Linking form

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Revision as of 20:34, 27 March 2013

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1 Background: intersection forms

After Poincaré and Lefschetz, a closed oriented manifold ${{Stub}} <wikitex>; ==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 $$I_N \colon H_k(N;\mathbb{Z}) \times H_{n-k}(N;\mathbb{Z}) \to \mathbb{Z}; ([p],[q]) \mapsto \langle p, q \rangle$$ and is such that $$I_N(x,y) = (-)^{k(n-k)}I_N(y,x).$$ == 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 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: $$L_N \colon TH_{\ell}(N;\mathbb{Z}) \times TH_{n-\ell-1}(N;\mathbb{Z}) \to \mathbb{Q}/\mathbb{Z}$$ such that $$L_N(x,y) = (-)^{\ell(n-\ell-1)}L_N(y,x)$$ 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: $$L_N([x],[y]) := \langle x, w \rangle/s \in \mathbb{Q}/\mathbb{Z}.$$ The resulting element is independent of the choices of $x,y,w$ and $s$. ==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

$\displaystyle I_N \colon H_k(N;\mathbb{Z}) \times H_{n-k}(N;\mathbb{Z}) \to \mathbb{Z}; ([p],[q]) \mapsto \langle p, q \rangle$ $\displaystyle I_N \colon H_k(N;\mathbb{Z}) \times H_{n-k}(N;\mathbb{Z}) \to \mathbb{Z}; ([p],[q]) \mapsto \langle p, q \rangle$

and is such that

$\displaystyle I_N(x,y) = (-)^{k(n-k)}I_N(y,x).$ $\displaystyle I_N(x,y) = (-)^{k(n-k)}I_N(y,x).$

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 $P$ $P$ is the set

$\displaystyle TP:= \{p \in P \,|\, ap=0 \text{ for some } a \in \mathbb{Z}\}.$ $\displaystyle TP:= \{p \in P \,|\, ap=0 \text{ for some } a \in \mathbb{Z}\}.$

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:

$\displaystyle L_N \colon TH_{\ell}(N;\mathbb{Z}) \times TH_{n-\ell-1}(N;\mathbb{Z}) \to \mathbb{Q}/\mathbb{Z}$ $\displaystyle L_N \colon TH_{\ell}(N;\mathbb{Z}) \times TH_{n-\ell-1}(N;\mathbb{Z}) \to \mathbb{Q}/\mathbb{Z}$

such that

$\displaystyle L_N(x,y) = (-)^{\ell(n-\ell-1)}L_N(y,x)$ $\displaystyle L_N(x,y) = (-)^{\ell(n-\ell-1)}L_N(y,x)$

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 $w \in C_{\ell+1}(N;\mathbb{Z})$ be such that $\partial w = sy$ , for some $s \in \mathbb{Z}$ . Then we define:

$\displaystyle L_N([x],[y]) := \langle x, w \rangle/s \in \mathbb{Q}/\mathbb{Z}.$ $\displaystyle L_N([x],[y]) := \langle x, w \rangle/s \in \mathbb{Q}/\mathbb{Z}.$

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 Poincar\'{e} duality isomorphisms

$\displaystyle PD \colon TH_{\ell}(N;\mathbb{Z}) \xrightarrow{\cong} TH^{n-\ell}(N;\mathbb{Z})$ $\displaystyle PD \colon TH_{\ell}(N;\mathbb{Z}) \xrightarrow{\cong} TH^{n-\ell}(N;\mathbb{Z})$

and

$\displaystyle PD \colon TH_{n-\ell-1}(N;\mathbb{Z}) \xrightarrow{\cong} TH^{\ell+1}(N;\mathbb{Z}).$ $\displaystyle PD \colon TH_{n-\ell-1}(N;\mathbb{Z}) \xrightarrow{\cong} TH^{\ell+1}(N;\mathbb{Z}).$

Associated to the short exact sequence of coefficients

$\displaystyle 0 \to \mathbb{Z} \to \mathbb{Q} \to \mathbb{Q}/\mathbb{Z} \to 0$ $\displaystyle 0 \to \mathbb{Z} \to \mathbb{Q} \to \mathbb{Q}/\mathbb{Z} \to 0$

is the Bockstein long exact sequence in cohomology.

$\displaystyle H^{n-\ell-1}(N;\mathbb{Q}) \to H^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z}) \xrightarrow{\beta} H^{n-\ell}(N;\mathbb{Z}) \to H^{n-\ell-1}(N;\mathbb{Q}).$ $\displaystyle H^{n-\ell-1}(N;\mathbb{Q}) \to H^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z}) \xrightarrow{\beta} H^{n-\ell}(N;\mathbb{Z}) \to H^{n-\ell-1}(N;\mathbb{Q}).$

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:

$\displaystyle \cup \colon H^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z}) \otimes H^{\ell+1}(N;\mathbb{Z}) \to H^{n}(N;\mathbb{Q}/\mathbb{Z}).$ $\displaystyle \cup \colon H^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z}) \otimes H^{\ell+1}(N;\mathbb{Z}) \to H^{n}(N;\mathbb{Q}/\mathbb{Z}).$

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

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

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

$\displaystyle L_{\mathbb{RP}^3}(\theta,\theta) = L_{\mathbb{RP}^3}([x],[y]) = \langle x,y \rangle/2 = 1/2.$ $\displaystyle L_{\mathbb{RP}^3}(\theta,\theta) = L_{\mathbb{RP}^3}([x],[y]) = \langle x,y \rangle/2 = 1/2.$

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 ${k}$ ${k}$ --chain $p \in C_{k}(N;\mathbb{Z})$ $p \in C_{k}(N;\mathbb{Z})$ and an $(n-k)$ $(n-k)$ --chain $q \in C_{n-k}(N;\mathbb{Z})$ $q \in C_{n-k}(N;\mathbb{Z})$ which is transverse to $q$ $q$ , the signed count of the intersections between $p$ $p$ and $q$ $q$ gives an intersection number $\langle\, p \, , \, q\, \rangle \in \mathbb{Z}$ $\langle\, p \, , \, q\, \rangle \in \mathbb{Z}$ .

