Linking form
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1 Background: intersection forms
After Poincaré and Lefschetz, 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
.
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](/images/math/e/b/d/ebddd6200fa14a95e472da816b20076d.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:
![\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/b/7/2b7173b23b688d4f71b77357ca02b638.png)
such that
![\displaystyle L_N(x,y) = (-)^{\ell(n-\ell-1)}L_N(y,x)](/images/math/2/5/a/25ab1e865334dc117a87d082aea40a83.png)
and 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/d/f/0/df0eb2fbf65c63b2e911c48228983104.png)
The resulting element is independent of the choices of and
.
3 Definition via homology
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-1}(N;\mathbb{Q}).](/images/math/a/c/f/acf0edd39720201b0317cdabe99a1d1f.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)
Compute . Then the Kronecker pairing:
![\displaystyle \langle a,[N] \rangle \in \mathbb{Q}/\mathbb{Z}](/images/math/2/8/a/28a841ba31d7a473ec5cf00d1469d2ec.png)
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 linking form is an exact sequence:
![\displaystyle 0 \to F \xrightarrow{\Phi} F^* \to TH_1(N;\mathbb{Z}) \xrightarrow{\partial} 0,](/images/math/f/6/6/f669705ff8a6260750ff6503cb2a086b.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')).](/images/math/f/2/f/f2ff25a0c880f6224ca03bc5686d836c.png)
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
[Lickorish1963], [Wallace1960]. This is sometimes called a surgery presentation for
. Suppose that
is a rational homology 3-sphere. The matrix
of (self-) linking numbers of the link determines a presentation of the linking form of
. Taking the number of link components in the surgery presentation for
as the rank of
, the linking matrix
determines a map
as above. 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 Poincar\'{e} duality.
7 Classification of 5-manifolds
Linking forms are central to the classification of simply connected 5-manifolds. See this 5-manifolds page. This page also described the classification of anti-symmetric linking forms.
8 References
- [Lickorish1963] Template:Lickorish1963
- [Wallace1960] A. H. Wallace, Modifications and cobounding manifolds, Canad. J. Math. 12 (1960), 503–528. MR0125588 (23 #A2887) Zbl 0116.40401