Linking form
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[edit] 1 Definition
After Seifert [Seifert1933], a closed oriented -manifold has a bilinear linking form
Here the torsion part of an abelian group is the subgroup
The linking form is the analogue for the torsion part of the homology of the intersection product which vanishes on the torsion part.
The linking form is defined as follows. Take and represented by cycles and . There is such that for some . Define
For fixed the resulting `residie modulo 1' is independent of the choices of and .
We have
[edit] 2 Definition via cohomology
Let and let . Note that we have Poincaré duality isomorphisms
and
Associated to the short exact sequence of coefficients
is the Bockstein long exact sequence in cohomology:
Choose such that . This is always possible since torsion elements in map to zero in . There is a cup product:
Then the Kronecker pairing,
of with the fundamental class of yields .
[edit] 3 Examples of 3-dimensional projective and lens spaces
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
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.
[edit] 4 Presentations of linking forms
A presentation for a middle dimensional linking form on
is an exact sequence:
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
The linking form of is then given by:
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 , 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.
For example, the lens space is the boundary of the bundle over with Euler number , so the presentation presents the linking form of [Gompf&Stipsicz1999, Example 5.3.2].
[edit] 5 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].
[edit] 6 Algebraic classification
An algebraic linking form is a non-singular bi-linear pairing
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: see [Kawauchi&Kojima1980, Theorem 4.1].
[edit] 7 References
- [Alexander&Hamrick&Vick1976] J. P. Alexander, G. C. Hamrick and J. W. Vick, Linking forms and maps of odd prime order, Trans. Amer. Math. Soc. 221 (1976), no.1, 169–185. MR0402786 (53 #6600) Zbl 0357.57009
- [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
- [Gompf&Stipsicz1999] R. E. Gompf and A. I. Stipsicz, -manifolds and Kirby calculus, American Mathematical Society, 1999. MR1707327 (2000h:57038) Zbl 0933.57020
- [Gordon&Litherland1978] C. M. Gordon and R. A. Litherland, On the signature of a link, Invent. Math. 47 (1978), no.1, 53–69. MR0500905 (58 #18407) Zbl 0391.57004
- [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
[edit] 8 External links
- The Wikipedia page on Poincaré duality