The intersection form is defined by

$\displaystyle I_N \colon H_k(N;\mathbb{Z}) \times H_{n-k}(N;\mathbb{Z}) \to \mathbb{Z}; ([p],[q]) \mapsto \langle p, q \rangle$ $\displaystyle I_N \colon H_k(N;\mathbb{Z}) \times H_{n-k}(N;\mathbb{Z}) \to \mathbb{Z}; ([p],[q]) \mapsto \langle p, q \rangle$

and is such that

$\displaystyle I_N(x,y) = (-)^{k(n-k)}I_N(y,x).$ $\displaystyle I_N(x,y) = (-)^{k(n-k)}I_N(y,x).$

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 $P$ $P$ is the set

$\displaystyle TP:= \{p \in P \,|\, ap=0 \text{ for some } a \in \mathbb{Z}\}.$ $\displaystyle TP:= \{p \in P \,|\, ap=0 \text{ for some } a \in \mathbb{Z}\}.$

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:

$\displaystyle L_N \colon TH_{\ell}(N;\mathbb{Z}) \times TH_{n-\ell-1}(N;\mathbb{Z}) \to \mathbb{Q}/\mathbb{Z}$ $\displaystyle L_N \colon TH_{\ell}(N;\mathbb{Z}) \times TH_{n-\ell-1}(N;\mathbb{Z}) \to \mathbb{Q}/\mathbb{Z}$

such that

$\displaystyle L_N(x,y) = (-)^{\ell(n-\ell-1)}L_N(y,x)$ $\displaystyle L_N(x,y) = (-)^{\ell(n-\ell-1)}L_N(y,x)$

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 $w \in C_{\ell+1}(N;\mathbb{Z})$ be such that $\partial w = sy$ , for some $s \in \mathbb{Z}$ . Then we define:

$\displaystyle L_N([x],[y]) := \langle x, w \rangle/s \in \mathbb{Q}/\mathbb{Z}.$ $\displaystyle L_N([x],[y]) := \langle x, w \rangle/s \in \mathbb{Q}/\mathbb{Z}.$

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 Poincar\'{e} duality isomorphisms

$\displaystyle PD \colon TH_{\ell}(N;\mathbb{Z}) \xrightarrow{\cong} TH^{n-\ell}(N;\mathbb{Z})$ $\displaystyle PD \colon TH_{\ell}(N;\mathbb{Z}) \xrightarrow{\cong} TH^{n-\ell}(N;\mathbb{Z})$

and

$\displaystyle PD \colon TH_{n-\ell-1}(N;\mathbb{Z}) \xrightarrow{\cong} TH^{\ell+1}(N;\mathbb{Z}).$ $\displaystyle PD \colon TH_{n-\ell-1}(N;\mathbb{Z}) \xrightarrow{\cong} TH^{\ell+1}(N;\mathbb{Z}).$

Associated to the short exact sequence of coefficients

$\displaystyle 0 \to \mathbb{Z} \to \mathbb{Q} \to \mathbb{Q}/\mathbb{Z} \to 0$ $\displaystyle 0 \to \mathbb{Z} \to \mathbb{Q} \to \mathbb{Q}/\mathbb{Z} \to 0$

is the Bockstein long exact sequence in cohomology.

$\displaystyle H^{n-\ell-1}(N;\mathbb{Q}) \to H^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z}) \xrightarrow{\beta} H^{n-\ell}(N;\mathbb{Z}) \to H^{n-\ell-1}(N;\mathbb{Q}).$ $\displaystyle H^{n-\ell-1}(N;\mathbb{Q}) \to H^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z}) \xrightarrow{\beta} H^{n-\ell}(N;\mathbb{Z}) \to H^{n-\ell-1}(N;\mathbb{Q}).$

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:

$\displaystyle \cup \colon H^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z}) \otimes H^{\ell+1}(N;\mathbb{Z}) \to H^{n}(N;\mathbb{Q}/\mathbb{Z}).$ $\displaystyle \cup \colon H^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z}) \otimes H^{\ell+1}(N;\mathbb{Z}) \to H^{n}(N;\mathbb{Q}/\mathbb{Z}).$

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

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

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

$\displaystyle L_{\mathbb{RP}^3}(\theta,\theta) = L_{\mathbb{RP}^3}([x],[y]) = \langle x,y \rangle/2 = 1/2.$ $\displaystyle L_{\mathbb{RP}^3}(\theta,\theta) = L_{\mathbb{RP}^3}([x],[y]) = \langle x,y \rangle/2 = 1/2.$

5 References

@@ Line 24: / Line 24: @@
 $$L_N([x],[y]) := \langle x, w \rangle/s \in \mathbb{Q}/\mathbb{Z}.$$
 The resulting element is independent of the choices of $x,y,w$ and $s$.
+==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==
@@ Line 29: / Line 45: @@
 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
 $$L_{\mathbb{RP}^3}(\theta,\theta) = L_{\mathbb{RP}^3}([x],[y]) = \langle x,y \rangle/2 = 1/2.$$

Linking form

Revision as of 20:34, 27 March 2013

1 Background: intersection forms

2 Definition of the linking form

3 Definition via homology

4 Example of 3-dimensional projective space

5 References

2 Definition of the linking form

3 Definition via homology

4 Example of 3-dimensional projective space

5 References

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