Linking form

This page has not been refereed. The information given here might be incomplete or provisional.

1 Background: intersection forms

After Poincaré, a closed oriented manifold $N^{n}$${{Stub}} ==Background: intersection forms== <wikitex>; After Poincaré, a closed oriented manifold $N^{n}$ has a bilinear ''intersection form'' defined on its homology. Given a ${k}$-chain $p \in C_{k}(N;\mathbb{Z})$ and an $(n-k)$-chain $q \in C_{n-k}(N;\mathbb{Z})$ which is transverse to $q$, the signed count of the intersections between $p$ and $q$ gives an intersection number \begin{equation} \label{eq:intersection} \langle\, p \, , \, q\, \rangle \in \mathbb{Z}. \end{equation} The intersection form is defined by $$I_N \colon H_k(N;\mathbb{Z}) \times H_{n-k}(N;\mathbb{Z}) \to \mathbb{Z}; \quad ([p],[q]) \mapsto \langle p, q \rangle$$ and is such that $$I_N(x,y) = (-)^{k(n-k)}I_N(y,x).$$ </wikitex> == Definition of the linking form== <wikitex>; By bilinearity, the intersection form vanishes on the torsion part of the homology. The torsion part of an abelian group $P$ is the subgroup $$TP:= \{p \in P \,|\, ap=0 \text{ for some } a \in \mathbb{Z}\}.$$ The analogue of the intersection pairing for the torsion part of the homology of a closed oriented manifold $N^n$ is the bilinear $\mathbb{Q}/\mathbb{Z}$-valued linking form, which is due to Seifert \cite{Seifert1933}: $$L_N \colon TH_{\ell}(N;\mathbb{Z}) \times TH_{n-\ell-1}(N;\mathbb{Z}) \to \mathbb{Q}/\mathbb{Z}.$$ The linking form is such that $$L_N(x,y) = (-)^{\ell(n-\ell)}L_N(y,x)$$ and is computed as follows. Given $[x] \in TH_\ell(N;\mathbb{Z})$ and $[y] \in TH_{n-\ell-1}(N;\mathbb{Z})$ represented by cycles $x \in C_\ell(N;\mathbb{Z})$ and $y \in C_{n-\ell-1}(N,\mathbb{Z})$, let $w \in C_{n-\ell}(N;\mathbb{Z})$ be such that $\partial w = sy$, for some $s \in \mathbb{Z}$. Then we define: $$L_N([x],[y]) := \langle x, w \rangle/s \in \mathbb{Q}/\mathbb{Z},$$ where $\langle x, w \rangle$ is defined in (\ref{eq:intersection}) above. The resulting element is independent of the choices of $x,y,w$ and $s$. </wikitex> ==Definition via cohomology== <wikitex>; Let $x \in TH_{\ell}(N;\mathbb{Z})$ and let $y \in TH_{n-\ell-1}(N;\mathbb{Z})$. Note that we have Poincaré duality isomorphisms $$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}).$$ Associated to the short exact sequence of coefficients $<script type="math/tex">N^{n}$ has a bilinear intersection form defined on its homology. Given a ${k}$${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

(1)$\langle\, p \, , \, q\, \rangle \in \mathbb{Z}.$$\langle\, p \, , \, q\, \rangle \in \mathbb{Z}.$

For an equivalent definition not involving the concept of transversality see [Intersection].

The intersection form is defined by

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

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 subgroup

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

The analogue of the intersection pairing for the torsion part of the homology of a closed oriented manifold $N^n$$N^n$ is the bilinear $\mathbb{Q}/\mathbb{Z}$$\mathbb{Q}/\mathbb{Z}$-valued linking form, which is due to Seifert [Seifert1933]:

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

The linking form is such that

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

and is computed as follows. Given $[x] \in TH_\ell(N;\mathbb{Z})$$[x] \in TH_\ell(N;\mathbb{Z})$ and $[y] \in TH_{n-\ell-1}(N;\mathbb{Z})$$[y] \in TH_{n-\ell-1}(N;\mathbb{Z})$ represented by cycles $x \in C_\ell(N;\mathbb{Z})$$x \in C_\ell(N;\mathbb{Z})$ and $y \in C_{n-\ell-1}(N,\mathbb{Z})$$y \in C_{n-\ell-1}(N,\mathbb{Z})$, let $w \in C_{n-\ell}(N;\mathbb{Z})$$w \in C_{n-\ell}(N;\mathbb{Z})$ be such that $\partial w = sy$$\partial w = sy$, for some $s \in \mathbb{Z}$$s \in \mathbb{Z}$. Then we define:

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

where $\langle x, w \rangle$$\langle x, w \rangle$ is defined in (1) above. The resulting element is independent of the choices of $x,y,w$$x,y,w$ and $s$$s$.

3 Definition via cohomology

Let $x \in TH_{\ell}(N;\mathbb{Z})$$x \in TH_{\ell}(N;\mathbb{Z})$ and let $y \in TH_{n-\ell-1}(N;\mathbb{Z})$$y \in TH_{n-\ell-1}(N;\mathbb{Z})$. Note that we have Poincaré 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:

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

Choose $z \in TH^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z})$$z \in TH^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z})$ such that $\beta(z) = PD(x)$$\beta(z) = PD(x)$. This is always possible since torsion elements in $H^{n-\ell}(N;\mathbb{Z})$$H^{n-\ell}(N;\mathbb{Z})$ map to zero in $H^{n-\ell}(N;\mathbb{Q})$$H^{n-\ell}(N;\mathbb{Q})$. There is a cup product:

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

Then the Kronecker pairing,

$\displaystyle \langle z \cup PD(y),[N] \rangle \in \mathbb{Q}/\mathbb{Z},$ $$\displaystyle \langle z \cup PD(y),[N] \rangle \in \mathbb{Q}/\mathbb{Z},$$

of $z \cup PD(y)$$z \cup PD(y)$ with the fundamental class of $N$$N$ yields $L_N(x,y)$$L_N(x,y)$.

4 Example of 3-dimensional projective space

As an example, let $N = \mathbb{RP}^3$$N = \mathbb{RP}^3$, so that $\ell=1$$\ell=1$ and $n=3$$n=3$. Now $H_1(\mathbb{RP}^3;\mathbb{Z}) \cong \mathbb{Z}_2$$H_1(\mathbb{RP}^3;\mathbb{Z}) \cong \mathbb{Z}_2$. Let $\theta \in H_1(\mathbb{RP}^3;\mathbb{Z})$$\theta \in H_1(\mathbb{RP}^3;\mathbb{Z})$ be the non-trivial element. To compute the linking $L_{\mathbb{RP}^3}(\theta,\theta)$$L_{\mathbb{RP}^3}(\theta,\theta)$, consider $\mathbb{RP}^3$$\mathbb{RP}^3$ modelled as $D^3/\sim$$D^3/\sim$, with antipodal points on $\partial D^2$$\partial D^2$ identified, and choose two representative $1$$1$-chains

Tex syntax error

$x$ and

Tex syntax error

$y$ for $\theta$$\theta$. Let

Tex syntax error

$x$ be the straight line between north and south poles and let

Tex syntax error

$y$ be half of the equator. Now $2y = \partial w$$2y = \partial w$, where $w \in C_2(\mathbb{RP}^3;\mathbb{Z})$$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$$\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 Example of lens spaces

Generalising the above example, the 3-dimensional lens space $N_{p,q} := L(p; q, 1)$$N_{p,q} := L(p; q, 1)$ has $H_1(N_{p,q};\mathbb{Z}) \cong \mathbb{Z}_p$$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})$$\theta \in H_1(N_{p,q};\mathbb{Z})$ by $L_{N_{p,q}}(\theta,\theta) = q/p$$L_{N_{p,q}}(\theta,\theta) = q/p$. Note that $N_{2,1} \cong \mathbb{RP}^3$$N_{2,1} \cong \mathbb{RP}^3$, so this is consistent with the above example.

