Linking form

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Contents

1 Background: intersection forms

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}.


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 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 is the bilinear \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}) 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:

\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 and s.

3 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

\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}).

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

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

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

4 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 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.

5 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.

6 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 is given by:

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

\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 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 is the bilinear \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}) 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:

\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 and s.

3 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

\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}).

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

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

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

4 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 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.

5 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.

6 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 is given by:

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


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 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 is the bilinear \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}) 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:

\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 and s.

3 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

\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}).

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

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

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

4 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 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.

5 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.

6 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 is given by:

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

\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 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 is the bilinear \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}) 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:

\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 and s.

3 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

\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}).

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

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

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

4 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 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.

5 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.

6 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 is given by:

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

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