Linking form
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
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
and is such that
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]:
such that
and computed as follows. Given and represented by cycles and , let be such that , for some . Then we define:
The resulting element is independent of the choices of and .
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:
Compute . Then the Kronecker pairing:
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
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
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 is given by:
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 Poincar\'{e} 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 Classification of 5-manifolds
Linking forms are central to the classification of simply connected 5-manifolds. See this 5-manifolds page, which also describes the classification of anti-symmetric linking forms.
8 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
- [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
- [Wallace1960] A. H. Wallace, Modifications and cobounding manifolds, Canad. J. Math. 12 (1960), 503–528. MR0125588 (23 #A2887) Zbl 0116.40401
The intersection form is defined by
and is such that
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]:
such that
and computed as follows. Given and represented by cycles and , let be such that , for some . Then we define:
The resulting element is independent of the choices of and .
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:
Compute . Then the Kronecker pairing:
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
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
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 is given by:
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 Poincar\'{e} 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 Classification of 5-manifolds
Linking forms are central to the classification of simply connected 5-manifolds. See this 5-manifolds page, which also describes the classification of anti-symmetric linking forms.
8 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
- [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
- [Wallace1960] A. H. Wallace, Modifications and cobounding manifolds, Canad. J. Math. 12 (1960), 503–528. MR0125588 (23 #A2887) Zbl 0116.40401
The intersection form is defined by
and is such that
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]:
such that
and computed as follows. Given and represented by cycles and , let be such that , for some . Then we define:
The resulting element is independent of the choices of and .
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:
Compute . Then the Kronecker pairing:
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
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
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 is given by:
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 Poincar\'{e} 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 Classification of 5-manifolds
Linking forms are central to the classification of simply connected 5-manifolds. See this 5-manifolds page, which also describes the classification of anti-symmetric linking forms.
8 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
- [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
- [Wallace1960] A. H. Wallace, Modifications and cobounding manifolds, Canad. J. Math. 12 (1960), 503–528. MR0125588 (23 #A2887) Zbl 0116.40401
The intersection form is defined by
and is such that
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]:
such that
and computed as follows. Given and represented by cycles and , let be such that , for some . Then we define:
The resulting element is independent of the choices of and .
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:
Compute . Then the Kronecker pairing:
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
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
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 is given by:
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 Poincar\'{e} 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 Classification of 5-manifolds
Linking forms are central to the classification of simply connected 5-manifolds. See this 5-manifolds page, which also describes the classification of anti-symmetric linking forms.
8 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
- [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
- [Wallace1960] A. H. Wallace, Modifications and cobounding manifolds, Canad. J. Math. 12 (1960), 503–528. MR0125588 (23 #A2887) Zbl 0116.40401