersection product which vanishes on the torsion part.

The linking form is defined as follows. Take [x] \in TH_\ell(N;\Zz) and [y] \in TH_{n-\ell-1}(N;\Zz) represented by cycles x \in C_\ell(N;\Zz) and y \in C_{n-\ell-1}(N,\Zz). There is Y\in C_{n-\ell}(N;\Z) such that \partial Y = sy for some s \in \Zz. Define

\displaystyle L_N([x],[y]):= \langle x, Y \rangle/s \in \mathbb{Q}/\Zz
to be the intersection number of x and Y divided by s and taken modulo 1.

For fixed N,[x],[y] the resulting `residie modulo 1' is independent of the choices of x,y,Y and s.

We have

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

[edit] 2 Definition via cohomology

Let x \in TH_{\ell}(N;\mathbb{Z}) and let y \in TH_{n-\ell-1}(N;\mathbb{Z}). Note that we have Poincaré duality isomorphisms

\displaystyle PD \colon TH_{\ell}(N;\mathbb{Z}) \xrightarrow{\cong} TH^{n-\ell}(N;\mathbb{Z})

and

\displaystyle PD \colon TH_{n-\ell-1}(N;\mathbb{Z}) \xrightarrow{\cong} TH^{\ell+1}(N;\mathbb{Z}).

Associated to the short exact sequence of coefficients

\displaystyle 0 \to \mathbb{Z} \to \mathbb{Q} \to \mathbb{Q}/\mathbb{Z} \to 0

is the Bockstein long exact sequence in cohomology:

\displaystyle H^{n-\ell-1}(N;\mathbb{Q}) \to H^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z}) \xrightarrow{\beta} H^{n-\ell}(N;\mathbb{Z}) \to H^{n-\ell}(N;\mathbb{Q}).

Choose z \in TH^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z}) such that \beta(z) = PD(x). This is always possible since torsion elements in H^{n-\ell}(N;\mathbb{Z}) map to zero in H^{n-\ell}(N;\mathbb{Q}). There is a cup product:

\displaystyle \cup \colon  H^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z}) \otimes H^{\ell+1}(N;\mathbb{Z}) \to  H^{n}(N;\mathbb{Q}/\mathbb{Z}).

Then the Kronecker pairing,

\displaystyle \langle z \cup PD(y),[N] \rangle \in \mathbb{Q}/\mathbb{Z},

of z \cup  PD(y) with the fundamental class of N yields L_N(x,y).

[edit] 3 Examples of 3-dimensional projective and lens spaces

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.

Generalising the above example, the 3-dimensional lens space N_{p,q} := L(p; q, 1) has H_1(N_{p,q};\mathbb{Z}) \cong \mathbb{Z}_p. The linking form is given on a generator \theta \in H_1(N_{p,q};\mathbb{Z}) by L_{N_{p,q}}(\theta,\theta) = q/p. Note that N_{2,1} \cong \mathbb{RP}^3, so this is consistent with the above example.

[edit] 4 Presentations of linking forms

A presentation for a middle dimensional linking form on N^{2\ell +1}

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

is an exact sequence:

\displaystyle 0 \to F \xrightarrow{\Phi} F^* \to TH_{\ell}(N;\mathbb{Z}) \xrightarrow{\partial} 0,

where F is a free abelain group and the linking L_N(x,y) can be computed as follows. Let x',y' \in F^* be such that \partial(x')=x and \partial(y')=y. Then we can tensor with \mathbb{Q} to obtain an isomorphism

\displaystyle \Phi \otimes \mathop{\mathrm{Id}} \colon F \otimes_{\mathbb{Z}} \mathbb{Q} \xrightarrow{\cong} F^* \otimes_{\mathbb{Z}} \mathbb{Q}.

The linking form of N is then given by:

\displaystyle L_N(x,y) = -(x' \otimes 1)((\Phi\otimes  \mathop{\mathrm{Id}})^{-1}(y'\otimes 1)).

This formula, in particular the appearance of the sign, is explained in [Gordon&Litherland1978, Section 3] and [Alexander&Hamrick&Vick1976, Proof of Theorem 2.1].

