Linking form
This page has not been refereed. The information given here might be incomplete or provisional. |
Introduction to linking forms
After Poincaré and Lefschetz, a closed oriented manifold has a bilinear intersection form defined on its homology:
such that
Given a --chain and an --chain which is transverse to , the signed count of the intersections between and gives an intersection number .
By bilinearity, the intersection form vanishes on the torsion part of the homology. 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:
such that
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 .
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