6 Presentations of linking forms

A presentation for a middle dimensional linking form on $N^{2\ell +1}$$N^{2\ell +1}$

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

is an exact sequence:

$\displaystyle 0 \to F \xrightarrow{\Phi} F^* \to TH_{\ell}(N;\mathbb{Z}) \xrightarrow{\partial} 0,$ $$\displaystyle 0 \to F \xrightarrow{\Phi} F^* \to TH_{\ell}(N;\mathbb{Z}) \xrightarrow{\partial} 0,$$

where $F$$F$ is a free abelain group and the linking $L_N(x,y)$$L_N(x,y)$ can be computed as follows. Let $x',y' \in F^*$$x',y' \in F^*$ be such that $\partial(x')=x$$\partial(x')=x$ and $\partial(y')=y$$\partial(y')=y$. Then we can tensor with $\mathbb{Q}$$\mathbb{Q}$ to obtain an isomorphism

$\displaystyle \Phi \otimes \mathop{\mathrm{Id}} \colon F \otimes_{\mathbb{Z}} \mathbb{Q} \xrightarrow{\cong} F^* \otimes_{\mathbb{Z}} \mathbb{Q}.$ $$\displaystyle \Phi \otimes \mathop{\mathrm{Id}} \colon F \otimes_{\mathbb{Z}} \mathbb{Q} \xrightarrow{\cong} F^* \otimes_{\mathbb{Z}} \mathbb{Q}.$$

The linking form of $N$$N$ is then given by:

$\displaystyle L_N(x,y) = -(x' \otimes 1)((\Phi\otimes \mathop{\mathrm{Id}})^{-1}(y'\otimes 1)).$ $$\displaystyle L_N(x,y) = -(x' \otimes 1)((\Phi\otimes \mathop{\mathrm{Id}})^{-1}(y'\otimes 1)).$$

This formula, in particular the appearance of the sign, is explained in [Gordon&Litherland1978, Section 3] and [Alexander&Hamrick&Vick1976, Proof of Theorem 2.1].

Let $\ell = 1$$\ell = 1$, so $2\ell + 1 = 3$$2\ell + 1 = 3$. Every 3-manifold $N$$N$ is the boundary of a simply connected 4-manifold, which is obtained by glueing 2-handles to an integrally framed link in $S^3$$S^3$ [Lickorish1962], [Wallace1960]. This is sometimes called a surgery presentation for $N$$N$. Suppose that $N$$N$ is a rational homology 3-sphere. Let $A$$A$ be the matrix of (self-) linking numbers of the surgery presentation link. Taking the number of link components in the surgery presentation for $N$$N$ as the rank of $F$$F$, the linking matrix $A$$A$ determines a map $\Phi$$\Phi$ as above, which presents the linking form of $N$$N$. The intersection form on a simply connected 4-manifold $W$$W$ whose boundary is $N$$N$ presents the linking form of $N$$N$. This follows from the long exact sequence of the pair $(W,N)$$(W,N)$ and Poincaré duality. See [Boyer1986] for more details and the use of such presentations for the classification of simply connected 4-manifolds with a given boundary.

For example, the lens space $L(p,1)$$L(p,1)$ is the boundary of the $D^2$$D^2$ bundle over $S^2$$S^2$ with Euler number $-p$$-p$, so the presentation $\Phi = (-p) \colon \mathbb{Z} \to \mathbb{Z}$$\Phi = (-p) \colon \mathbb{Z} \to \mathbb{Z}$ presents the linking form of $L(p,1)$$L(p,1)$ [Gompf&Stipsicz1999, Example 5.3.2].

7 Role in the classification of odd-dimensional manifolds

Linking forms play an important role in the classification of odd-dimensional manifolds. For closed simply connected $5$$5$-manifolds $M$$M$, the linking form is a complete invariant if $H_2(M; \Zz) = TH_2(M; \Zz)$$H_2(M; \Zz) = TH_2(M; \Zz)$. For more information in dimension $5$$5$, see the page on simply-connected 5-manifolds.

For the role of linking forms in the classification of smooth $(q-1)$$(q-1)$-connected $(2q+1)$$(2q+1)$ manifolds with boundary a homotopy sphere, see [Wall1967, Theorem 7].

8 Algebraic classification

An algebraic linking form is a non-singular bi-linear pairing

$\displaystyle b \colon T \times T \to \Qq/\Zz$ $$\displaystyle b \colon T \times T \to \Qq/\Zz$$

on a finite abelian group

Tex syntax error

$T$. It is called symmetric if $b(x, y) = b(y, x)$$b(x, y) = b(y, x)$ and skew-symmetric

if $b(x, y) = - b(y, x)$$b(x, y) = - b(y, x)$.

The classification of skew-symmetric linking forms is rather simple and is due to Wall, [Wall1963, Theorem 3]. It is described in detail in the page on simply-connected 5-manifolds.

The classification of symmetric linking forms is rather intricate. It was begun in [Wall1963] and completed by Kawauchi and Kojima: see [Kawauchi&Kojima1980, Theorem 4.1].

9 References

• [Alexander&Hamrick&Vick1976] J. P. Alexander, G. C. Hamrick and J. W. Vick, Linking forms and maps of odd prime order, Trans. Amer. Math. Soc. 221 (1976), no.1, 169–185. MR0402786 (53 #6600) Zbl 0357.57009
• [Boyer1986] S. Boyer, Simply-connected $4$$4$-manifolds with a given boundary, Trans. Amer. Math. Soc. 298 (1986), no.1, 331–357. MR857447 (88b:57023) Zbl 0790.57009
• [Gompf&Stipsicz1999] R. E. Gompf and A. I. Stipsicz, $4$$4$-manifolds and Kirby calculus, American Mathematical Society, 1999. MR1707327 (2000h:57038) Zbl 0933.57020
• [Gordon&Litherland1978] C. M. Gordon and R. A. Litherland, On the signature of a link, Invent. Math. 47 (1978), no.1, 53–69. MR0500905 (58 #18407) Zbl 0391.57004
• [Intersection] Intersection form, http://www.map.mpim-bonn.mpg.de/Intersection_form
• [Kawauchi&Kojima1980] A. Kawauchi and S. Kojima, Algebraic classification of linking pairings on $3$$3$-manifolds, Math. Ann. 253 (1980), no.1, 29–42. MR594531 (82b:57007) Zbl 0427.57001
• [Lickorish1962] W. B. R. Lickorish, A representation of orientable combinatorial $3$$3$-manifolds, Ann. of Math. (2) 76 (1962), 531–540. MR0151948 (27 #1929) Zbl 0106.37102
• [Seifert1933] H. Seifert, Verschlingungsinvarianten, Sitzungsber. Preuß. Akad. Wiss., Phys.-Math. Kl. 1933, No.26-29, (1933) 811-828. Zbl 0008.18101
• [Wall1963] C. T. C. Wall, Quadratic forms on finite groups, and related topics, Topology 2 (1963), 281–298. MR0156890 (28 #133) Zbl 0215.39903
• [Wall1967] C. T. C. Wall, Classification problems in differential topology. VI. Classification of $(s-1)$$(s-1)$-connected $(2s+1)$$(2s+1)$-manifolds, Topology 6 (1967), 273–296. MR0216510 (35 #7343) Zbl 0173.26102
• [Wallace1960] A. H. Wallace, Modifications and cobounding manifolds, Canad. J. Math. 12 (1960), 503–528. MR0125588 (23 #A2887) Zbl 0116.40401

