Linking form
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[edit] 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
The intersection form is defined by
and is such that
[edit] 2 Definition of the linking form
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]:
The linking form is such that
and is computed as follows. Given and represented by cycles and , let be such that , for some . Then we define:
where is defined in (1) above. The resulting element is independent of the choices of and .
[edit] 3 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] 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
[edit] 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.
[edit] 6 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] 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].
[edit] 8 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] 9 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] 10 External links
- The Wikipedia page on Poincaré duality