# Linking form

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## 1 Background: intersection forms

After Poincaré and Lefschetz, a closed oriented manifold $N^{n}$${{Stub}} ==Background: intersection forms== ; After Poincaré and Lefschetz, a closed oriented manifold N^{n} has a bilinear [[Intersection form|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}. The intersection form is defined by 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 and is such that I_N(x,y) = (-)^{k(n-k)}I_N(y,x). == Definition of the linking form== ; By bilinearity, the intersection form vanishes on the torsion part of the homology. The torsion part of an abelian group P is the subgroup 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 \cite{Seifert1933}: L_N \colon TH_{\ell}(N;\mathbb{Z}) \times TH_{n-\ell-1}(N;\mathbb{Z}) \to \mathbb{Q}/\mathbb{Z} such that L_N(x,y) = (-)^{\ell(n-\ell-1)}L_N(y,x) 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_{n-\ell}(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 \mathbb{Q}/\mathbb{Z}. The resulting element is independent of the choices of x,y,w and s. ==Definition via homology== ; 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 PD \colon TH_{\ell}(N;\mathbb{Z}) \xrightarrow{\cong} TH^{n-\ell}(N;\mathbb{Z}) and 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 has a bilinear intersection form defined on its homology. Given a ${k}$${k}$--chain $p \in C_{k}(N;\mathbb{Z})$$p \in C_{k}(N;\mathbb{Z})$ and an $(n-k)$$(n-k)$--chain $q \in C_{n-k}(N;\mathbb{Z})$$q \in C_{n-k}(N;\mathbb{Z})$ which is transverse to $q$$q$, the signed count of the intersections between $p$$p$ and $q$$q$ gives an intersection number $\langle\, p \, , \, q\, \rangle \in \mathbb{Z}$$\langle\, p \, , \, q\, \rangle \in \mathbb{Z}$.

The intersection form is defined by

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

and is such that

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

## 2 Definition of the linking form

By bilinearity, the intersection form vanishes on the torsion part of the homology. The torsion part of an abelian group $P$$P$ is the subgroup
$\displaystyle TP:= \{p \in P \,|\, ap=0 \text{ for some } a \in \mathbb{Z}\}.$

The analogue of the intersection pairing for the torsion part of the homology of a closed oriented manifold $N^n$$N^n$ is the bilinear $\mathbb{Q}/\mathbb{Z}$$\mathbb{Q}/\mathbb{Z}$--valued linking form, which is due to Seifert [Seifert1933]:

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

such that

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

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

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

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

## 3 Definition via homology

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

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

Compute $a:= z \cup PD(y)$$a:= z \cup PD(y)$. Then the Kronecker pairing:

$\displaystyle \langle a,[N] \rangle \in \mathbb{Q}/\mathbb{Z}$

of $a$$a$ with the fundamental class of $N$$N$ yields $L_N(x,y)$$L_N(x,y)$.

## 4 Example of 3-dimensional projective space

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

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

## 5 Example of lens spaces

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

## 6 Presentations of linking forms

A presentation for a middle dimensional linking form on $N^{2\ell +1}$$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$$F$ is a free abelain group and the linking $L_N(x,y)$$L_N(x,y)$ can be computed as follows. Let $x',y' \in F^*$$x',y' \in F^*$ be such that $\partial(x')=x$$\partial(x')=x$ and $\partial(y')=y$$\partial(y')=y$. Then we can tensor with $\mathbb{Q}$$\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 is given by:

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

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

## 7 Classification of 5-manifolds

Linking forms are central to the classification of simply connected 5-manifolds. See this 5-manifolds page, which also describes the classification of anti-symmetric linking forms.