Let \ell = 1, so 2\ell + 1 = 3. Every 3-manifold N is the boundary of a simply connected 4-manifold, which is obtained by glueing 2-handles to an integrally framed link in S^3 [Lickorish1962], [Wallace1960]. This is sometimes called a surgery presentation for N. Suppose that N is a rational homology 3-sphere. Let A be the matrix of (self-) linking numbers of the surgery presentation link. Taking the number of link components in the surgery presentation for N as the rank of F, the linking matrix A determines a map \Phi as above, which presents the linking form of N. The intersection form on a simply connected 4-manifold W whose boundary is N presents the linking form of N. This follows from the long exact sequence of the pair (W,N) and Poincaré duality. See [Boyer1986] for more details and the use of such presentations for the classification of simply connected 4-manifolds with a given boundary.

For example, the lens space L(p,1) is the boundary of the D^2 bundle over S^2 with Euler number -p, so the presentation \Phi = (-p) \colon \mathbb{Z} \to \mathbb{Z} presents the linking form of L(p,1) [Gompf&Stipsicz1999, Example 5.3.2].

[edit] 5 Role in the classification of odd-dimensional manifolds

Linking forms play an important role in the classification of odd-dimensional manifolds.

For closed simply connected
Tex syntax error
-manifolds M, the linking form is a complete invariant if H_2(M; \Zz) = TH_2(M; \Zz). For more information in dimension
Tex syntax error
, see the page on simply-connected 5-manifolds.

For the role of linking forms in the classification of smooth (q-1)-connected (2q+1) manifolds with boundary a homotopy sphere, see [Wall1967, Theorem 7].

[edit] 6 Algebraic classification

An algebraic linking form is a non-singular bi-linear pairing

\displaystyle  b \colon T \times T \to \Qq/\Zz

on a finite abelian group T. It is called symmetric if b(x, y) = b(y, x) and skew-symmetric if b(x, y) = - b(y, x).

The classification of skew-symmetric linking forms is rather simple and is due to Wall, [Wall1963, Theorem 3]. It is described in detail in the page on simply-connected 5-manifolds.

The classification of symmetric linking forms is rather intricate. It was begun in [Wall1963] and completed by Kawauchi and Kojima: see [Kawauchi&Kojima1980, Theorem 4.1].

[edit] 7 References

[edit] 8 External links

\to \mathbb{Z} \to \mathbb{Q} \to \mathbb{Q}/\mathbb{Z} \to 0$$ is the Bockstein long exact sequence in cohomology. $$H^{n-\ell-1}(N;\mathbb{Q}) \to H^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z}) \xrightarrow{\beta} H^{n-\ell}(N;\mathbb{Z}) \to H^{n-\ell-1}(N;\mathbb{Q}).$$ Choose $z \in TH^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z})$ such that $\beta(z) = PD(x)$. This is always possible since torsion elements in $H^{n-\ell}(N;\mathbb{Z})$ map to zero in $H^{n-\ell}(N;\mathbb{Q})$. There is a cup product: $$\cup \colon H^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z}) \otimes H^{\ell+1}(N;\mathbb{Z}) \to H^{n}(N;\mathbb{Q}/\mathbb{Z}).$$ Compute $a:= z \cup PD(y)$. Then the Kronecker pairing: $$\langle a,[N] \rangle \in \mathbb{Q}/\mathbb{Z}$$ yields $L_N(x,y)$. ==Example of 3-dimensional projective space== ; 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-manifold N has a bilinear linking form

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

Here the torsion part of an abelian group P is the subgroup

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

The linking form is the analogue for the torsion part of the homology of the intersection product which vanishes on the torsion part.

The linking form is defined as follows. Take [x] \in TH_\ell(N;\Zz) and [y] \in TH_{n-\ell-1}(N;\Zz) represented by cycles x \in C_\ell(N;\Zz) and y \in C_{n-\ell-1}(N,\Zz). There is Y\in C_{n-\ell}(N;\Z) such that \partial Y = sy for some s \in \Zz. Define

\displaystyle L_N([x],[y]):= \langle x, Y \rangle/s \in \mathbb{Q}/\Zz
to be the intersection number of x and Y divided by s and taken modulo 1.

For fixed N,[x],[y] the resulting `residie modulo 1' is independent of the choices of x,y,Y and s.

