Linking form
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− | + | ==Background: intersection forms== | |
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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 $\langle\, p \, , \, q\, \rangle \in \mathbb{Z}$. | 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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and is such that | and is such that | ||
$$I_N(x,y) = (-)^{k(n-k)}I_N(y,x).$$ | $$I_N(x,y) = (-)^{k(n-k)}I_N(y,x).$$ | ||
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== Definition of the linking form== | == Definition of the linking form== | ||
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By bilinearity, the intersection form vanishes on the torsion part of the homology. The torsion part of an abelian group $P$ is the set $$TP:= \{p \in P \,|\, ap=0 \text{ for some } a \in \mathbb{Z}\}.$$ | By bilinearity, the intersection form vanishes on the torsion part of the homology. The torsion part of an abelian group $P$ is the set $$TP:= \{p \in P \,|\, ap=0 \text{ for some } a \in \mathbb{Z}\}.$$ | ||
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$$L_N([x],[y]) := \langle x, w \rangle/s \in \mathbb{Q}/\mathbb{Z}.$$ | $$L_N([x],[y]) := \langle x, w \rangle/s \in \mathbb{Q}/\mathbb{Z}.$$ | ||
The resulting element is independent of the choices of $x,y,w$ and $s$. | The resulting element is independent of the choices of $x,y,w$ and $s$. | ||
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==Definition via homology== | ==Definition via homology== | ||
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Let $x \in TH_{\ell}(N;\mathbb{Z})$ and let $y \in TH_{n-\ell-1}(N;\mathbb{Z})$. Note that we have Poincar\'{e} 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 Poincar\'{e} duality isomorphisms | ||
$$PD \colon TH_{\ell}(N;\mathbb{Z}) \xrightarrow{\cong} TH^{n-\ell}(N;\mathbb{Z})$$ | $$PD \colon TH_{\ell}(N;\mathbb{Z}) \xrightarrow{\cong} TH^{n-\ell}(N;\mathbb{Z})$$ | ||
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$$\langle a,[N] \rangle \in \mathbb{Q}/\mathbb{Z}$$ | $$\langle a,[N] \rangle \in \mathbb{Q}/\mathbb{Z}$$ | ||
yields $L_N(x,y)$. | yields $L_N(x,y)$. | ||
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==Example of 3-dimensional projective space== | ==Example of 3-dimensional projective space== | ||
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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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==Example of lens spaces== | ==Example of lens spaces== | ||
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Generalising the above example, the 3-dimensional lens space $L_{p,q}$ has $H_1(L_{p,q};\mathbb{Z}) \cong \mathbb{Z}_p$. The linking form is given on a generator $\theta \in H_1(L_{p,q};\mathbb{Z})$ by L_{L_{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 $L_{p,q}$ has $H_1(L_{p,q};\mathbb{Z}) \cong \mathbb{Z}_p$. The linking form is given on a generator $\theta \in H_1(L_{p,q};\mathbb{Z})$ by L_{L_{p,q}}(\theta,\theta) = p/q. Note that $L_{2,1} \cong \mathbb{RP}^3$, so this is consistent with the above example. | ||
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== References == | == References == | ||
{{#RefList:}} | {{#RefList:}} | ||
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[[Category:Definitions]] | [[Category:Definitions]] |
Revision as of 21:24, 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 Poincar\'{e} 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 L_{L_{p,q}}(\theta,\theta) = p/q. Note that , so this is consistent with the above example.
6 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-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^{n} 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 Poincar\'{e} 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 L_{L_{p,q}}(\theta,\theta) = p/q. Note that , so this is consistent with the above example.
6 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 $L_{p,q}$ has $H_1(L_{p,q};\mathbb{Z}) \cong \mathbb{Z}_p$. The linking form is given on a generator $\theta \in H_1(L_{p,q};\mathbb{Z})$ by L_{L_{p,q}}(\theta,\theta) = p/q. Note that $L_{2,1} \cong \mathbb{RP}^3$, so this is consistent with the above example. == References == {{#RefList:}} [[Category:Theory]] [[Category:Definitions]]N^{n} 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 Poincar\'{e} 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 L_{L_{p,q}}(\theta,\theta) = p/q. Note that , so this is consistent with the above example.