10 External links

• The Wikipedia page on Poincaré duality

\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}(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}).$$ Then the Kronecker pairing, $$\langle z \cup PD(y),[N] \rangle \in \mathbb{Q}/\mathbb{Z},$$ of $z \cup PD(y)$ with the fundamental class of $N$ yields $L_N(x,y)$. ==Example of 3-dimensional projective space== ; As an example, let $N = \mathbb{RP}^3$, so that $\ell=1$ and $n=3$. Now $H_1(\mathbb{RP}^3;\mathbb{Z}) \cong \mathbb{Z}_2$. Let $\theta \in H_1(\mathbb{RP}^3;\mathbb{Z})$ be the non-trivial element. To compute the linking $L_{\mathbb{RP}^3}(\theta,\theta)$, consider $\mathbb{RP}^3$ modelled as $D^3/\sim$, with antipodal points on $\partial D^2$ identified, and choose two representative N^{n} has a bilinear intersection form defined on its homology. Given a ${k}$${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

(1)$\langle\, p \, , \, q\, \rangle \in \mathbb{Z}.$$\langle\, p \, , \, q\, \rangle \in \mathbb{Z}.$

For an equivalent definition not involving the concept of transversality see [Intersection].

The intersection form is defined by

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

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 subgroup

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

The analogue of the intersection pairing for the torsion part of the homology of a closed oriented manifold $N^n$$N^n$ is the bilinear $\mathbb{Q}/\mathbb{Z}$$\mathbb{Q}/\mathbb{Z}$-valued linking form, which is due to Seifert [Seifert1933]:

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

The linking form is such that

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

and is computed as follows. Given $[x] \in TH_\ell(N;\mathbb{Z})$$[x] \in TH_\ell(N;\mathbb{Z})$ and $[y] \in TH_{n-\ell-1}(N;\mathbb{Z})$$[y] \in TH_{n-\ell-1}(N;\mathbb{Z})$ represented by cycles $x \in C_\ell(N;\mathbb{Z})$$x \in C_\ell(N;\mathbb{Z})$ and $y \in C_{n-\ell-1}(N,\mathbb{Z})$$y \in C_{n-\ell-1}(N,\mathbb{Z})$, let $w \in C_{n-\ell}(N;\mathbb{Z})$$w \in C_{n-\ell}(N;\mathbb{Z})$ be such that $\partial w = sy$$\partial w = sy$, for some $s \in \mathbb{Z}$$s \in \mathbb{Z}$. Then we define:

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

where $\langle x, w \rangle$$\langle x, w \rangle$ is defined in (1) above. The resulting element is independent of the choices of $x,y,w$$x,y,w$ and $s$$s$.

3 Definition via cohomology

Let $x \in TH_{\ell}(N;\mathbb{Z})$$x \in TH_{\ell}(N;\mathbb{Z})$ and let $y \in TH_{n-\ell-1}(N;\mathbb{Z})$$y \in TH_{n-\ell-1}(N;\mathbb{Z})$. Note that we have Poincaré 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:

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

Choose $z \in TH^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z})$$z \in TH^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z})$ such that $\beta(z) = PD(x)$$\beta(z) = PD(x)$. This is always possible since torsion elements in $H^{n-\ell}(N;\mathbb{Z})$$H^{n-\ell}(N;\mathbb{Z})$ map to zero in $H^{n-\ell}(N;\mathbb{Q})$$H^{n-\ell}(N;\mathbb{Q})$. There is a cup product:

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

Then the Kronecker pairing,

$\displaystyle \langle z \cup PD(y),[N] \rangle \in \mathbb{Q}/\mathbb{Z},$ $$\displaystyle \langle z \cup PD(y),[N] \rangle \in \mathbb{Q}/\mathbb{Z},$$

of $z \cup PD(y)$$z \cup PD(y)$ with the fundamental class of $N$$N$ yields $L_N(x,y)$$L_N(x,y)$.

4 Example of 3-dimensional projective space

As an example, let $N = \mathbb{RP}^3$$N = \mathbb{RP}^3$, so that $\ell=1$$\ell=1$ and $n=3$$n=3$. Now $H_1(\mathbb{RP}^3;\mathbb{Z}) \cong \mathbb{Z}_2$$H_1(\mathbb{RP}^3;\mathbb{Z}) \cong \mathbb{Z}_2$. Let $\theta \in H_1(\mathbb{RP}^3;\mathbb{Z})$$\theta \in H_1(\mathbb{RP}^3;\mathbb{Z})$ be the non-trivial element. To compute the linking $L_{\mathbb{RP}^3}(\theta,\theta)$$L_{\mathbb{RP}^3}(\theta,\theta)$, consider $\mathbb{RP}^3$$\mathbb{RP}^3$ modelled as $D^3/\sim$$D^3/\sim$, with antipodal points on $\partial D^2$$\partial D^2$ identified, and choose two representative $1$$1$-chains

Tex syntax error

$x$ and

Tex syntax error

$y$ for $\theta$$\theta$. Let

Tex syntax error

$x$ be the straight line between north and south poles and let

Tex syntax error

$y$ be half of the equator. Now $2y = \partial w$$2y = \partial w$, where $w \in C_2(\mathbb{RP}^3;\mathbb{Z})$$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$$\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 Example of lens spaces

Generalising the above example, the 3-dimensional lens space $N_{p,q} := L(p; q, 1)$$N_{p,q} := L(p; q, 1)$ has $H_1(N_{p,q};\mathbb{Z}) \cong \mathbb{Z}_p$$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})$$\theta \in H_1(N_{p,q};\mathbb{Z})$ by $L_{N_{p,q}}(\theta,\theta) = q/p$$L_{N_{p,q}}(\theta,\theta) = q/p$. Note that $N_{2,1} \cong \mathbb{RP}^3$$N_{2,1} \cong \mathbb{RP}^3$, so this is consistent with the above example.

