Intersection number of immersions
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Let $M$ be an oriented $m$-dimensional manifold. The '''homology intersection pairing''' of $M$, $$\lambda: H_n(M)\times H_{m-n}(M) \to \Z; \quad (x,y) \mapsto \lambda(x,y),$$ | Let $M$ be an oriented $m$-dimensional manifold. The '''homology intersection pairing''' of $M$, $$\lambda: H_n(M)\times H_{m-n}(M) \to \Z; \quad (x,y) \mapsto \lambda(x,y),$$ | ||
− | is defined analogously to the case $m=2n$, when it is called the [[Intersection_form| | + | is defined analogously to the case $m=2n$, when it is called the [[Intersection_form|intersection form]]. |
Besides generalizing that simple direct definition, we can use the concept of the ''cup product'' and define the pairing by | Besides generalizing that simple direct definition, we can use the concept of the ''cup product'' and define the pairing by | ||
$$\lambda(x,y) = \langle x^*\cup y^*,[M]\rangle \in \Z,$$ | $$\lambda(x,y) = \langle x^*\cup y^*,[M]\rangle \in \Z,$$ |
Revision as of 12:22, 29 March 2019
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
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.
2 Definition
is defined analogously to the case , when it is called the intersection form. Besides generalizing that simple direct definition, we can use the concept of the cup product and define the pairing by
where , are the Poincaré duals of , and is the fundamental class.
The homology intersection pairing is bilinear and satisfies
for all , . These properties are easy to check using the simple direct definition; they also follow from simple properties of the cup product.
The algebraic intersection number of immersions of oriented manifolds , in a connected oriented manifold, is the homology intersection of the homology classes , :
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
4 Equivalence of definitions
The algebraic and geometric intersection numbers agree,
For a proof of this clasical fact see [Scorpan2005, Section 3.2] or [Ranicki2002, Proposition 7.22].
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