We have

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

[edit] 2 Definition via cohomology

Let x \in TH_{\ell}(N;\mathbb{Z}) and let y \in TH_{n-\ell-1}(N;\mathbb{Z}). Note that we have Poincaré duality isomorphisms

\displaystyle PD \colon TH_{\ell}(N;\mathbb{Z}) \xrightarrow{\cong} TH^{n-\ell}(N;\mathbb{Z})

and

\displaystyle PD \colon TH_{n-\ell-1}(N;\mathbb{Z}) \xrightarrow{\cong} TH^{\ell+1}(N;\mathbb{Z}).

Associated to the short exact sequence of coefficients

\displaystyle 0 \to \mathbb{Z} \to \mathbb{Q} \to \mathbb{Q}/\mathbb{Z} \to 0

is the Bockstein long exact sequence in cohomology:

\displaystyle H^{n-\ell-1}(N;\mathbb{Q}) \to H^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z}) \xrightarrow{\beta} H^{n-\ell}(N;\mathbb{Z}) \to H^{n-\ell}(N;\mathbb{Q}).

Choose z \in TH^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z}) such that \beta(z) = PD(x). This is always possible since torsion elements in H^{n-\ell}(N;\mathbb{Z}) map to zero in H^{n-\ell}(N;\mathbb{Q}). There is a cup product:

\displaystyle \cup \colon  H^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z}) \otimes H^{\ell+1}(N;\mathbb{Z}) \to  H^{n}(N;\mathbb{Q}/\mathbb{Z}).

Then the Kronecker pairing,

\displaystyle \langle z \cup PD(y),[N] \rangle \in \mathbb{Q}/\mathbb{Z},

of z \cup  PD(y) with the fundamental class of N yields L_N(x,y).

[edit] 3 Examples of 3-dimensional projective and lens spaces

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.

Generalising the above example, the 3-dimensional lens space N_{p,q} := L(p; q, 1) has H_1(N_{p,q};\mathbb{Z}) \cong \mathbb{Z}_p. The linking form is given on a generator \theta \in H_1(N_{p,q};\mathbb{Z}) by L_{N_{p,q}}(\theta,\theta) = q/p. Note that N_{2,1} \cong \mathbb{RP}^3, so this is consistent with the above example.

[edit] 4 Presentations of linking forms

A presentation for a middle dimensional linking form on N^{2\ell +1}

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

is an exact sequence:

\displaystyle 0 \to F \xrightarrow{\Phi} F^* \to TH_{\ell}(N;\mathbb{Z}) \xrightarrow{\partial} 0,

where F is a free abelain group and the linking L_N(x,y) can be computed as follows. Let x',y' \in F^* be such that \partial(x')=x and \partial(y')=y. Then we can tensor with \mathbb{Q} to obtain an isomorphism

\displaystyle \Phi \otimes \mathop{\mathrm{Id}} \colon F \otimes_{\mathbb{Z}} \mathbb{Q} \xrightarrow{\cong} F^* \otimes_{\mathbb{Z}} \mathbb{Q}.

The linking form of N is then given by:

\displaystyle L_N(x,y) = -(x' \otimes 1)((\Phi\otimes  \mathop{\mathrm{Id}})^{-1}(y'\otimes 1)).

This formula, in particular the appearance of the sign, is explained in [Gordon&Litherland1978, Section 3] and [Alexander&Hamrick&Vick1976, Proof of Theorem 2.1].

Let \ell = 1, so 2\ell + 1 = 3. Every 3-manifold N is the boundary of a simply connected 4-manifold, which is obtained by glueing 2-handles to an integrally framed link in S^3 [Lickorish1962], [Wallace1960]. This is sometimes called a surgery presentation for N. Suppose that N is a rational homology 3-sphere. Let A be the matrix of (self-) linking numbers of the surgery presentation link. Taking the number of link components in the surgery presentation for N as the rank of F, the linking matrix A determines a map \Phi as above, which presents the linking form of N. The intersection form on a simply connected 4-manifold W whose boundary is N presents the linking form of N. This follows from the long exact sequence of the pair (W,N) and Poincaré duality. See [Boyer1986] for more details and the use of such presentations for the classification of simply connected 4-manifolds with a given boundary.

For example, the lens space L(p,1) is the boundary of the D^2 bundle over S^2 with Euler number -p, so the presentation \Phi = (-p) \colon \mathbb{Z} \to \mathbb{Z} presents the linking form of L(p,1) [Gompf&Stipsicz1999, Example 5.3.2].