6 Presentations of linking forms

A presentation for a middle dimensional linking form on $N^{2\ell +1}$$N^{2\ell +1}$

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

is an exact sequence:

$\displaystyle 0 \to F \xrightarrow{\Phi} F^* \to TH_{\ell}(N;\mathbb{Z}) \xrightarrow{\partial} 0,$ $$\displaystyle 0 \to F \xrightarrow{\Phi} F^* \to TH_{\ell}(N;\mathbb{Z}) \xrightarrow{\partial} 0,$$

where $F$$F$ is a free abelain group and the linking $L_N(x,y)$$L_N(x,y)$ can be computed as follows. Let $x',y' \in F^*$$x',y' \in F^*$ be such that $\partial(x')=x$$\partial(x')=x$ and $\partial(y')=y$$\partial(y')=y$. Then we can tensor with $\mathbb{Q}$$\mathbb{Q}$ to obtain an isomorphism

$\displaystyle \Phi \otimes \mathop{\mathrm{Id}} \colon F \otimes_{\mathbb{Z}} \mathbb{Q} \xrightarrow{\cong} F^* \otimes_{\mathbb{Z}} \mathbb{Q}.$ $$\displaystyle \Phi \otimes \mathop{\mathrm{Id}} \colon F \otimes_{\mathbb{Z}} \mathbb{Q} \xrightarrow{\cong} F^* \otimes_{\mathbb{Z}} \mathbb{Q}.$$

The linking form of $N$$N$ is then given by:

$\displaystyle L_N(x,y) = -(x' \otimes 1)((\Phi\otimes \mathop{\mathrm{Id}})^{-1}(y'\otimes 1)).$ $$\displaystyle L_N(x,y) = -(x' \otimes 1)((\Phi\otimes \mathop{\mathrm{Id}})^{-1}(y'\otimes 1)).$$

This formula, in particular the appearance of the sign, is explained in [Gordon&Litherland1978, Section 3] and [Alexander&Hamrick&Vick1976, Proof of Theorem 2.1].

Let $\ell = 1$$\ell = 1$, so $2\ell + 1 = 3$$2\ell + 1 = 3$. Every 3-manifold $N$$N$ is the boundary of a simply connected 4-manifold, which is obtained by glueing 2-handles to an integrally framed link in $S^3$$S^3$ [Lickorish1962], [Wallace1960]. This is sometimes called a surgery presentation for $N$$N$. Suppose that $N$$N$ is a rational homology 3-sphere. Let $A$$A$ be the matrix of (self-) linking numbers of the surgery presentation link. Taking the number of link components in the surgery presentation for $N$$N$ as the rank of $F$$F$, the linking matrix $A$$A$ determines a map $\Phi$$\Phi$ as above, which presents the linking form of $N$$N$. The intersection form on a simply connected 4-manifold $W$$W$ whose boundary is $N$$N$ presents the linking form of $N$$N$. This follows from the long exact sequence of the pair $(W,N)$$(W,N)$ and Poincaré duality. See [Boyer1986] for more details and the use of such presentations for the classification of simply connected 4-manifolds with a given boundary.

For example, the lens space $L(p,1)$$L(p,1)$ is the boundary of the $D^2$$D^2$ bundle over $S^2$$S^2$ with Euler number $-p$$-p$, so the presentation $\Phi = (-p) \colon \mathbb{Z} \to \mathbb{Z}$$\Phi = (-p) \colon \mathbb{Z} \to \mathbb{Z}$ presents the linking form of $L(p,1)$$L(p,1)$ [Gompf&Stipsicz1999, Example 5.3.2].

7 Role in the classification of odd-dimensional manifolds

Linking forms play an important role in the classification of odd-dimensional manifolds. For closed simply connected $5$$5$-manifolds $M$$M$, the linking form is a complete invariant if $H_2(M; \Zz) = TH_2(M; \Zz)$$H_2(M; \Zz) = TH_2(M; \Zz)$. For more information in dimension $5$$5$, see the page on simply-connected 5-manifolds.

For the role of linking forms in the classification of smooth $(q-1)$$(q-1)$-connected $(2q+1)$$(2q+1)$ manifolds with boundary a homotopy sphere, see [Wall1967, Theorem 7].

8 Algebraic classification

An algebraic linking form is a non-singular bi-linear pairing

$\displaystyle b \colon T \times T \to \Qq/\Zz$ $$\displaystyle b \colon T \times T \to \Qq/\Zz$$

on a finite abelian group

Tex syntax error

$T$. It is called symmetric if $b(x, y) = b(y, x)$$b(x, y) = b(y, x)$ and skew-symmetric

if $b(x, y) = - b(y, x)$$b(x, y) = - b(y, x)$.

The classification of skew-symmetric linking forms is rather simple and is due to Wall, [Wall1963, Theorem 3]. It is described in detail in the page on simply-connected 5-manifolds.

The classification of symmetric linking forms is rather intricate. It was begun in [Wall1963] and completed by Kawauchi and Kojima: see [Kawauchi&Kojima1980, Theorem 4.1].

9 References

• [Alexander&Hamrick&Vick1976] J. P. Alexander, G. C. Hamrick and J. W. Vick, Linking forms and maps of odd prime order, Trans. Amer. Math. Soc. 221 (1976), no.1, 169–185. MR0402786 (53 #6600) Zbl 0357.57009
• [Boyer1986] S. Boyer, Simply-connected $4$$4$-manifolds with a given boundary, Trans. Amer. Math. Soc. 298 (1986), no.1, 331–357. MR857447 (88b:57023) Zbl 0790.57009
• [Gompf&Stipsicz1999] R. E. Gompf and A. I. Stipsicz, $4$$4$-manifolds and Kirby calculus, American Mathematical Society, 1999. MR1707327 (2000h:57038) Zbl 0933.57020
• [Gordon&Litherland1978] C. M. Gordon and R. A. Litherland, On the signature of a link, Invent. Math. 47 (1978), no.1, 53–69. MR0500905 (58 #18407) Zbl 0391.57004
• [Intersection] Intersection form, http://www.map.mpim-bonn.mpg.de/Intersection_form
• [Kawauchi&Kojima1980] A. Kawauchi and S. Kojima, Algebraic classification of linking pairings on $3$$3$-manifolds, Math. Ann. 253 (1980), no.1, 29–42. MR594531 (82b:57007) Zbl 0427.57001
• [Lickorish1962] W. B. R. Lickorish, A representation of orientable combinatorial $3$$3$-manifolds, Ann. of Math. (2) 76 (1962), 531–540. MR0151948 (27 #1929) Zbl 0106.37102
• [Seifert1933] H. Seifert, Verschlingungsinvarianten, Sitzungsber. Preuß. Akad. Wiss., Phys.-Math. Kl. 1933, No.26-29, (1933) 811-828. Zbl 0008.18101
• [Wall1963] C. T. C. Wall, Quadratic forms on finite groups, and related topics, Topology 2 (1963), 281–298. MR0156890 (28 #133) Zbl 0215.39903
• [Wall1967] C. T. C. Wall, Classification problems in differential topology. VI. Classification of $(s-1)$$(s-1)$-connected $(2s+1)$$(2s+1)$-manifolds, Topology 6 (1967), 273–296. MR0216510 (35 #7343) Zbl 0173.26102
• [Wallace1960] A. H. Wallace, Modifications and cobounding manifolds, Canad. J. Math. 12 (1960), 503–528. MR0125588 (23 #A2887) Zbl 0116.40401