\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}(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}$$of a with the fundamental class of N yields L_N(x,y). ==Example of 3-dimensional projective space== ; As an example, let N = \mathbb{RP}^3, so that \ell=1 and n=3. Now H_1(\mathbb{RP}^3;\mathbb{Z}) \cong \mathbb{Z}_2. Let \theta \in H_1(\mathbb{RP}^3;\mathbb{Z}) be the non-trivial element. To compute the linking L_{\mathbb{RP}^3}(\theta,\theta), consider \mathbb{RP}^3 modelled as D^3/\sim, with antipodal points on \partial D^2 identified, and choose two representative N^{n} has a bilinear intersection form defined on its homology. Given a ${k}$${k}$--chain $p \in C_{k}(N;\mathbb{Z})$$p \in C_{k}(N;\mathbb{Z})$ and an $(n-k)$$(n-k)$--chain $q \in C_{n-k}(N;\mathbb{Z})$$q \in C_{n-k}(N;\mathbb{Z})$ which is transverse to $q$$q$, the signed count of the intersections between $p$$p$ and $q$$q$ gives an intersection number $\langle\, p \, , \, q\, \rangle \in \mathbb{Z}$$\langle\, p \, , \, q\, \rangle \in \mathbb{Z}$. The intersection form is defined by $\displaystyle I_N \colon H_k(N;\mathbb{Z}) \times H_{n-k}(N;\mathbb{Z}) \to \mathbb{Z}; ([p],[q]) \mapsto \langle p, q \rangle$ and is such that $\displaystyle I_N(x,y) = (-)^{k(n-k)}I_N(y,x).$ ## 2 Definition of the linking form By bilinearity, the intersection form vanishes on the torsion part of the homology. The torsion part of an abelian group $P$$P$ is the subgroup $\displaystyle TP:= \{p \in P \,|\, ap=0 \text{ for some } a \in \mathbb{Z}\}.$ The analogue of the intersection pairing for the torsion part of the homology of a closed oriented manifold $N^n$$N^n$ is the bilinear $\mathbb{Q}/\mathbb{Z}$$\mathbb{Q}/\mathbb{Z}$--valued linking form, which is due to Seifert [Seifert1933]: $\displaystyle L_N \colon TH_{\ell}(N;\mathbb{Z}) \times TH_{n-\ell-1}(N;\mathbb{Z}) \to \mathbb{Q}/\mathbb{Z}$ such that $\displaystyle L_N(x,y) = (-)^{\ell(n-\ell-1)}L_N(y,x)$ and computed as follows. Given $[x] \in TH_\ell(N;\mathbb{Z})$$[x] \in TH_\ell(N;\mathbb{Z})$ and $[y] \in TH_{n-\ell-1}(N;\mathbb{Z})$$[y] \in TH_{n-\ell-1}(N;\mathbb{Z})$ represented by cycles $x \in C_\ell(N;\mathbb{Z})$$x \in C_\ell(N;\mathbb{Z})$ and $y \in C_{n-\ell-1}(N,\mathbb{Z})$$y \in C_{n-\ell-1}(N,\mathbb{Z})$, let $w \in C_{n-\ell}(N;\mathbb{Z})$$w \in C_{n-\ell}(N;\mathbb{Z})$ be such that $\partial w = sy$$\partial w = sy$, for some $s \in \mathbb{Z}$$s \in \mathbb{Z}$. Then we define: $\displaystyle L_N([x],[y]) := \langle x, w \rangle/s \in \mathbb{Q}/\mathbb{Z}.$ The resulting element is independent of the choices of $x,y,w$$x,y,w$ and $s$$s$. ## 3 Definition via homology Let $x \in TH_{\ell}(N;\mathbb{Z})$$x \in TH_{\ell}(N;\mathbb{Z})$ and let $y \in TH_{n-\ell-1}(N;\mathbb{Z})$$y \in TH_{n-\ell-1}(N;\mathbb{Z})$. Note that we have Poincaré duality isomorphisms $\displaystyle PD \colon TH_{\ell}(N;\mathbb{Z}) \xrightarrow{\cong} TH^{n-\ell}(N;\mathbb{Z})$ and $\displaystyle PD \colon TH_{n-\ell-1}(N;\mathbb{Z}) \xrightarrow{\cong} TH^{\ell+1}(N;\mathbb{Z}).$ Associated to the short exact sequence of coefficients $\displaystyle 0 \to \mathbb{Z} \to \mathbb{Q} \to \mathbb{Q}/\mathbb{Z} \to 0$ is the Bockstein long exact sequence in cohomology. $\displaystyle H^{n-\ell-1}(N;\mathbb{Q}) \to H^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z}) \xrightarrow{\beta} H^{n-\ell}(N;\mathbb{Z}) \to H^{n-\ell}(N;\mathbb{Q}).