[edit] 5 Role in the classification of odd-dimensional manifolds

Linking forms play an important role in the classification of odd-dimensional manifolds.

For closed simply connected
Tex syntax error
-manifolds M, the linking form is a complete invariant if H_2(M; \Zz) = TH_2(M; \Zz). For more information in dimension
Tex syntax error
, see the page on simply-connected 5-manifolds.

For the role of linking forms in the classification of smooth (q-1)-connected (2q+1) manifolds with boundary a homotopy sphere, see [Wall1967, Theorem 7].

[edit] 6 Algebraic classification

An algebraic linking form is a non-singular bi-linear pairing

\displaystyle  b \colon T \times T \to \Qq/\Zz

on a finite abelian group T. It is called symmetric if b(x, y) = b(y, x) and skew-symmetric if b(x, y) = - b(y, x).

The classification of skew-symmetric linking forms is rather simple and is due to Wall, [Wall1963, Theorem 3]. It is described in detail in the page on simply-connected 5-manifolds.

The classification of symmetric linking forms is rather intricate. It was begun in [Wall1963] and completed by Kawauchi and Kojima: see [Kawauchi&Kojima1980, Theorem 4.1].

[edit] 7 References

[edit] 8 External links

$-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 $$L_{\mathbb{RP}^3}(\theta,\theta) = L_{\mathbb{RP}^3}([x],[y]) = \langle x,y \rangle/2 = 1/2.$$
==Example of lens spaces== ; Generalising the above example, the 3-dimensional lens space $N_{p,q}$ has $H_1(N_{p,q};\mathbb{Z}) \cong \mathbb{Z}_p$. The linking form is given on a generator $\theta \in H_1(N_{p,q};\mathbb{Z})$ by $L_{N_{p,q}}(\theta,\theta) = p/q$. Note that $L_{2,1} \cong \mathbb{RP}^3$, so this is consistent with the above example. ==Presentations of linking forms== ; A presentation for a linking form is an exact sequence: $-manifold N has a bilinear linking form

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

Here the torsion part of an abelian group P is the subgroup

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

The linking form is the analogue for the torsion part of the homology of the intersection product which vanishes on the torsion part.

The linking form is defined as follows. Take [x] \in TH_\ell(N;\Zz) and [y] \in TH_{n-\ell-1}(N;\Zz) represented by cycles x \in C_\ell(N;\Zz) and y \in C_{n-\ell-1}(N,\Zz). There is Y\in C_{n-\ell}(N;\Z) such that \partial Y = sy for some s \in \Zz. Define

\displaystyle L_N([x],[y]):= \langle x, Y \rangle/s \in \mathbb{Q}/\Zz
to be the intersection number of x and Y divided by s and taken modulo 1.

For fixed N,[x],[y] the resulting `residie modulo 1' is independent of the choices of x,y,Y and s.

We have

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

[edit] 2 Definition via cohomology

Let x \in TH_{\ell}(N;\mathbb{Z}) and let y \in TH_{n-\ell-1}(N;\mathbb{Z}). Note that we have Poincaré duality isomorphisms

\displaystyle PD \colon TH_{\ell}(N;\mathbb{Z}) \xrightarrow{\cong} TH^{n-\ell}(N;\mathbb{Z})

and

\displaystyle PD \colon TH_{n-\ell-1}(N;\mathbb{Z}) \xrightarrow{\cong} TH^{\ell+1}(N;\mathbb{Z}).

Associated to the short exact sequence of coefficients

\displaystyle 0 \to \mathbb{Z} \to \mathbb{Q} \to \mathbb{Q}/\mathbb{Z} \to 0

is the Bockstein long exact sequence in cohomology:

\displaystyle H^{n-\ell-1}(N;\mathbb{Q}) \to H^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z}) \xrightarrow{\beta} H^{n-\ell}(N;\mathbb{Z}) \to H^{n-\ell}(N;\mathbb{Q}).

Choose z \in TH^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z}) such that \beta(z) = PD(x). This is always possible since torsion elements in H^{n-\ell}(N;\mathbb{Z}) map to zero in H^{n-\ell}(N;\mathbb{Q}). There is a cup product:

\displaystyle \cup \colon  H^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z}) \otimes H^{\ell+1}(N;\mathbb{Z}) \to  H^{n}(N;\mathbb{Q}/\mathbb{Z}).