10 External links

• The Wikipedia page on Poincaré duality

$-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 spaces|lens space]] $N_{p,q} := L(p; q, 1)$ 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 $L_{N_{p,q}}(\theta,\theta) = q/p$. Note that $N_{2,1} \cong \mathbb{RP}^3$, so this is consistent with the above example. ==Presentations of linking forms== ; A presentation for a middle dimensional linking form on $N^{2\ell +1}$ $$L_N \colon TH_{\ell}(N;\mathbb{Z}) \times TH_{\ell}(N;\mathbb{Z}) \to \mathbb{Q}/\mathbb{Z}$$ is an exact sequence: $$N^{n}$ has a bilinear intersection form defined on its homology. Given a ${k}$${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

(1)$\langle\, p \, , \, q\, \rangle \in \mathbb{Z}.$$\langle\, p \, , \, q\, \rangle \in \mathbb{Z}.$

For an equivalent definition not involving the concept of transversality see [Intersection].

The intersection form is defined by

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

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 subgroup

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

The analogue of the intersection pairing for the torsion part of the homology of a closed oriented manifold $N^n$$N^n$ is the bilinear $\mathbb{Q}/\mathbb{Z}$$\mathbb{Q}/\mathbb{Z}$-valued linking form, which is due to Seifert [Seifert1933]:

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

The linking form is such that

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

and is computed as follows. Given $[x] \in TH_\ell(N;\mathbb{Z})$$[x] \in TH_\ell(N;\mathbb{Z})$ and $[y] \in TH_{n-\ell-1}(N;\mathbb{Z})$$[y] \in TH_{n-\ell-1}(N;\mathbb{Z})$ represented by cycles $x \in C_\ell(N;\mathbb{Z})$$x \in C_\ell(N;\mathbb{Z})$ and $y \in C_{n-\ell-1}(N,\mathbb{Z})$$y \in C_{n-\ell-1}(N,\mathbb{Z})$, let $w \in C_{n-\ell}(N;\mathbb{Z})$$w \in C_{n-\ell}(N;\mathbb{Z})$ be such that $\partial w = sy$$\partial w = sy$, for some $s \in \mathbb{Z}$$s \in \mathbb{Z}$. Then we define:

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

where $\langle x, w \rangle$$\langle x, w \rangle$ is defined in (1) above. The resulting element is independent of the choices of $x,y,w$$x,y,w$ and $s$$s$.

3 Definition via cohomology

Let $x \in TH_{\ell}(N;\mathbb{Z})$$x \in TH_{\ell}(N;\mathbb{Z})$ and let $y \in TH_{n-\ell-1}(N;\mathbb{Z})$$y \in TH_{n-\ell-1}(N;\mathbb{Z})$. Note that we have Poincaré 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:

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

Choose $z \in TH^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z})$$z \in TH^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z})$ such that $\beta(z) = PD(x)$$\beta(z) = PD(x)$. This is always possible since torsion elements in $H^{n-\ell}(N;\mathbb{Z})$$H^{n-\ell}(N;\mathbb{Z})$ map to zero in $H^{n-\ell}(N;\mathbb{Q})$$H^{n-\ell}(N;\mathbb{Q})$. There is a cup product:

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

Then the Kronecker pairing,

$\displaystyle \langle z \cup PD(y),[N] \rangle \in \mathbb{Q}/\mathbb{Z},$ $$\displaystyle \langle z \cup PD(y),[N] \rangle \in \mathbb{Q}/\mathbb{Z},$$

of $z \cup PD(y)$$z \cup PD(y)$ with the fundamental class of $N$$N$ yields $L_N(x,y)$$L_N(x,y)$.

4 Example of 3-dimensional projective space

As an example, let $N = \mathbb{RP}^3$$N = \mathbb{RP}^3$, so that $\ell=1$$\ell=1$ and $n=3$$n=3$. Now $H_1(\mathbb{RP}^3;\mathbb{Z}) \cong \mathbb{Z}_2$$H_1(\mathbb{RP}^3;\mathbb{Z}) \cong \mathbb{Z}_2$. Let $\theta \in H_1(\mathbb{RP}^3;\mathbb{Z})$$\theta \in H_1(\mathbb{RP}^3;\mathbb{Z})$ be the non-trivial element. To compute the linking $L_{\mathbb{RP}^3}(\theta,\theta)$$L_{\mathbb{RP}^3}(\theta,\theta)$, consider $\mathbb{RP}^3$$\mathbb{RP}^3$ modelled as $D^3/\sim$$D^3/\sim$, with antipodal points on $\partial D^2$$\partial D^2$ identified, and choose two representative $1$$1$-chains

Tex syntax error

$x$ and

Tex syntax error

$y$ for $\theta$$\theta$. Let

Tex syntax error

$x$ be the straight line between north and south poles and let

Tex syntax error

$y$ be half of the equator. Now $2y = \partial w$$2y = \partial w$, where $w \in C_2(\mathbb{RP}^3;\mathbb{Z})$$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$$\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 Example of lens spaces

Generalising the above example, the 3-dimensional lens space $N_{p,q} := L(p; q, 1)$$N_{p,q} := L(p; q, 1)$ has $H_1(N_{p,q};\mathbb{Z}) \cong \mathbb{Z}_p$$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})$$\theta \in H_1(N_{p,q};\mathbb{Z})$ by $L_{N_{p,q}}(\theta,\theta) = q/p$$L_{N_{p,q}}(\theta,\theta) = q/p$. Note that $N_{2,1} \cong \mathbb{RP}^3$$N_{2,1} \cong \mathbb{RP}^3$, so this is consistent with the above example.

6 Presentations of linking forms

A presentation for a middle dimensional linking form on $N^{2\ell +1}$$N^{2\ell +1}$

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

is an exact sequence:

$\displaystyle 0 \to F \xrightarrow{\Phi} F^* \to TH_{\ell}(N;\mathbb{Z}) \xrightarrow{\partial} 0,$ $$\displaystyle 0 \to F \xrightarrow{\Phi} F^* \to TH_{\ell}(N;\mathbb{Z}) \xrightarrow{\partial} 0,$$

where $F$$F$ is a free abelain group and the linking $L_N(x,y)$$L_N(x,y)$ can be computed as follows. Let $x',y' \in F^*$$x',y' \in F^*$ be such that $\partial(x')=x$$\partial(x')=x$ and $\partial(y')=y$$\partial(y')=y$. Then we can tensor with $\mathbb{Q}$$\mathbb{Q}$ to obtain an isomorphism

$\displaystyle \Phi \otimes \mathop{\mathrm{Id}} \colon F \otimes_{\mathbb{Z}} \mathbb{Q} \xrightarrow{\cong} F^* \otimes_{\mathbb{Z}} \mathbb{Q}.$ $$\displaystyle \Phi \otimes \mathop{\mathrm{Id}} \colon F \otimes_{\mathbb{Z}} \mathbb{Q} \xrightarrow{\cong} F^* \otimes_{\mathbb{Z}} \mathbb{Q}.$$

The linking form of $N$$N$ is then given by:

$\displaystyle L_N(x,y) = -(x' \otimes 1)((\Phi\otimes \mathop{\mathrm{Id}})^{-1}(y'\otimes 1)).$ $$\displaystyle L_N(x,y) = -(x' \otimes 1)((\Phi\otimes \mathop{\mathrm{Id}})^{-1}(y'\otimes 1)).$$

This formula, in particular the appearance of the sign, is explained in [Gordon&Litherland1978, Section 3] and [Alexander&Hamrick&Vick1976, Proof of Theorem 2.1].

