Intersection number of immersions
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[edit] 1 Introduction
Let be a connected oriented manifold of dimension and , immersions of oriented - and -manifolds. The intersection number of and 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.
This page is based on [Ranicki2002], see also [Broecker&Jaenich1982, Excercise 14.9.6].
[edit] 2 Statement
Let
be the homology intersection pairing (or product) of .
The double point set of and is defined by
where the diagonal.
A double point of and is transverse if the linear map
is an isomorphism. Immersions and are transverse (or have transverse intersection) if is finite and every double point is transverse.
The index, or the sign of a transverse double point is
Theorem 2.1. If and are transverse, then
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]. Unless this equality is a definition, the left- and right- hand sides of the equality can be called algebraic and geometric intersection number of and .
[edit] 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