Then the Kronecker pairing,

\displaystyle \langle z \cup PD(y),[N] \rangle \in \mathbb{Q}/\mathbb{Z},

of z \cup  PD(y) with the fundamental class of N yields L_N(x,y).

[edit] 3 Examples of 3-dimensional projective and lens spaces

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.

Generalising the above example, the 3-dimensional lens space N_{p,q} := L(p; q, 1) has H_1(N_{p,q};\mathbb{Z}) \cong \mathbb{Z}_p. The linking form is given on a generator \theta \in H_1(N_{p,q};\mathbb{Z}) by L_{N_{p,q}}(\theta,\theta) = q/p. Note that N_{2,1} \cong \mathbb{RP}^3, so this is consistent with the above example.

[edit] 4 Presentations of linking forms

A presentation for a middle dimensional linking form on N^{2\ell +1}

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

is an exact sequence:

\displaystyle 0 \to F \xrightarrow{\Phi} F^* \to TH_{\ell}(N;\mathbb{Z}) \xrightarrow{\partial} 0,

where F is a free abelain group and the linking L_N(x,y) can be computed as follows. Let x',y' \in F^* be such that \partial(x')=x and \partial(y')=y. Then we can tensor with \mathbb{Q} to obtain an isomorphism

\displaystyle \Phi \otimes \mathop{\mathrm{Id}} \colon F \otimes_{\mathbb{Z}} \mathbb{Q} \xrightarrow{\cong} F^* \otimes_{\mathbb{Z}} \mathbb{Q}.

The linking form of N is then given by:

\displaystyle L_N(x,y) = -(x' \otimes 1)((\Phi\otimes  \mathop{\mathrm{Id}})^{-1}(y'\otimes 1)).

This formula, in particular the appearance of the sign, is explained in [Gordon&Litherland1978, Section 3] and [Alexander&Hamrick&Vick1976, Proof of Theorem 2.1].

Let \ell = 1, so 2\ell + 1 = 3. Every 3-manifold N is the boundary of a simply connected 4-manifold, which is obtained by glueing 2-handles to an integrally framed link in S^3 [Lickorish1962], [Wallace1960]. This is sometimes called a surgery presentation for N. Suppose that N is a rational homology 3-sphere. Let A be the matrix of (self-) linking numbers of the surgery presentation link. Taking the number of link components in the surgery presentation for N as the rank of F, the linking matrix A determines a map \Phi as above, which presents the linking form of N. The intersection form on a simply connected 4-manifold W whose boundary is N presents the linking form of N. This follows from the long exact sequence of the pair (W,N) and Poincaré duality. See [Boyer1986] for more details and the use of such presentations for the classification of simply connected 4-manifolds with a given boundary.

For example, the lens space L(p,1) is the boundary of the D^2 bundle over S^2 with Euler number -p, so the presentation \Phi = (-p) \colon \mathbb{Z} \to \mathbb{Z} presents the linking form of L(p,1) [Gompf&Stipsicz1999, Example 5.3.2].

[edit] 5 Role in the classification of odd-dimensional manifolds

Linking forms play an important role in the classification of odd-dimensional manifolds.

For closed simply connected
Tex syntax error
-manifolds M, the linking form is a complete invariant if H_2(M; \Zz) = TH_2(M; \Zz). For more information in dimension
Tex syntax error
, see the page on simply-connected 5-manifolds.

For the role of linking forms in the classification of smooth (q-1)-connected (2q+1) manifolds with boundary a homotopy sphere, see [Wall1967, Theorem 7].

[edit] 6 Algebraic classification

An algebraic linking form is a non-singular bi-linear pairing

\displaystyle  b \colon T \times T \to \Qq/\Zz

on a finite abelian group T. It is called symmetric if b(x, y) = b(y, x) and skew-symmetric if b(x, y) = - b(y, x).

The classification of skew-symmetric linking forms is rather simple and is due to Wall, [Wall1963, Theorem 3]. It is described in detail in the page on simply-connected 5-manifolds.

The classification of symmetric linking forms is rather intricate. It was begun in [Wall1963] and completed by Kawauchi and Kojima: see [Kawauchi&Kojima1980, Theorem 4.1].

