Intersection number of immersions
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[edit] 1 Introduction
This page is based on [Ranicki2002]. Let , be immersions of oriented manifolds in a connected oriented manifold. The intersection number has both an algebraic and geometric formulation; roughly speaking it counts with sign the number of intersection points that the two immersions have. The intersection number is an obstruction to perturbing the immersions into being disjoint. When it vanishes this perturbation can often be achieved using the Whitney trick. The intersection number of immersions is closely related to the intersection form of a -dimensional manifold and in turn its signature: important invariants used in the classification of manifolds.
[edit] 2 Definition
As a consequence of the properties of the cup product the homology intersection pairing is bilinear and satisfies
for all , .
The algebraic intersection number of immersions of oriented manifolds , in a connected oriented manifold, , is the homology intersection of the homology classes , :[edit] 3 Alternative description
The double point set of maps is defined by
with the diagonal subspace.
A double point of immersions is transverse if the linear mapImmersions have transverse intersections (or are transverse) if each double point is transverse and is finite.
The index of a transverse double point isThe geometric intersection number of transverse immersions is
[edit] 4 Equivalence of definitions
[edit] 5 References
- [Ranicki2002] A. Ranicki, Algebraic and geometric surgery, The Clarendon Press Oxford University Press, Oxford, 2002. MR2061749 (2005e:57075) Zbl 1003.57001
- [Scorpan2005] A. Scorpan, The wild world of 4-manifolds, American Mathematical Society, 2005. MR2136212 (2006h:57018) Zbl 1075.57001