1 Definition

After Seifert [Seifert1933], a closed oriented $n$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}n$-manifold $N$$N$ has a bilinear linking form

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

Here the torsion part of an abelian group $P$$P$ is the subgroup

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

The linking form is the analogue for the torsion part of the homology of the intersection product which vanishes on the torsion part.

The linking form is defined as follows. Take $[x] \in TH_\ell(N;\Zz)$$[x] \in TH_\ell(N;\Zz)$ and $[y] \in TH_{n-\ell-1}(N;\Zz)$$[y] \in TH_{n-\ell-1}(N;\Zz)$ represented by cycles $x \in C_\ell(N;\Zz)$$x \in C_\ell(N;\Zz)$ and $y \in C_{n-\ell-1}(N,\Zz)$$y \in C_{n-\ell-1}(N,\Zz)$. There is $Y\in C_{n-\ell}(N;\Z)$$Y\in C_{n-\ell}(N;\Z)$ such that $\partial Y = sy$$\partial Y = sy$ for some $s \in \Zz$$s \in \Zz$. Define

$\displaystyle L_N([x],[y]):= \langle x, Y \rangle/s \in \mathbb{Q}/\Zz$
to be the intersection number of $x$$x$ and $Y$$Y$ divided by $s$$s$ and taken modulo 1.

For fixed $N,[x],[y]$$N,[x],[y]$ the resulting `residie modulo 1' is independent of the choices of $x,y,Y$$x,y,Y$ and $s$$s$.

We have

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

2 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

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

and

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

Associated to the short exact sequence of coefficients

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

is the Bockstein long exact sequence in cohomology:

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

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:

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

Then the Kronecker pairing,

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

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)$.

3 Examples of 3-dimensional projective and lens spaces

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 $x$$x$ and $y$$y$ for $\theta$$\theta$. Let $x$$x$ be the straight line between north and south poles and let $y$$y$ be half of the equator. Now $2y = \partial w$$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

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

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.

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

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

is an exact sequence:

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

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

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

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

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

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].

5 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].

6 Algebraic classification

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

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

on a finite abelian group $T$$T$. It is called symmetric if $b(x, y) = b(y, x)$$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].