[edit] 7 References

[edit] 8 External links

\to F \xrightarrow{\Phi} F^* \to TH_1(N;\mathbb{Z}) \xrightarrow{\partial} 0,$$ where $F$ is a free abelain group and the linking $L_N(x,y)$ can be computed as follows. Let $x',y' \in F^*$ be such that $\partial(x')=x$ and $\partial(y')=y$. Then we can tensor with $\mathbb{Q}$ to obtain an isomorphism $$\Phi \otimes \mathop{\mathrm{Id}} \colon F \otimes_{\mathbb{Z}} \mathbb{Q} \xrightarrow{\cong} F^* \otimes_{\mathbb{Z}} \mathbb{Q}.$$ The linking form is given by: $$L_N(x,y) = (x' \otimes 1)((\Phi\otimes \mathop{\mathrm{Id}})^{-1}(y')).$$ Every 3-manifold $N$ is the boundary of a simply connected 4-manifold, which is obtained by glueing 2-handles to an integrally framed link in $S^3$ \cite{Lickorish1963}, \cite{Wallace1960}. This is sometimes called a surgery presentation for $N$. Suppose that $N$ is a rational homology 3-sphere. The matrix $A$ of (self-) linking numbers of the link determines a presentation of the linking form of $N$. Taking the number of link components in the surgery presentation for $N$ as the rank of $F$, the linking matrix $A$ determines a map $\Phi$ as above. The intersection form on a simply connected 4-manifold $W$ whose boundary is $N$ presents the linking form of $N$. This follows from the long exact sequence of the pair $W,N$ and Poincar\'{e} duality.
==Classification of 5-manifolds== Linking forms are central to the classification of simply connected 5-manifolds. See this [[5-manifolds: 1-connected|5-manifolds]] page. This page also described the classification of anti-symmetric linking forms. == References == {{#RefList:}} [[Category:Definitions]]n-manifold N has a bilinear linking form

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

Here the torsion part of an abelian group P is the subgroup

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

The linking form is the analogue for the torsion part of the homology of the intersection product which vanishes on the torsion part.

The linking form is defined as follows. Take [x] \in TH_\ell(N;\Zz) and [y] \in TH_{n-\ell-1}(N;\Zz) represented by cycles x \in C_\ell(N;\Zz) and y \in C_{n-\ell-1}(N,\Zz). There is Y\in C_{n-\ell}(N;\Z) such that \partial Y = sy for some s \in \Zz. Define

\displaystyle L_N([x],[y]):= \langle x, Y \rangle/s \in \mathbb{Q}/\Zz
to be the intersection number of x and Y divided by s and taken modulo 1.

For fixed N,[x],[y] the resulting `residie modulo 1' is independent of the choices of x,y,Y and s.

We have

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

[edit] 2 Definition via cohomology

Let x \in TH_{\ell}(N;\mathbb{Z}) and let y \in TH_{n-\ell-1}(N;\mathbb{Z}). Note that we have Poincaré duality isomorphisms

\displaystyle PD \colon TH_{\ell}(N;\mathbb{Z}) \xrightarrow{\cong} TH^{n-\ell}(N;\mathbb{Z})

and

\displaystyle PD \colon TH_{n-\ell-1}(N;\mathbb{Z}) \xrightarrow{\cong} TH^{\ell+1}(N;\mathbb{Z}).

Associated to the short exact sequence of coefficients

\displaystyle 0 \to \mathbb{Z} \to \mathbb{Q} \to \mathbb{Q}/\mathbb{Z} \to 0

is the Bockstein long exact sequence in cohomology:

\displaystyle H^{n-\ell-1}(N;\mathbb{Q}) \to H^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z}) \xrightarrow{\beta} H^{n-\ell}(N;\mathbb{Z}) \to H^{n-\ell}(N;\mathbb{Q}).

Choose z \in TH^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z}) such that \beta(z) = PD(x). This is always possible since torsion elements in H^{n-\ell}(N;\mathbb{Z}) map to zero in H^{n-\ell}(N;\mathbb{Q}). There is a cup product:

\displaystyle \cup \colon  H^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z}) \otimes H^{\ell+1}(N;\mathbb{Z}) \to  H^{n}(N;\mathbb{Q}/\mathbb{Z}).