$ Choose $z \in TH^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z})$$z \in TH^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z})$ such that $\beta(z) = PD(x)$$\beta(z) = PD(x)$. This is always possible since torsion elements in $H^{n-\ell}(N;\mathbb{Z})$$H^{n-\ell}(N;\mathbb{Z})$ map to zero in $H^{n-\ell}(N;\mathbb{Q})$$H^{n-\ell}(N;\mathbb{Q})$. There is a cup product: $\displaystyle \cup \colon H^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z}) \otimes H^{\ell+1}(N;\mathbb{Z}) \to H^{n}(N;\mathbb{Q}/\mathbb{Z}).$ Compute $a:= z \cup PD(y)$$a:= z \cup PD(y)$. Then the Kronecker pairing: $\displaystyle \langle a,[N] \rangle \in \mathbb{Q}/\mathbb{Z}$ of $a$$a$ with the fundamental class of $N$$N$ yields $L_N(x,y)$$L_N(x,y)$. ## 4 Example of 3-dimensional projective space As an example, let $N = \mathbb{RP}^3$$N = \mathbb{RP}^3$, so that $\ell=1$$\ell=1$ and $n=3$$n=3$. Now $H_1(\mathbb{RP}^3;\mathbb{Z}) \cong \mathbb{Z}_2$$H_1(\mathbb{RP}^3;\mathbb{Z}) \cong \mathbb{Z}_2$. Let $\theta \in H_1(\mathbb{RP}^3;\mathbb{Z})$$\theta \in H_1(\mathbb{RP}^3;\mathbb{Z})$ be the non-trivial element. To compute the linking $L_{\mathbb{RP}^3}(\theta,\theta)$$L_{\mathbb{RP}^3}(\theta,\theta)$, consider $\mathbb{RP}^3$$\mathbb{RP}^3$ modelled as $D^3/\sim$$D^3/\sim$, with antipodal points on $\partial D^2$$\partial D^2$ identified, and choose two representative $1$$1$-chains $x$$x$ and $y$$y$ for $\theta$$\theta$. Let $x$$x$ be the straight line between north and south poles and let $y$$y$ be half of the equator. Now $2y = \partial w$$2y = \partial w$, where $w \in C_2(\mathbb{RP}^3;\mathbb{Z})$$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$$\langle x,w \rangle = 1$, so that $\displaystyle L_{\mathbb{RP}^3}(\theta,\theta) = L_{\mathbb{RP}^3}([x],[y]) = \langle x,y \rangle/2 = 1/2.$ ## 5 Example of lens spaces Generalising the above example, the 3-dimensional lens space $N_{p,q}$$N_{p,q}$ has $H_1(N_{p,q};\mathbb{Z}) \cong \mathbb{Z}_p$$H_1(N_{p,q};\mathbb{Z}) \cong \mathbb{Z}_p$. The linking form is given on a generator $\theta \in H_1(N_{p,q};\mathbb{Z})$$\theta \in H_1(N_{p,q};\mathbb{Z})$ by $L_{N_{p,q}}(\theta,\theta) = q/p$$L_{N_{p,q}}(\theta,\theta) = q/p$. Note that $L_{2,1} \cong \mathbb{RP}^3$$L_{2,1} \cong \mathbb{RP}^3$, so this is consistent with the above example. ## 6 Presentations of linking forms A presentation for a middle dimensional linking form on $N^{2\ell +1}$$N^{2\ell +1}$ $\displaystyle L_N \colon TH_{\ell}(N;\mathbb{Z}) \times TH_{\ell}(N;\mathbb{Z}) \to \mathbb{Q}/\mathbb{Z}$ is an exact sequence: $\displaystyle 0 \to F \xrightarrow{\Phi} F^* \to TH_{\ell}(N;\mathbb{Z}) \xrightarrow{\partial} 0,$ where $F$$F$ is a free abelain group and the linking $L_N(x,y)$$L_N(x,y)$ can be computed as follows. Let $x',y' \in F^*$$x',y' \in F^*$ be such that $\partial(x')=x$$\partial(x')=x$ and $\partial(y')=y$$\partial(y')=y$. Then we can tensor with $\mathbb{Q}$$\mathbb{Q}$ to obtain an isomorphism $\displaystyle \Phi \otimes \mathop{\mathrm{Id}} \colon F \otimes_{\mathbb{Z}} \mathbb{Q} \xrightarrow{\cong} F^* \otimes_{\mathbb{Z}} \mathbb{Q}.$ The linking form is given by: $\displaystyle L_N(x,y) = (x' \otimes 1)((\Phi\otimes \mathop{\mathrm{Id}})^{-1}(y'\otimes 1)).