Linking form
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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. | 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> | </wikitex> | ||
+ | |||
+ | |||
+ | ==Presentations of linking forms== | ||
+ | |||
+ | A presentation for a linking form is an exact sequence: | ||
+ | |||
+ | $$0 \to F \xrightarrow{\Phi} F^* \to TH_1(N;\mathbb{Z}) \xrightarrow{\partial} 0,$$ | ||
+ | |||
+ | 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== | ==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. | 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 == | == References == |
Revision as of 22:28, 27 March 2013
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Background: intersection forms
After Poincaré and Lefschetz, a closed oriented manifold has a bilinear intersection form defined on its homology. Given a --chain and an --chain which is transverse to , the signed count of the intersections between and gives an intersection number .
The intersection form is defined by
and is such that
2 Definition of the linking form
The analogue of the intersection pairing for the torsion part of the homology of a closed oriented manifold is the bilinear --valued linking form, which is due to Seifert:
such that
and computed as follows. Given and represented by cycles and , let be such that , for some . Then we define:
The resulting element is independent of the choices of and .
3 Definition via homology
Let and let . Note that we have Poincaré duality isomorphisms
and
Associated to the short exact sequence of coefficients
is the Bockstein long exact sequence in cohomology.
Choose such that . This is always possible since torsion elements in map to zero in . There is a cup product:
Compute . Then the Kronecker pairing:
yields .
4 Example of 3-dimensional projective space
As an example, let , so that and . Now . Let be the non-trivial element. To compute the linking , consider modelled as , with antipodal points on identified, and choose two representative -chains and for . Let be the straight line between north and south poles and let be half of the equator. Now , where is the 2-disk whose boundary is the equator. We see that , so that
5 Example of lens spaces
Generalising the above example, the 3-dimensional lens space has . The linking form is given on a generator by . Note that , so this is consistent with the above example.
6 Presentations of linking forms
A presentation for a linking form is an exact sequence:
$$0 \to F \xrightarrow{\Phi} F^* \to TH_1(N;\mathbb{Z}) \xrightarrow{\partial} 0,$$
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 $S^3$ [Lickorish1963], [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.
7 Classification of 5-manifolds
Linking forms are central to the classification of simply connected 5-manifolds. See this 5-manifolds page. This page also described the classification of anti-symmetric linking forms.
8 References
- [Lickorish1963] Template:Lickorish1963
- [Wallace1960] A. H. Wallace, Modifications and cobounding manifolds, Canad. J. Math. 12 (1960), 503–528. MR0125588 (23 #A2887) Zbl 0116.40401
The intersection form is defined by
and is such that
2 Definition of the linking form
The analogue of the intersection pairing for the torsion part of the homology of a closed oriented manifold is the bilinear --valued linking form, which is due to Seifert:
such that
and computed as follows. Given and represented by cycles and , let be such that , for some . Then we define:
The resulting element is independent of the choices of and .
3 Definition via homology
Let and let . Note that we have Poincaré duality isomorphisms
and
Associated to the short exact sequence of coefficients
is the Bockstein long exact sequence in cohomology.
Choose such that . This is always possible since torsion elements in map to zero in . There is a cup product:
Compute . Then the Kronecker pairing:
yields .
4 Example of 3-dimensional projective space
As an example, let , so that and . Now . Let be the non-trivial element. To compute the linking , consider modelled as , with antipodal points on identified, and choose two representative -chains and for . Let be the straight line between north and south poles and let be half of the equator. Now , where is the 2-disk whose boundary is the equator. We see that , so that
5 Example of lens spaces
Generalising the above example, the 3-dimensional lens space has . The linking form is given on a generator by . Note that , so this is consistent with the above example.
6 Presentations of linking forms
A presentation for a linking form is an exact sequence:
$$0 \to F \xrightarrow{\Phi} F^* \to TH_1(N;\mathbb{Z}) \xrightarrow{\partial} 0,$$
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 $S^3$ [Lickorish1963], [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.
7 Classification of 5-manifolds
Linking forms are central to the classification of simply connected 5-manifolds. See this 5-manifolds page. This page also described the classification of anti-symmetric linking forms.
8 References
- [Lickorish1963] Template:Lickorish1963
- [Wallace1960] A. H. Wallace, Modifications and cobounding manifolds, Canad. J. Math. 12 (1960), 503–528. MR0125588 (23 #A2887) Zbl 0116.40401
The intersection form is defined by
and is such that
2 Definition of the linking form
The analogue of the intersection pairing for the torsion part of the homology of a closed oriented manifold is the bilinear --valued linking form, which is due to Seifert:
such that
and computed as follows. Given and represented by cycles and , let be such that , for some . Then we define:
The resulting element is independent of the choices of and .
3 Definition via homology
Let and let . Note that we have Poincaré duality isomorphisms
and
Associated to the short exact sequence of coefficients
is the Bockstein long exact sequence in cohomology.
Choose such that . This is always possible since torsion elements in map to zero in . There is a cup product:
Compute . Then the Kronecker pairing:
yields .
4 Example of 3-dimensional projective space
As an example, let , so that and . Now . Let be the non-trivial element. To compute the linking , consider modelled as , with antipodal points on identified, and choose two representative -chains and for . Let be the straight line between north and south poles and let be half of the equator. Now , where is the 2-disk whose boundary is the equator. We see that , so that
5 Example of lens spaces
Generalising the above example, the 3-dimensional lens space has . The linking form is given on a generator by . Note that , so this is consistent with the above example.
6 Presentations of linking forms
A presentation for a linking form is an exact sequence:
$$0 \to F \xrightarrow{\Phi} F^* \to TH_1(N;\mathbb{Z}) \xrightarrow{\partial} 0,$$
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 $S^3$ [Lickorish1963], [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.
7 Classification of 5-manifolds
Linking forms are central to the classification of simply connected 5-manifolds. See this 5-manifolds page. This page also described the classification of anti-symmetric linking forms.
8 References
- [Lickorish1963] Template:Lickorish1963
- [Wallace1960] A. H. Wallace, Modifications and cobounding manifolds, Canad. J. Math. 12 (1960), 503–528. MR0125588 (23 #A2887) Zbl 0116.40401