Intersection number of immersions
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1 Introduction
This page is based on [Ranicki2002], see also [Broecker&Jaenich1982, Excercise 14.9.6]. 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 manifold (and so to the signature of a -dimensional manifold) and to characteristic classes. These are important invariants used in the classification of manifolds.
2 Statement
Let be a connected oriented manifold of dimension and
the homology intersection pairing (or product) of .
Theorem 2.1. For any immersions , of oriented - and -manifolds the number equals to defined below.
This this clasical fact is either a theorem or a definition depending on which definition of homology intersection pairing one accepts. For a proof see [Scorpan2005, Section 3.2] or [Ranicki2002, Proposition 7.22].
The number can be called algebraic intersection number of and .
3 Alternative description
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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 intersection (or are transverse) if each double point is transverse and is finite.
The index, or the sign of a transverse double point is
The geometric intersection number of transverse immersions is defined as
4 References
- [Broecker&Jaenich1982] Th. Br\"ocker and K. J\"anich Introduction to Differential Topology, Cambridge University Press, 1982. ISBN-13: 978-0521284707, ISBN-10: 0521284708.
- [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