$ Let $\ell = 1$$\ell = 1$, so $2\ell + 1 = 3$$2\ell + 1 = 3$. Every 3-manifold $N$$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$$S^3$ [Lickorish1962], [Wallace1960]. This is sometimes called a surgery presentation for $N$$N$. Suppose that $N$$N$ is a rational homology 3-sphere. Let $A$$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$$N$ as the rank of $F$$F$, the linking matrix $A$$A$ determines a map $\Phi$$\Phi$ as above, which presents the linking form of $N$$N$. The intersection form on a simply connected 4-manifold $W$$W$ whose boundary is $N$$N$ presents the linking form of $N$$N$. This follows from the long exact sequence of the pair~$(W,N)$$(W,N)$ and Poincar\'{e} duality. See [Boyer1986] for more details and the use of such presentations for the classification of simply connected 4-manifolds with a given boundary. ## 7 Classification of 5-manifolds Linking forms are central to the classification of simply connected 5-manifolds. See this 5-manifolds page, which also describes the classification of anti-symmetric linking forms. ## 8 References -chains x and y for \theta. Let x be the straight line between north and south poles and let y be half of the equator. Now y = \partial w, where w \in C_2(\mathbb{RP}^3;\mathbb{Z}) is the 2-disk whose boundary is the equator. We see that \langle x,w \rangle = 1, so that$$L_{\mathbb{RP}^3}(\theta,\theta) = L_{\mathbb{RP}^3}([x],[y]) = \langle x,y \rangle/2 = 1/2.$$==Example of lens spaces== ; Generalising the above example, the 3-dimensional lens space N_{p,q} has H_1(N_{p,q};\mathbb{Z}) \cong \mathbb{Z}_p. The linking form is given on a generator \theta \in H_1(N_{p,q};\mathbb{Z}) by L_{N_{p,q}}(\theta,\theta) = q/p. Note that L_{2,1} \cong \mathbb{RP}^3, so this is consistent with the above example. ==Presentations of linking forms== ; A presentation for a middle dimensional linking form on N^{2\ell +1}$$L_N \colon TH_{\ell}(N;\mathbb{Z}) \times TH_{\ell}(N;\mathbb{Z}) \to \mathbb{Q}/\mathbb{Z}$$is an exact sequence: $N^{n}$ has a bilinear intersection form defined on its homology. Given a ${k}$${k}$--chain $p \in C_{k}(N;\mathbb{Z})$$p \in C_{k}(N;\mathbb{Z})$ and an $(n-k)$$(n-k)$--chain $q \in C_{n-k}(N;\mathbb{Z})$$q \in C_{n-k}(N;\mathbb{Z})$ which is transverse to $q$$q$, the signed count of the intersections between $p$$p$ and $q$$q$ gives an intersection number $\langle\, p \, , \, q\, \rangle \in \mathbb{Z}$$\langle\, p \, , \, q\, \rangle \in \mathbb{Z}$. The intersection form is defined by $\displaystyle I_N \colon H_k(N;\mathbb{Z}) \times H_{n-k}(N;\mathbb{Z}) \to \mathbb{Z}; ([p],[q]) \mapsto \langle p, q \rangle$ and is such that $\displaystyle I_N(x,y) = (-)^{k(n-k)}I_N(y,x).$ ## 2 Definition of the linking form By bilinearity, the intersection form vanishes on the torsion part of the homology. The torsion part of an abelian group $P$$P$ is the subgroup $\displaystyle TP:= \{p \in P \,|\, ap=0 \text{ for some } a \in \mathbb{Z}\}.$ The analogue of the intersection pairing for the torsion part of the homology of a closed oriented manifold $N^n$$N^n$ is the bilinear $\mathbb{Q}/\mathbb{Z}$$\mathbb{Q}/\mathbb{Z}$--valued linking form, which is due to Seifert [Seifert1933]: $\displaystyle L_N \colon TH_{\ell}(N;\mathbb{Z}) \times TH_{n-\ell-1}(N;\mathbb{Z}) \to \mathbb{Q}/\mathbb{Z}$ such that $\displaystyle L_N(x,y) = (-)^{\ell(n-\ell-1)}L_N(y,x)$ and computed as follows. Given $[x] \in TH_\ell(N;\mathbb{Z})$$[x] \in TH_\ell(N;\mathbb{Z})$ and $[y] \in TH_{n-\ell-1}(N;\mathbb{Z})$$[y] \in TH_{n-\ell-1}(N;\mathbb{Z})$ represented by cycles $x \in C_\ell(N;\mathbb{Z})$$x \in C_\ell(N;\mathbb{Z})$ and $y \in C_{n-\ell-1}(N,\mathbb{Z})$$y \in C_{n-\ell-1}(N,\mathbb{Z})$, let $w \in C_{n-\ell}(N;\mathbb{Z})$$w \in C_{n-\ell}(N;\mathbb{Z})$ be such that $\partial w = sy$$\partial w = sy$, for some $s \in \mathbb{Z}$$s \in \mathbb{Z}$. Then we define: $\displaystyle L_N([x],[y]) := \langle x, w \rangle/s \in \mathbb{Q}/\mathbb{Z}.