Let $\ell = 1$$\ell = 1$, so $2\ell + 1 = 3$$2\ell + 1 = 3$. Every 3-manifold $N$$N$ is the boundary of a simply connected 4-manifold, which is obtained by glueing 2-handles to an integrally framed link in $S^3$$S^3$ [Lickorish1962], [Wallace1960]. This is sometimes called a surgery presentation for $N$$N$. Suppose that $N$$N$ is a rational homology 3-sphere. Let $A$$A$ be the matrix of (self-) linking numbers of the surgery presentation link. Taking the number of link components in the surgery presentation for $N$$N$ as the rank of $F$$F$, the linking matrix $A$$A$ determines a map $\Phi$$\Phi$ as above, which presents the linking form of $N$$N$. The intersection form on a simply connected 4-manifold $W$$W$ whose boundary is $N$$N$ presents the linking form of $N$$N$. This follows from the long exact sequence of the pair $(W,N)$$(W,N)$ and Poincaré duality. See [Boyer1986] for more details and the use of such presentations for the classification of simply connected 4-manifolds with a given boundary.

For example, the lens space $L(p,1)$$L(p,1)$ is the boundary of the $D^2$$D^2$ bundle over $S^2$$S^2$ with Euler number $-p$$-p$, so the presentation $\Phi = (-p) \colon \mathbb{Z} \to \mathbb{Z}$$\Phi = (-p) \colon \mathbb{Z} \to \mathbb{Z}$ presents the linking form of $L(p,1)$$L(p,1)$ [Gompf&Stipsicz1999, Example 5.3.2].

7 Role in the classification of odd-dimensional manifolds

Linking forms play an important role in the classification of odd-dimensional manifolds. For closed simply connected $5$$5$-manifolds $M$$M$, the linking form is a complete invariant if $H_2(M; \Zz) = TH_2(M; \Zz)$$H_2(M; \Zz) = TH_2(M; \Zz)$. For more information in dimension $5$$5$, see the page on simply-connected 5-manifolds.

For the role of linking forms in the classification of smooth $(q-1)$$(q-1)$-connected $(2q+1)$$(2q+1)$ manifolds with boundary a homotopy sphere, see [Wall1967, Theorem 7].

8 Algebraic classification

An algebraic linking form is a non-singular bi-linear pairing

$\displaystyle b \colon T \times T \to \Qq/\Zz$ $$\displaystyle b \colon T \times T \to \Qq/\Zz$$

on a finite abelian group

Tex syntax error

$T$. It is called symmetric if $b(x, y) = b(y, x)$$b(x, y) = b(y, x)$ and skew-symmetric

if $b(x, y) = - b(y, x)$$b(x, y) = - b(y, x)$.

The classification of skew-symmetric linking forms is rather simple and is due to Wall, [Wall1963, Theorem 3]. It is described in detail in the page on simply-connected 5-manifolds.

The classification of symmetric linking forms is rather intricate. It was begun in [Wall1963] and completed by Kawauchi and Kojima: see [Kawauchi&Kojima1980, Theorem 4.1].

9 References

• [Alexander&Hamrick&Vick1976] J. P. Alexander, G. C. Hamrick and J. W. Vick, Linking forms and maps of odd prime order, Trans. Amer. Math. Soc. 221 (1976), no.1, 169–185. MR0402786 (53 #6600) Zbl 0357.57009
• [Boyer1986] S. Boyer, Simply-connected $4$$4$-manifolds with a given boundary, Trans. Amer. Math. Soc. 298 (1986), no.1, 331–357. MR857447 (88b:57023) Zbl 0790.57009
• [Gompf&Stipsicz1999] R. E. Gompf and A. I. Stipsicz, $4$$4$-manifolds and Kirby calculus, American Mathematical Society, 1999. MR1707327 (2000h:57038) Zbl 0933.57020
• [Gordon&Litherland1978] C. M. Gordon and R. A. Litherland, On the signature of a link, Invent. Math. 47 (1978), no.1, 53–69. MR0500905 (58 #18407) Zbl 0391.57004
• [Intersection] Intersection form, http://www.map.mpim-bonn.mpg.de/Intersection_form
• [Kawauchi&Kojima1980] A. Kawauchi and S. Kojima, Algebraic classification of linking pairings on $3$$3$-manifolds, Math. Ann. 253 (1980), no.1, 29–42. MR594531 (82b:57007) Zbl 0427.57001
• [Lickorish1962] W. B. R. Lickorish, A representation of orientable combinatorial $3$$3$-manifolds, Ann. of Math. (2) 76 (1962), 531–540. MR0151948 (27 #1929) Zbl 0106.37102
• [Seifert1933] H. Seifert, Verschlingungsinvarianten, Sitzungsber. Preuß. Akad. Wiss., Phys.-Math. Kl. 1933, No.26-29, (1933) 811-828. Zbl 0008.18101
• [Wall1963] C. T. C. Wall, Quadratic forms on finite groups, and related topics, Topology 2 (1963), 281–298. MR0156890 (28 #133) Zbl 0215.39903
• [Wall1967] C. T. C. Wall, Classification problems in differential topology. VI. Classification of $(s-1)$$(s-1)$-connected $(2s+1)$$(2s+1)$-manifolds, Topology 6 (1967), 273–296. MR0216510 (35 #7343) Zbl 0173.26102
• [Wallace1960] A. H. Wallace, Modifications and cobounding manifolds, Canad. J. Math. 12 (1960), 503–528. MR0125588 (23 #A2887) Zbl 0116.40401

