Linking form
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== Definition of the linking form== | == Definition of the linking form== | ||
− | By bilinearity, the intersection form vanishes on the torsion part of the homology. 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: | + | By bilinearity, the intersection form vanishes on the torsion part of the homology. The torsion part of an abelian group $P$ is the set $$TP:= \{p \in P \,|\, ap=0 \text{ for some } a \in \mathbb{Z}\}.$$ |
+ | |||
+ | 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: | ||
$$L_N \colon TH_{\ell}(N;\mathbb{Z}) \times TH_{n-\ell-1}(N;\mathbb{Z}) \to \mathbb{Q}/\mathbb{Z}$$ | $$L_N \colon TH_{\ell}(N;\mathbb{Z}) \times TH_{n-\ell-1}(N;\mathbb{Z}) \to \mathbb{Q}/\mathbb{Z}$$ | ||
such that | such that | ||
− | $$L_N(x,y) = (-)^{\ell(n-\ell-1)}L_N(y,x) | + | $$L_N(x,y) = (-)^{\ell(n-\ell-1)}L_N(y,x)$$ |
− | + | and computed as follows. Given $[x] \in TH_\ell(N;\mathbb{Z})$ and $[y] \in TH_{n-\ell-1}(N;\mathbb{Z})$ represented by cycles $x \in C_\ell(N;\mathbb{Z})$ and $y \in C_{n-\ell-1}(N,\mathbb{Z})$, let $w \in C_{\ell+1}(N;\mathbb{Z})$ be such that $\partial w = sy$, for some $s \in \mathbb{Z}$. Then we define: | |
− | + | ||
− | Given $[x] \in TH_\ell(N;\mathbb{Z})$ and $[y] \in TH_{n-\ell-1}(N;\mathbb{Z})$ represented by cycles $x \in C_\ell(N;\mathbb{Z})$ and $y \in C_{n-\ell-1}(N,\mathbb{Z})$, let $w \in C_{\ell+1}(N;\mathbb{Z})$ be such that $\partial w = sy$, for some $s \in \mathbb{Z}$. Then we define: | + | |
$$L_N([x],[y]) := \langle x, w \rangle/s \in \mathbb{Q}/\mathbb{Z}.$$ | $$L_N([x],[y]) := \langle x, w \rangle/s \in \mathbb{Q}/\mathbb{Z}.$$ | ||
The resulting element is independent of the choices of $x,y,w$ and $s$. | The resulting element is independent of the choices of $x,y,w$ and $s$. |
Revision as of 16:00, 27 March 2013
This page has not been refereed. The information given here might be incomplete or provisional. |
1 Background: intersection forms
After Poincaré and Lefschetz, a closed oriented manifold has a bilinear intersection form defined on its homology. Given a --chain and an --chain which is transverse to , the signed count of the intersections between and gives an intersection number .
The intersection form is defined by
and is such that
2 Definition of the linking form
By bilinearity, the intersection form vanishes on the torsion part of the homology. The torsion part of an abelian group is the setThe analogue of the intersection pairing for the torsion part of the homology of a closed oriented manifold is the bilinear --valued linking form, which is due to Seifert:
such that
and computed as follows. Given and represented by cycles and , let be such that , for some . Then we define:
The resulting element is independent of the choices of and .
As an example, let , so that and . Now . Let be the non--trivial element. To compute the linking , consider modelled as , with antipodal points on identified, and choose two representative --chains and for . Let be the straight line between north and south poles and let be half of the equator. Now , where is the 2-disk whose boundary is the equator. We see that , so that
3 References
$--chains $x$ and $y$ for $\theta$. Let $x$ be the straight line between north and south poles and let $y$ be half of the equator. Now y = \partial w$, where $w \in C_2(\mathbb{RP}^3;\mathbb{Z})$ is the 2-disk whose boundary is the equator. We see that $\langle x,w \rangle = 1$, so that $$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 .
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