$ The resulting element is independent of the choices of $x,y,w$$x,y,w$ and $s$$s$. ## 3 Definition via homology Let $x \in TH_{\ell}(N;\mathbb{Z})$$x \in TH_{\ell}(N;\mathbb{Z})$ and let $y \in TH_{n-\ell-1}(N;\mathbb{Z})$$y \in TH_{n-\ell-1}(N;\mathbb{Z})$. Note that we have Poincaré duality isomorphisms $\displaystyle PD \colon TH_{\ell}(N;\mathbb{Z}) \xrightarrow{\cong} TH^{n-\ell}(N;\mathbb{Z})$ and $\displaystyle PD \colon TH_{n-\ell-1}(N;\mathbb{Z}) \xrightarrow{\cong} TH^{\ell+1}(N;\mathbb{Z}).$ Associated to the short exact sequence of coefficients $\displaystyle 0 \to \mathbb{Z} \to \mathbb{Q} \to \mathbb{Q}/\mathbb{Z} \to 0$ is the Bockstein long exact sequence in cohomology. $\displaystyle H^{n-\ell-1}(N;\mathbb{Q}) \to H^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z}) \xrightarrow{\beta} H^{n-\ell}(N;\mathbb{Z}) \to H^{n-\ell}(N;\mathbb{Q}).$ Choose $z \in TH^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z})$$z \in TH^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z})$ such that $\beta(z) = PD(x)$$\beta(z) = PD(x)$. This is always possible since torsion elements in $H^{n-\ell}(N;\mathbb{Z})$$H^{n-\ell}(N;\mathbb{Z})$ map to zero in $H^{n-\ell}(N;\mathbb{Q})$$H^{n-\ell}(N;\mathbb{Q})$. There is a cup product: $\displaystyle \cup \colon H^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z}) \otimes H^{\ell+1}(N;\mathbb{Z}) \to H^{n}(N;\mathbb{Q}/\mathbb{Z}).$ Compute $a:= z \cup PD(y)$$a:= z \cup PD(y)$. Then the Kronecker pairing: $\displaystyle \langle a,[N] \rangle \in \mathbb{Q}/\mathbb{Z}$ of $a$$a$ with the fundamental class of $N$$N$ yields $L_N(x,y)$$L_N(x,y)$. ## 4 Example of 3-dimensional projective space As an example, let $N = \mathbb{RP}^3$$N = \mathbb{RP}^3$, so that $\ell=1$$\ell=1$ and $n=3$$n=3$. Now $H_1(\mathbb{RP}^3;\mathbb{Z}) \cong \mathbb{Z}_2$$H_1(\mathbb{RP}^3;\mathbb{Z}) \cong \mathbb{Z}_2$. Let $\theta \in H_1(\mathbb{RP}^3;\mathbb{Z})$$\theta \in H_1(\mathbb{RP}^3;\mathbb{Z})$ be the non-trivial element. To compute the linking $L_{\mathbb{RP}^3}(\theta,\theta)$$L_{\mathbb{RP}^3}(\theta,\theta)$, consider $\mathbb{RP}^3$$\mathbb{RP}^3$ modelled as $D^3/\sim$$D^3/\sim$, with antipodal points on $\partial D^2$$\partial D^2$ identified, and choose two representative $1$$1$-chains $x$$x$ and $y$$y$ for $\theta$$\theta$. Let $x$$x$ be the straight line between north and south poles and let $y$$y$ be half of the equator. Now $2y = \partial w$$2y = \partial w$, where $w \in C_2(\mathbb{RP}^3;\mathbb{Z})$$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$$\langle x,w \rangle = 1$, so that $\displaystyle L_{\mathbb{RP}^3}(\theta,\theta) = L_{\mathbb{RP}^3}([x],[y]) = \langle x,y \rangle/2 = 1/2.