10 External links

• The Wikipedia page on Poincaré duality

\to F \xrightarrow{\Phi} F^* \to TH_{\ell}(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}.$$ The linking form of $N$ is then given by: $$L_N(x,y) = -(x' \otimes 1)((\Phi\otimes \mathop{\mathrm{Id}})^{-1}(y'\otimes 1)).$$ This formula, in particular the appearance of the sign, is explained in \cite[Section 3]{Gordon&Litherland1978} and \cite[Proof of Theorem 2.1]{Alexander&Hamrick&Vick1976}. Let $\ell = 1$, so \ell + 1 = 3$. 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{Lickorish1962}, \cite{Wallace1960}. This is sometimes called a surgery presentation for $N$. Suppose that $N$ is a rational homology 3-sphere. Let $A$ be the matrix of (self-) linking numbers of the surgery presentation link. 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, which presents the linking form of $N$. 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 Poincaré duality. See \cite{Boyer1986} for more details and the use of such presentations for the classification of simply connected 4-manifolds with a given boundary. For example, the lens space $L(p,1)$ is the boundary of the $D^2$ bundle over $S^2$ with Euler number $-p$, so the presentation $\Phi = (-p) \colon \mathbb{Z} \to \mathbb{Z}$ presents the linking form of $L(p,1)$ \cite[Example 5.3.2]{Gompf&Stipsicz1999}. == Role in the classification of odd-dimensional manifolds == ; Linking forms play an important role in the classification of odd-dimensional manifolds. For closed simply connected $-manifolds $M$, the linking form is a complete invariant if $H_2(M; \Zz) = TH_2(M; \Zz)$. For more information in dimension $, see the page on [[5-manifolds: 1-connected|simply-connected 5-manifolds]]. For the role of linking forms in the classification of smooth $(q-1)$-connected $(2q+1)$ manifolds with boundary a homotopy sphere, see \cite[Theorem 7]{Wall1967}. == Algebraic classification == ; An algebraic linking form is a non-singular bi-linear pairing $$ b \colon T \times T \to \Qq/\Zz$$ on a finite abelian group $T$. It is called symmetric if $b(x, y) = b(y, x)$ and skew-symmetric if $b(x, y) = - b(y, x)$. The classification of skew-symmetric linking forms is rather simple and is due to Wall, \cite[Theorem 3]{Wall1963}. It is described in detail in the page on [[5-manifolds: 1-connected#Linking_forms|simply-connected 5-manifolds]]. The classification of symmetric linking forms is rather intricate. It was begun in \cite{Wall1963} and completed by Kawauchi and Kojima: see \cite[Theorem 4.1]{Kawauchi&Kojima1980}. == References == {{#RefList:}} == External links == * The Wikipedia page on [[Wikipedia:Poincare duality#Bilinear_pairings_formulation|Poincaré duality]] [[Category:Definitions]]N^{n} has a bilinear intersection form defined on its homology. Given a ${k}$${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

(1)$\langle\, p \, , \, q\, \rangle \in \mathbb{Z}.$$\langle\, p \, , \, q\, \rangle \in \mathbb{Z}.$

For an equivalent definition not involving the concept of transversality see [Intersection].

The intersection form is defined by

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

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 subgroup

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

The analogue of the intersection pairing for the torsion part of the homology of a closed oriented manifold $N^n$$N^n$ is the bilinear $\mathbb{Q}/\mathbb{Z}$$\mathbb{Q}/\mathbb{Z}$-valued linking form, which is due to Seifert [Seifert1933]:

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

The linking form is such that

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

and is computed as follows. Given $[x] \in TH_\ell(N;\mathbb{Z})$$[x] \in TH_\ell(N;\mathbb{Z})$ and $[y] \in TH_{n-\ell-1}(N;\mathbb{Z})$$[y] \in TH_{n-\ell-1}(N;\mathbb{Z})$ represented by cycles $x \in C_\ell(N;\mathbb{Z})$$x \in C_\ell(N;\mathbb{Z})$ and $y \in C_{n-\ell-1}(N,\mathbb{Z})$$y \in C_{n-\ell-1}(N,\mathbb{Z})$, let $w \in C_{n-\ell}(N;\mathbb{Z})$$w \in C_{n-\ell}(N;\mathbb{Z})$ be such that $\partial w = sy$$\partial w = sy$, for some $s \in \mathbb{Z}$$s \in \mathbb{Z}$. Then we define:

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

where $\langle x, w \rangle$$\langle x, w \rangle$ is defined in (1) above. The resulting element is independent of the choices of $x,y,w$$x,y,w$ and $s$$s$.

3 Definition via cohomology

Let $x \in TH_{\ell}(N;\mathbb{Z})$$x \in TH_{\ell}(N;\mathbb{Z})$ and let $y \in TH_{n-\ell-1}(N;\mathbb{Z})$$y \in TH_{n-\ell-1}(N;\mathbb{Z})$. Note that we have Poincaré 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:

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

Choose $z \in TH^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z})$$z \in TH^{n-\ell-1}(N;\mathbb{Q}/\mathbb{Z})$ such that $\beta(z) = PD(x)$$\beta(z) = PD(x)$. This is always possible since torsion elements in $H^{n-\ell}(N;\mathbb{Z})$$H^{n-\ell}(N;\mathbb{Z})$ map to zero in $H^{n-\ell}(N;\mathbb{Q})$$H^{n-\ell}(N;\mathbb{Q})$. There is a cup product:

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

Then the Kronecker pairing,

$\displaystyle \langle z \cup PD(y),[N] \rangle \in \mathbb{Q}/\mathbb{Z},$ $$\displaystyle \langle z \cup PD(y),[N] \rangle \in \mathbb{Q}/\mathbb{Z},$$

of $z \cup PD(y)$$z \cup PD(y)$ with the fundamental class of $N$$N$ yields $L_N(x,y)$$L_N(x,y)$.

4 Example of 3-dimensional projective space

As an example, let $N = \mathbb{RP}^3$$N = \mathbb{RP}^3$, so that $\ell=1$$\ell=1$ and $n=3$$n=3$. Now $H_1(\mathbb{RP}^3;\mathbb{Z}) \cong \mathbb{Z}_2$$H_1(\mathbb{RP}^3;\mathbb{Z}) \cong \mathbb{Z}_2$. Let $\theta \in H_1(\mathbb{RP}^3;\mathbb{Z})$$\theta \in H_1(\mathbb{RP}^3;\mathbb{Z})$ be the non-trivial element. To compute the linking $L_{\mathbb{RP}^3}(\theta,\theta)$$L_{\mathbb{RP}^3}(\theta,\theta)$, consider $\mathbb{RP}^3$$\mathbb{RP}^3$ modelled as $D^3/\sim$$D^3/\sim$, with antipodal points on $\partial D^2$$\partial D^2$ identified, and choose two representative $1$$1$-chains

Tex syntax error

$x$ and

Tex syntax error

$y$ for $\theta$$\theta$. Let

Tex syntax error

$x$ be the straight line between north and south poles and let

Tex syntax error

$y$ be half of the equator. Now $2y = \partial w$$2y = \partial w$, where $w \in C_2(\mathbb{RP}^3;\mathbb{Z})$$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$$\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 Example of lens spaces

Generalising the above example, the 3-dimensional lens space $N_{p,q} := L(p; q, 1)$$N_{p,q} := L(p; q, 1)$ has $H_1(N_{p,q};\mathbb{Z}) \cong \mathbb{Z}_p$$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})$$\theta \in H_1(N_{p,q};\mathbb{Z})$ by $L_{N_{p,q}}(\theta,\theta) = q/p$$L_{N_{p,q}}(\theta,\theta) = q/p$. Note that $N_{2,1} \cong \mathbb{RP}^3$$N_{2,1} \cong \mathbb{RP}^3$, so this is consistent with the above example.

