Linking form

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1 Background: intersection forms

After Poincaré and Lefschetz, 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 $\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 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:

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

such that

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

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

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

The resulting element is independent of the choices of $x,y,w$$x,y,w$ and $s$$s$.

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

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