$ ## 5 Example of lens spaces Generalising the above example, the 3-dimensional lens space $N_{p,q}$$N_{p,q}$ has $H_1(N_{p,q};\mathbb{Z}) \cong \mathbb{Z}_p$$H_1(N_{p,q};\mathbb{Z}) \cong \mathbb{Z}_p$. The linking form is given on a generator $\theta \in H_1(N_{p,q};\mathbb{Z})$$\theta \in H_1(N_{p,q};\mathbb{Z})$ by $L_{N_{p,q}}(\theta,\theta) = q/p$$L_{N_{p,q}}(\theta,\theta) = q/p$. Note that $L_{2,1} \cong \mathbb{RP}^3$$L_{2,1} \cong \mathbb{RP}^3$, so this is consistent with the above example. ## 6 Presentations of linking forms A presentation for a middle dimensional linking form on $N^{2\ell +1}$$N^{2\ell +1}$ $\displaystyle L_N \colon TH_{\ell}(N;\mathbb{Z}) \times TH_{\ell}(N;\mathbb{Z}) \to \mathbb{Q}/\mathbb{Z}$ is an exact sequence: $\displaystyle 0 \to F \xrightarrow{\Phi} F^* \to TH_{\ell}(N;\mathbb{Z}) \xrightarrow{\partial} 0,$ where $F$$F$ is a free abelain group and the linking $L_N(x,y)$$L_N(x,y)$ can be computed as follows. Let $x',y' \in F^*$$x',y' \in F^*$ be such that $\partial(x')=x$$\partial(x')=x$ and $\partial(y')=y$$\partial(y')=y$. Then we can tensor with $\mathbb{Q}$$\mathbb{Q}$ to obtain an isomorphism $\displaystyle \Phi \otimes \mathop{\mathrm{Id}} \colon F \otimes_{\mathbb{Z}} \mathbb{Q} \xrightarrow{\cong} F^* \otimes_{\mathbb{Z}} \mathbb{Q}.$ The linking form is given by: $\displaystyle L_N(x,y) = (x' \otimes 1)((\Phi\otimes \mathop{\mathrm{Id}})^{-1}(y'\otimes 1)).$ Let $\ell = 1$$\ell = 1$, so $2\ell + 1 = 3$$2\ell + 1 = 3$. Every 3-manifold $N$$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$$S^3$ [Lickorish1962], [Wallace1960]. This is sometimes called a surgery presentation for $N$$N$. Suppose that $N$$N$ is a rational homology 3-sphere. Let $A$$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$$N$ as the rank of $F$$F$, the linking matrix $A$$A$ determines a map $\Phi$$\Phi$ as above, which presents the linking form of $N$$N$. The intersection form on a simply connected 4-manifold $W$$W$ whose boundary is $N$$N$ presents the linking form of $N$$N$. This follows from the long exact sequence of the pair~$(W,N)$$(W,N)$ and Poincar\'{e} duality. See [Boyer1986] for more details and the use of such presentations for the classification of simply connected 4-manifolds with a given boundary. ## 7 Classification of 5-manifolds Linking forms are central to the classification of simply connected 5-manifolds. See this 5-manifolds page, which also describes the classification of anti-symmetric linking forms. ## 8 References \to F \xrightarrow{\Phi} F^* \to TH_{\ell}(N;\mathbb{Z}) \xrightarrow{\partial} 0,$$ 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 $$\Phi \otimes \mathop{\mathrm{Id}} \colon F \otimes_{\mathbb{Z}} \mathbb{Q} \xrightarrow{\cong} F^* \otimes_{\mathbb{Z}} \mathbb{Q}.$$ The linking form is given by: $$L_N(x,y) = (x' \otimes 1)((\Phi\otimes \mathop{\mathrm{Id}})^{-1}(y'\otimes 1)).$$ Let $\ell = 1$, so \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 Poincar\'{e} 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.
==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, which also describes the classification of anti-symmetric linking forms. == References == {{#RefList:}} [[Category:Definitions]]N^{n} has a bilinear intersection form defined on its homology. Given a ${k}$${k}$--chain $p \in C_{k}(N;\mathbb{Z})$$p \in C_{k}(N;\mathbb{Z})$ and an $(n-k)$$(n-k)$--chain $q \in C_{n-k}(N;\mathbb{Z})$$q \in C_{n-k}(N;\mathbb{Z})$ which is transverse to $q$$q$, the signed count of the intersections between $p$$p$ and $q$$q$ gives an intersection number $\langle\, p \, , \, q\, \rangle \in \mathbb{Z}$$\langle\, p \, , \, q\, \rangle \in \mathbb{Z}$.