6 Presentations of linking forms

A presentation for a middle dimensional linking form on $N^{2\ell +1}$$N^{2\ell +1}$

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

is an exact sequence:

$\displaystyle 0 \to F \xrightarrow{\Phi} F^* \to TH_{\ell}(N;\mathbb{Z}) \xrightarrow{\partial} 0,$ $$\displaystyle 0 \to F \xrightarrow{\Phi} F^* \to TH_{\ell}(N;\mathbb{Z}) \xrightarrow{\partial} 0,$$

where $F$$F$ is a free abelain group and the linking $L_N(x,y)$$L_N(x,y)$ can be computed as follows. Let $x',y' \in F^*$$x',y' \in F^*$ be such that $\partial(x')=x$$\partial(x')=x$ and $\partial(y')=y$$\partial(y')=y$. Then we can tensor with $\mathbb{Q}$$\mathbb{Q}$ to obtain an isomorphism

$\displaystyle \Phi \otimes \mathop{\mathrm{Id}} \colon F \otimes_{\mathbb{Z}} \mathbb{Q} \xrightarrow{\cong} F^* \otimes_{\mathbb{Z}} \mathbb{Q}.$ $$\displaystyle \Phi \otimes \mathop{\mathrm{Id}} \colon F \otimes_{\mathbb{Z}} \mathbb{Q} \xrightarrow{\cong} F^* \otimes_{\mathbb{Z}} \mathbb{Q}.$$

The linking form of $N$$N$ is then given by:

$\displaystyle L_N(x,y) = -(x' \otimes 1)((\Phi\otimes \mathop{\mathrm{Id}})^{-1}(y'\otimes 1)).$ $$\displaystyle L_N(x,y) = -(x' \otimes 1)((\Phi\otimes \mathop{\mathrm{Id}})^{-1}(y'\otimes 1)).$$

This formula, in particular the appearance of the sign, is explained in [Gordon&Litherland1978, Section 3] and [Alexander&Hamrick&Vick1976, Proof of Theorem 2.1].

Let $\ell = 1$$\ell = 1$, so $2\ell + 1 = 3$$2\ell + 1 = 3$. Every 3-manifold $N$$N$ is the boundary of a simply connected 4-manifold, which is obtained by glueing 2-handles to an integrally framed link in $S^3$$S^3$ [Lickorish1962], [Wallace1960]. This is sometimes called a surgery presentation for $N$$N$. Suppose that $N$$N$ is a rational homology 3-sphere. Let $A$$A$ be the matrix of (self-) linking numbers of the surgery presentation link. Taking the number of link components in the surgery presentation for $N$$N$ as the rank of $F$$F$, the linking matrix $A$$A$ determines a map $\Phi$$\Phi$ as above, which presents the linking form of $N$$N$. The intersection form on a simply connected 4-manifold $W$$W$ whose boundary is $N$$N$ presents the linking form of $N$$N$. This follows from the long exact sequence of the pair $(W,N)$$(W,N)$ and Poincaré duality. See [Boyer1986] for more details and the use of such presentations for the classification of simply connected 4-manifolds with a given boundary.

For example, the lens space $L(p,1)$$L(p,1)$ is the boundary of the $D^2$$D^2$ bundle over $S^2$$S^2$ with Euler number $-p$$-p$, so the presentation $\Phi = (-p) \colon \mathbb{Z} \to \mathbb{Z}$$\Phi = (-p) \colon \mathbb{Z} \to \mathbb{Z}$ presents the linking form of $L(p,1)$$L(p,1)$ [Gompf&Stipsicz1999, Example 5.3.2].

7 Role in the classification of odd-dimensional manifolds

Linking forms play an important role in the classification of odd-dimensional manifolds. For closed simply connected $5$$5$-manifolds $M$$M$, the linking form is a complete invariant if $H_2(M; \Zz) = TH_2(M; \Zz)$$H_2(M; \Zz) = TH_2(M; \Zz)$. For more information in dimension $5$$5$, see the page on simply-connected 5-manifolds.

For the role of linking forms in the classification of smooth $(q-1)$$(q-1)$-connected $(2q+1)$$(2q+1)$ manifolds with boundary a homotopy sphere, see [Wall1967, Theorem 7].

8 Algebraic classification

An algebraic linking form is a non-singular bi-linear pairing

$\displaystyle b \colon T \times T \to \Qq/\Zz$ $$\displaystyle b \colon T \times T \to \Qq/\Zz$$

on a finite abelian group

Tex syntax error

$T$. It is called symmetric if $b(x, y) = b(y, x)$$b(x, y) = b(y, x)$ and skew-symmetric

if $b(x, y) = - b(y, x)$$b(x, y) = - b(y, x)$.

The classification of skew-symmetric linking forms is rather simple and is due to Wall, [Wall1963, Theorem 3]. It is described in detail in the page on simply-connected 5-manifolds.

The classification of symmetric linking forms is rather intricate. It was begun in [Wall1963] and completed by Kawauchi and Kojima: see [Kawauchi&Kojima1980, Theorem 4.1].

9 References

• [Alexander&Hamrick&Vick1976] J. P. Alexander, G. C. Hamrick and J. W. Vick, Linking forms and maps of odd prime order, Trans. Amer. Math. Soc. 221 (1976), no.1, 169–185. MR0402786 (53 #6600) Zbl 0357.57009
• [Boyer1986] S. Boyer, Simply-connected $4$$4$-manifolds with a given boundary, Trans. Amer. Math. Soc. 298 (1986), no.1, 331–357. MR857447 (88b:57023) Zbl 0790.57009
• [Gompf&Stipsicz1999] R. E. Gompf and A. I. Stipsicz, $4$$4$-manifolds and Kirby calculus, American Mathematical Society, 1999. MR1707327 (2000h:57038) Zbl 0933.57020
• [Gordon&Litherland1978] C. M. Gordon and R. A. Litherland, On the signature of a link, Invent. Math. 47 (1978), no.1, 53–69. MR0500905 (58 #18407) Zbl 0391.57004
• [Intersection] Intersection form, http://www.map.mpim-bonn.mpg.de/Intersection_form
• [Kawauchi&Kojima1980] A. Kawauchi and S. Kojima, Algebraic classification of linking pairings on $3$$3$-manifolds, Math. Ann. 253 (1980), no.1, 29–42. MR594531 (82b:57007) Zbl 0427.57001
• [Lickorish1962] W. B. R. Lickorish, A representation of orientable combinatorial $3$$3$-manifolds, Ann. of Math. (2) 76 (1962), 531–540. MR0151948 (27 #1929) Zbl 0106.37102
• [Seifert1933] H. Seifert, Verschlingungsinvarianten, Sitzungsber. Preuß. Akad. Wiss., Phys.-Math. Kl. 1933, No.26-29, (1933) 811-828. Zbl 0008.18101
• [Wall1963] C. T. C. Wall, Quadratic forms on finite groups, and related topics, Topology 2 (1963), 281–298. MR0156890 (28 #133) Zbl 0215.39903
• [Wall1967] C. T. C. Wall, Classification problems in differential topology. VI. Classification of $(s-1)$$(s-1)$-connected $(2s+1)$$(2s+1)$-manifolds, Topology 6 (1967), 273–296. MR0216510 (35 #7343) Zbl 0173.26102
• [Wallace1960] A. H. Wallace, Modifications and cobounding manifolds, Canad. J. Math. 12 (1960), 503–528. MR0125588 (23 #A2887) Zbl 0116.40401

10 External links

• The Wikipedia page on Poincaré duality