Then the Kronecker pairing,

\displaystyle \langle z \cup PD(y),[N] \rangle \in \mathbb{Q}/\mathbb{Z},

of z \cup  PD(y) with the fundamental class of N yields L_N(x,y).

[edit] 3 Examples of 3-dimensional projective and lens spaces

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.

Generalising the above example, the 3-dimensional lens space N_{p,q} := L(p; q, 1) has H_1(N_{p,q};\mathbb{Z}) \cong \mathbb{Z}_p. The linking form is given on a generator \theta \in H_1(N_{p,q};\mathbb{Z}) by L_{N_{p,q}}(\theta,\theta) = q/p. Note that N_{2,1} \cong \mathbb{RP}^3, so this is consistent with the above example.

[edit] 4 Presentations of linking forms

A presentation for a middle dimensional linking form on N^{2\ell +1}

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

is an exact sequence:

\displaystyle 0 \to F \xrightarrow{\Phi} F^* \to TH_{\ell}(N;\mathbb{Z}) \xrightarrow{\partial} 0,

where F is a free abelain group and the linking L_N(x,y) can be computed as follows. Let x',y' \in F^* be such that \partial(x')=x and \partial(y')=y. Then we can tensor with \mathbb{Q} to obtain an isomorphism

\displaystyle \Phi \otimes \mathop{\mathrm{Id}} \colon F \otimes_{\mathbb{Z}} \mathbb{Q} \xrightarrow{\cong} F^* \otimes_{\mathbb{Z}} \mathbb{Q}.

The linking form of N is then given by:

\displaystyle L_N(x,y) = -(x' \otimes 1)((\Phi\otimes  \mathop{\mathrm{Id}})^{-1}(y'\otimes 1)).

This formula, in particular the appearance of the sign, is explained in [Gordon&Litherland1978, Section 3] and [Alexander&Hamrick&Vick1976, Proof of Theorem 2.1].

Let \ell = 1, so 2\ell + 1 = 3. Every 3-manifold N is the boundary of a simply connected 4-manifold, which is obtained by glueing 2-handles to an integrally framed link in S^3 [Lickorish1962], [Wallace1960]. This is sometimes called a surgery presentation for N. Suppose that N is a rational homology 3-sphere. Let A be the matrix of (self-) linking numbers of the surgery presentation link. Taking the number of link components in the surgery presentation for N as the rank of F, the linking matrix A determines a map \Phi as above, which presents the linking form of N. The intersection form on a simply connected 4-manifold W whose boundary is N presents the linking form of N. This follows from the long exact sequence of the pair (W,N) and Poincaré duality. See [Boyer1986] for more details and the use of such presentations for the classification of simply connected 4-manifolds with a given boundary.

For example, the lens space L(p,1) is the boundary of the D^2 bundle over S^2 with Euler number -p, so the presentation \Phi = (-p) \colon \mathbb{Z} \to \mathbb{Z} presents the linking form of L(p,1) [Gompf&Stipsicz1999, Example 5.3.2].

[edit] 5 Role in the classification of odd-dimensional manifolds

Linking forms play an important role in the classification of odd-dimensional manifolds.

For closed simply connected
Tex syntax error
-manifolds M, the linking form is a complete invariant if H_2(M; \Zz) = TH_2(M; \Zz). For more information in dimension
Tex syntax error
, see the page on simply-connected 5-manifolds.

For the role of linking forms in the classification of smooth (q-1)-connected (2q+1) manifolds with boundary a homotopy sphere, see [Wall1967, Theorem 7].

[edit] 6 Algebraic classification

An algebraic linking form is a non-singular bi-linear pairing

\displaystyle  b \colon T \times T \to \Qq/\Zz

on a finite abelian group T. It is called symmetric if b(x, y) = b(y, x) and skew-symmetric if b(x, y) = - b(y, x).

The classification of skew-symmetric linking forms is rather simple and is due to Wall, [Wall1963, Theorem 3]. It is described in detail in the page on simply-connected 5-manifolds.

The classification of symmetric linking forms is rather intricate. It was begun in [Wall1963] and completed by Kawauchi and Kojima: see [Kawauchi&Kojima1980, Theorem 4.1].

[edit] 7 References

[edit] 8 External links

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