1 Background: intersection forms

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

The intersection form is defined by

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

and is such that

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

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

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]:

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

The linking form is such that

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

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:

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

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

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

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)$.

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 $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

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

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.

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].

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

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

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].