Intersection number of immersions

Revision as of 14:52, 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> == Statement == <wikitex>; Let $M$ be a connected oriented manifold of dimension $m=n_1+n_2$ and $$I_M=\lambda_M=\lambda: H_{n_1}(M)\times H_{n_2}(M) \to \Z$$ the [[Intersection_form#Definition|''homology intersection pairing (or product)'']] of $M$. {{beginthm|Theorem}}\label{t:algeo} For any immersions $f_1:N_1\looparrowright M$, $f_2:N_2\looparrowright M$ of oriented $n_1$- and $n_2$-manifolds the number $\lambda(f_{1*}[N_1],f_{2*}[N_2])$ equals to $\lambda^{\mathrm{geo}}(N_1,N_2)$ defined below. {{endthm}} 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 \cite{Scorpan2005|Section 3.2} or \cite{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 $f_2$. <!-- $\lambda^{\mathrm{alg}}(f_1,f_2)\in\Z$ $$\lambda^{\mathrm{alg}}(N_1,N_2) := \lambda((f_1)_*[N_1],(f_2)_*[N_2]).$$ the homology intersection of the homology classes $(f_1)_*[N_1]\in H_{n_1}(M)$, $(f_2)_*[N_2]\in H_{n_2}(M)$: </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 $f_i:N_i\looparrowright M$ $(i=1,2)$ 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 $$I(x) = \left\{ \begin{array}{cc} +1, & \mathrm{if}\; df(x)\; \mathrm{preserves}\; \mathrm{orientations}\ -1, & \mathrm{otherwise}.\end{array}\right.$$ The ''geometric'' intersection number of transverse immersions $f_i:N_i\looparrowright M$ $(i=1,2)$ is defined as $$\lambda^{\mathrm{geo}}(N_1,N_2) = \sum_{x\in S_2(f_1,f_2)}{I(x)}\in \Z.$$ </wikitex> <!-- 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]]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 $M$.

Let $N_i$ be oriented $n_i$-manifolds $(i=1,2)$. The double point set of maps $f_i:N_i\to M$ $(i=1,2)$ is defined by

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

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 immersions $f_i:N_i\looparrowright M$ $(i=1,2)$ 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 $f_i:N_i\looparrowright M$ $(i=1,2)$ have transverse intersection (or are transverse) 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. For any transverse immersions $f_1:N_1\looparrowright M$, $f_2:N_2\looparrowright M$ of oriented $n_1$- and $n_2$-manifolds

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

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]. 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 $f_2$.

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