Linking form
Markpowell13 (Talk | contribs) |
Markpowell13 (Talk | contribs) |
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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 $s$. | ||
+ | |||
+ | ==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 Poincar\'{e} 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 | ||
+ | $$0 \to \mathbb{Z} \to \mathbb{Q} \to \mathbb{Q}/\mathbb{Z} \to 0$$ | ||
+ | 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== | ==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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Revision as of 20:34, 27 March 2013
This page has not been refereed. The information given here might be incomplete or provisional. |
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
By bilinearity, the intersection form vanishes on the torsion part of the homology. The torsion part of an abelian group is the setThe 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 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
By bilinearity, the intersection form vanishes on the torsion part of the homology. The torsion part of an abelian group is the setThe 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 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.$$ == 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
By bilinearity, the intersection form vanishes on the torsion part of the homology. The torsion part of an abelian group is the setThe 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