The intersection form is defined by

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

and is such that

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

## 2 Definition of the linking form

By bilinearity, the intersection form vanishes on the torsion part of the homology. The torsion part of an abelian group $P$$P$ is the subgroup
$\displaystyle TP:= \{p \in P \,|\, ap=0 \text{ for some } a \in \mathbb{Z}\}.$

The analogue of the intersection pairing for the torsion part of the homology of a closed oriented manifold $N^n$$N^n$ is the bilinear $\mathbb{Q}/\mathbb{Z}$$\mathbb{Q}/\mathbb{Z}$--valued linking form, which is due to Seifert [Seifert1933]:

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

such that

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

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

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

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

## 3 Definition via homology

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

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

Compute $a:= z \cup PD(y)$$a:= z \cup PD(y)$. Then the Kronecker pairing:

$\displaystyle \langle a,[N] \rangle \in \mathbb{Q}/\mathbb{Z}$

of $a$$a$ with the fundamental class of $N$$N$ yields $L_N(x,y)$$L_N(x,y)$.

## 4 Example of 3-dimensional projective space

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

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

## 5 Example of lens spaces

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

## 6 Presentations of linking forms

A presentation for a middle dimensional linking form on $N^{2\ell +1}$$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$$F$ is a free abelain group and the linking $L_N(x,y)$$L_N(x,y)$ can be computed as follows. Let $x',y' \in F^*$$x',y' \in F^*$ be such that $\partial(x')=x$$\partial(x')=x$ and $\partial(y')=y$$\partial(y')=y$. Then we can tensor with $\mathbb{Q}$$\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 is given by:

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

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

## 7 Classification of 5-manifolds

Linking forms are central to the classification of simply connected 5-manifolds. See this 5-manifolds page, which also describes the classification of anti-symmetric linking forms.