Intersection number of immersions

Latest revision as of 15:17, 2 April 2019

[edit] 1 Introduction

Let ${{Stub}} == Introduction == <wikitex>; Let $M$ be a connected oriented manifold of dimension $m=n_1+n_2$ and $f_1:N_1\looparrowright M$, $f_2:N_2\looparrowright M$ immersions of oriented $n_1$- and $n_2$-manifolds. The [[Intersection_form|intersection number]] of $f_1$ and $f_2$ 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 \cite{Ranicki2002}, see also \cite[Excercise 14.9.6]{Broecker&Jaenich1982}. </wikitex> == Statement == <wikitex>; Let $$I_M=\lambda_M=\lambda: H_{n_1}(M)\times H_{n_2}(M) \to \Z$$ be the [[Intersection_form#Definition|''homology intersection pairing (or product)'']] of $M$. The ''double point set'' of $f_1$ and $f_2$ is defined by $$S_2(f_1,f_2) = \{(x_1,x_2)\in N_1\times N_2 | f_1(x_1) = f_2(x_2)\in M\} = (f_1\times f_2)^{-1}(\Delta(M)),$$ where $\Delta(M) = \{(x,x)\in M\times M | x\in M\}$ the diagonal. A double point $x=(x_1,x_2)\in S_2(f_1,f_2)$ of $f_1$ and $f_2$ is ''transverse'' if the linear map $$df(x) = df_1(x_1)\oplus df_2(x_2): \tau_{N_1}(x_1)\oplus \tau_{N_2}(x_2) \to \tau_M(f(x))$$ is an isomorphism. Immersions $f_1$ and $f_2$ are ''transverse'' (or have ''transverse intersection'') if $S_2(f_1,f_2)$ is finite and every double point is transverse. 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 $$I(x) = \left\{ \begin{array}{cc} +1, & \mathrm{if}\; df(x)\; \mathrm{preserves}\; \mathrm{orientations}\ -1, & \mathrm{otherwise}.\end{array}\right.$$ {{beginthm|Theorem}}\label{t:algeo} If $f_1$ and $f_2$ are transverse, then $$\lambda(f_{1*}[N_1],f_{2*}[N_2])=\sum_{x\in S_2(f_1,f_2)}{I(x)}.$$ {{endthm}} This clasical fact is either a theorem or a definition depending on which definition of homology intersection pairing one accepts. For a proof see \cite{Scorpan2005|Section 3.2} or \cite{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 $f_1$ and $f_2$. <!-- $\lambda^{\mathrm{geo}}(N_1,N_2)$ defined below. $\lambda^{\mathrm{alg}}(f_1,f_2)\in\Z$ $$\lambda^{\mathrm{alg}}(N_1,N_2) := \lambda((f_1)_*[N_1],(f_2)_*[N_2]).$$ the homology intersection of the homology classes $(f_1)_*[N_1]\in H_{n_1}(M)$, $(f_2)_*[N_2]\in H_{n_2}(M)$: ==Alternative description== The ''geometric'' intersection number of transverse immersions $f_i:N_i\looparrowright M$ $(i=1,2)$ is defined as $$\lambda^{\mathrm{geo}}(N_1,N_2) = \sum_{x\in S_2(f_1,f_2)}{I(x)}\in \Z.$$ The algebraic and geometric intersection numbers agree, $$\lambda^{\mathrm{alg}}(N_1,N_2)=\lambda^{\mathrm{geo}}(N_1,N_2).$$ --> ==References== {{#RefList:}} [[Category:Definitions]] [[Category:Forgotten in Textbooks]]M$ be a connected oriented manifold of dimension $m=n_1+n_2$ and $f_1:N_1\looparrowright M$, $f_2:N_2\looparrowright M$ immersions of oriented $n_1$- and $n_2$-manifolds. The intersection number of $f_1$ and $f_2$ 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

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

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

The double point set of $f_1$ and $f_2$ is defined by

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

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

A double point $x=(x_1,x_2)\in S_2(f_1,f_2)$ of $f_1$ and $f_2$ is transverse if the linear map

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

is an isomorphism. Immersions $f_1$ and $f_2$ are transverse (or have transverse intersection) if $S_2(f_1,f_2)$ is finite and every double point is transverse.

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

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

Theorem 2.1. If $f_1$ and $f_2$ are transverse, then

$$\displaystyle \lambda(f_{1*}[N_1],f_{2*}[N_2])=\sum_{x\in S_2(f_1,f_2)}{I(x)}.$$

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 $f_1$ and $f_2$.

[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