Linking form
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1 Background: intersection forms
After Poincaré, a closed oriented manifold has a bilinear intersection form defined on its homology. Given a
-chain
and an
-chain
which is transverse to
, the signed count of the intersections between
and
gives an intersection number
![\langle\, p \, , \, q\, \rangle \in \mathbb{Z}.](/images/math/1/9/7/1976b7a3c96603b84c02f2fb64f708d3.png)
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](/images/math/0/9/5/095e080c258f510bf05313a5cdd17847.png)
and is such that
![\displaystyle I_N(x,y) = (-)^{k(n-k)}I_N(y,x).](/images/math/d/1/d/d1dabe06146f75691e295414051c4ff0.png)
2 Definition of the linking form
![P](/images/math/6/b/5/6b52835f794dc38160c3157e48761ad3.png)
![\displaystyle TP:= \{p \in P \,|\, ap=0 \text{ for some } a \in \mathbb{Z}\}.](/images/math/5/f/0/5f0c2de27bab64de76d170022c84fc1f.png)
The analogue of the intersection pairing for the torsion part of the homology of a closed oriented manifold is the bilinear
-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}.](/images/math/2/3/7/237387c26388354b0262bb07f8cafb6d.png)
The linking form is such that
![\displaystyle L_N(x,y) = (-)^{\ell(n-\ell-1)}L_N(y,x)](/images/math/2/5/a/25ab1e865334dc117a87d082aea40a83.png)
and is computed as follows. Given and
represented by cycles
and
, let
be such that
, for some
. Then we define:
![\displaystyle L_N([x],[y]) := \langle x, w \rangle/s \in \mathbb{Q}/\mathbb{Z},](/images/math/9/8/7/98751c668357b0a097afca17f743fdf9.png)
where is defined in (1) above. The resulting element is independent of the choices of
and
.
3 Definition via cohomology
Let and let
. Note that we have Poincaré duality isomorphisms
![\displaystyle PD \colon TH_{\ell}(N;\mathbb{Z}) \xrightarrow{\cong} TH^{n-\ell}(N;\mathbb{Z})](/images/math/7/5/9/75938638624b6bfb9365c914f7a3c5fa.png)
and
![\displaystyle PD \colon TH_{n-\ell-1}(N;\mathbb{Z}) \xrightarrow{\cong} TH^{\ell+1}(N;\mathbb{Z}).](/images/math/8/4/1/841c558c3d109c23a82a0d35561276d2.png)
Associated to the short exact sequence of coefficients
![\displaystyle 0 \to \mathbb{Z} \to \mathbb{Q} \to \mathbb{Q}/\mathbb{Z} \to 0](/images/math/6/1/4/61426f112ce3a7b346adc376cacc9316.png)
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}).](/images/math/2/1/b/21b09083285c18670be2922a705236ba.png)
Choose such that
. This is always possible since torsion elements in
map to zero in
. 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}).](/images/math/8/c/a/8cab7468890dfe1584b149976438ef06.png)
Then the Kronecker pairing,
![\displaystyle \langle z \cup PD(y),[N] \rangle \in \mathbb{Q}/\mathbb{Z},](/images/math/5/9/f/59f39628bc85fc39b6ac97035e7e5ea8.png)
of with the fundamental class of
yields
.
4 Example of 3-dimensional projective space
As an example, let , so that
and
. Now
. Let
be the non-trivial element. To compute the linking
, consider
modelled as
, with antipodal points on
identified, and choose two representative
-chains
and
for
. Let
be the straight line between north and south poles and let
be half of the equator. Now
, where
is the 2-disk whose boundary is the equator. We see that
, so that
![\displaystyle L_{\mathbb{RP}^3}(\theta,\theta) = L_{\mathbb{RP}^3}([x],[y]) = \langle x,y \rangle/2 = 1/2.](/images/math/0/5/7/05725e0e70a522c8cbd35b7582e9aedd.png)
5 Example of lens spaces
Generalising the above example, the 3-dimensional lens space has
. The linking form is given on a generator
by
. Note that
, so this is consistent with the above example.
6 Presentations of linking forms
A presentation for a middle dimensional linking form on
![\displaystyle L_N \colon TH_{\ell}(N;\mathbb{Z}) \times TH_{\ell}(N;\mathbb{Z}) \to \mathbb{Q}/\mathbb{Z}](/images/math/a/5/9/a59bb62ab7a553a7fb488ca309b5a592.png)
is an exact sequence:
![\displaystyle 0 \to F \xrightarrow{\Phi} F^* \to TH_{\ell}(N;\mathbb{Z}) \xrightarrow{\partial} 0,](/images/math/1/a/7/1a7cdbc1904e2e70a3f0fd8078be0482.png)
where is a free abelain group and the linking
can be computed as follows. Let
be such that
and
. Then we can tensor with
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}.](/images/math/a/2/9/a29e556673f0dddcb8ac18a4382f6fbd.png)
The linking form is given by:
![\displaystyle L_N(x,y) = (x' \otimes 1)((\Phi\otimes \mathop{\mathrm{Id}})^{-1}(y'\otimes 1)).](/images/math/e/3/3/e337321934f0c9f66b268957499cd9f4.png)
Let , so
. Every 3-manifold
is the boundary of a simply connected 4-manifold, which is obtained by glueing 2-handles to an integrally framed link in
[Lickorish1962], [Wallace1960]. This is sometimes called a surgery presentation for
. Suppose that
is a rational homology 3-sphere. Let
be the matrix of (self-) linking numbers of the surgery presentation link. Taking the number of link components in the surgery presentation for
as the rank of
, the linking matrix
determines a map
as above, which presents the linking form of
. The intersection form on a simply connected 4-manifold
whose boundary is
presents the linking form of
. This follows from the long exact sequence of the pair
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.
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 -manifolds
, the linking form is a complete invariant if
. For more information in dimension
, see the page on simply-connected 5-manifolds.
For the role of linking forms in the classification of smooth -connected
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](/images/math/2/8/0/2802ac7a3e0ab4856faaf2e3d8dbe8f1.png)
on a finite abelian group . It is called symmetric if
and skew-symmetric
if
.
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 in [Kawauchi&Kojima1980]; in particular, see Theorem 4.1.
9 References
- [Boyer1986] S. Boyer, Simply-connected
-manifolds with a given boundary, Trans. Amer. Math. Soc. 298 (1986), no.1, 331–357. MR857447 (88b:57023) Zbl 0790.57009
- [Kawauchi&Kojima1980] A. Kawauchi and S. Kojima, Algebraic classification of linking pairings on
-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
-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
-connected
-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