Intersection number of immersions

Revision as of 14:42, 2 April 2019

1 Introduction

This page is based on [Ranicki2002], see also [Broecker&Jaenich1982, Excercise 14.9.6]. Let ${{Stub}} == Introduction == <wikitex>; This page is based on \cite{Ranicki2002}, see also \cite[Excercise 14.9.6]{Broecker&Jaenich1982}. Let $f_1:N_1^{n_1} \looparrowright M^{n_1+n_2}$, $f_2:N_2^{n_2} \looparrowright M^{n_1+n_2}$ be immersions of oriented manifolds in a connected oriented manifold. The intersection number $\lambda([N_1],[N_2])\in\Z$ 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|intersection form]] of a manifold (and so to the [[Intersection_form#Definition|signature]] of a k$-dimensional manifold) and to [[Stiefel-Whitney_characteristic_classes|characteristic classes]]. These are important invariants used in the classification of [[Manifold|manifolds]]. </wikitex> == Definition == <wikitex>; Let $M$ be an oriented $m$-dimensional manifold. We use the [[Intersection_form#Definition|''homology intersection pairing (or product)'']] of $M$, $$I_M=\lambda_M=\lambda: H_n(M)\times H_{m-n}(M) \to \Z; \quad (x,y) \mapsto \lambda(x,y).$$ The '''algebraic intersection number''' $\lambda^{\mathrm{alg}}(f_1,f_2)\in\Z$ of immersions of oriented manifolds $f_1:N_1^{n_1} \looparrowright M^{n_1+n_2}$, $f_2:N_2^{n_2} \looparrowright M^{n_1+n_2}$ in a connected oriented manifold, is the homology intersection of the homology classes $(f_1)_*[N_1]\in H_{n_1}(M)$, $(f_2)_*[N_2]\in H_{n_2}(M)$: $$\lambda^{\mathrm{alg}}(N_1,N_2) := \lambda((f_1)_*[N_1],(f_2)_*[N_2]).$$ </wikitex> ==Alternative description== <wikitex>; The '''double point set''' of maps $f_i:N_i\to M$ $(i=1,2)$ is defined by $$S_2(f_1,f_2) = \{(x_1,x_2)\in N_1\times N_2 | f_1(x_1) = f_2(x_2)\in M\} = (f_1\times f_2)^{-1}(\Delta(M))$$ with $\Delta(M) = \{(x,x) | x\in M\}\subset M\times M$ the diagonal subspace. A double point $x=(x_1,x_2)\in S_2(f_1,f_2)$ of immersions $f_i:N_i^{n_i} \looparrowright M^{n_1+n_2}$ $(i=1,2)$ is '''transverse''' if the linear map $$df(x) = (df_1(x_1),df_2(x_2)): \tau_{N_1}(x_1)\oplus \tau_{N_2}(x_2) \to \tau_M(f(x))$$ is an isomorphism. Immersions $f_i:N_i^{n_i} \looparrowright M^{n_1+n_2}$ $(i=1,2)$ have '''transverse intersections''' (or are '''transverse''') if each double point is transverse and $S_2(f_1,f_2)$ is finite. The '''index''' $I(x)\in\Z$ of a transverse double point $x=(x_1,x_2)\in S_2(f_1,f_2)$ is $$I(x) = \left\{ \begin{array}{cc} +1, & \mathrm{if}\; df(x)\; \mathrm{preserves}\; \mathrm{orientations}\ -1, & \mathrm{otherwise}.\end{array}\right.$$ The '''geometric intersection number''' of transverse immersions $f_i:N_i^{n_i} \looparrowright M^{n_1+n_2}$ $(i=1,2)$ is $$\lambda^{\mathrm{geo}}(N_1,N_2) = \sum_{x\in S_2(f_1,f_2)}{I(x)}\in \Z.$$ </wikitex> == Equivalence of definitions == <wikitex>; The algebraic and geometric intersection numbers agree, $$\lambda^{\mathrm{alg}}(N_1,N_2)=\lambda^{\mathrm{geo}}(N_1,N_2).$$ For a proof of this clasical fact see \cite{Scorpan2005|Section 3.2} or \cite{Ranicki2002|Proposition 7.22}. </wikitex> ==References== {{#RefList:}} [[Category:Definitions]] [[Category:Forgotten in Textbooks]]f_1:N_1^{n_1} \looparrowright M^{n_1+n_2}$, $f_2:N_2^{n_2} \looparrowright M^{n_1+n_2}$ be immersions of oriented manifolds in a connected oriented manifold. The intersection number $\lambda([N_1],[N_2])\in\Z$ 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 $4k$-dimensional manifold) and to characteristic classes. These are important invariants used in the classification of manifolds.

2 Statement

Let $M$ be a connected oriented manifold of dimension $m=n_1+n_2$ and

$$\displaystyle I_M=\lambda_M=\lambda: H_{n_1}(M)\times H_{n_2}(M) \to \Z$$

the homology intersection pairing (or product) of $M$.

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 $\lambda(f_{1*}[N_1],f_{2*}[N_2])$ can be called algebraic intersection number of $f_1$ and $f_2$.

3 Alternative description

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The double point set of maps $f_i:N_i\to M$ $(i=1,2)$ is defined by

$$\displaystyle S_2(f_1,f_2) = \{(x_1,x_2)\in N_1\times N_2 | f_1(x_1) = f_2(x_2)\in M\} = (f_1\times f_2)^{-1}(\Delta(M))$$

with $\Delta(M) = \{(x,x) | x\in M\}\subset M\times M$ the diagonal subspace.

$$\displaystyle df(x) = (df_1(x_1),df_2(x_2)): \tau_{N_1}(x_1)\oplus \tau_{N_2}(x_2) \to \tau_M(f(x))$$

Immersions $f_i:N_i\looparrowright M$ $(i=1,2)$ have transverse intersection (or are transverse) if each double point is transverse and $S_2(f_1,f_2)$ is finite.

The index, or the sign $I(x)\in\Z$ of a transverse double point $x=(x_1,x_2)\in S_2(f_1,f_2)$ is

$$\displaystyle I(x) = \left\{ \begin{array}{cc} +1, & \mathrm{if}\; df(x)\; \mathrm{preserves}\; \mathrm{orientations}\\ -1, & \mathrm{otherwise}.\end{array}\right.$$

The geometric intersection number of transverse immersions $f_i:N_i\looparrowright M$ $(i=1,2)$ is defined as

$$\displaystyle \lambda^{\mathrm{geo}}(N_1,N_2) = \sum_{x\in S_2(f_1,f_2)}{I(x)}\in \Z.$$

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