Linking form

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[edit] 1 Background: intersection forms

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

(1)\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).

[edit] 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 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 is the bilinear \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-1)}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:

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

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

[edit] 3 Definition via cohomology

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

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

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}) 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}).

Then the Kronecker pairing,

\displaystyle \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).

[edit] 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.

[edit] 5 Example of lens spaces

Generalising the above example, the 3-dimensional 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.

[edit] 6 Presentations of linking forms

A presentation for a middle dimensional linking form on 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 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

\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 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, so 2\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 [Lickorish1962], [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 [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) [Gompf&Stipsicz1999, Example 5.3.2].

[edit] 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-manifolds M, the linking form is a complete invariant if H_2(M; \Zz) = TH_2(M; \Zz). For more information in dimension 5, see the page on 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 [Wall1967, Theorem 7].

[edit] 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. 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, [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].

[edit] 9 References

[edit] 10 External links

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