Intersection number of immersions

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== Introduction ==
== Introduction ==
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This page is based on \cite{Ranicki2002}. 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 $4k$-dimensional manifold and in turn its [[Signature|signature]]: important invariants used in the classification of [[Manifold|manifolds]].
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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]].
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== Definition ==
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Let $M$ be an oriented $m$-dimensional manifold. The '''homology intersection pairing''' of $M$, $$\lambda: H_n(M)\times H_{m-n}(M) \to \Z; \quad (x,y) \mapsto \lambda(x,y),$$
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is defined by $$\lambda(x,y) = \langle x^*\cup y^*,[M]\rangle \in \Z$$ where $x^*\in H^{m-n}(M)$, $y^*\in H^n(M)$ are the [[Poincaré duality|Poincaré duals]] of $x$, $y$ and $[M]$ is the [[fundamental class]].
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As a consequence of the properties of the cup product the homology intersection pairing is bilinear and satisfies
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$$\lambda(y,x) = (-1)^{n(m-n)}\lambda(x,y)$$
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for all $x\in H_n(M)$, $y\in H_{m-n}(M)$.
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The '''algebraic intersection number''' 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, $\lambda^{\mathrm{alg}}(N_1,N_2)\in\Z$, is the homology intersection of the homology classes $(f_1)_*[N_1]\in H_{n_1}(M)$, $(f_2)_*[N_2]\in H_{n_2}(M)$: $$\lambda^{\mathrm{alg}}(N_1,N_2) := \lambda((f_1)_*[N_1],(f_2)_*[N_2]).$$
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This page is based on \cite{Ranicki2002}, see also \cite[Excercise 14.9.6]{Broecker&Jaenich1982}.
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==Alternative description==
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== Statement ==
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The '''double point set''' of maps $f_i:N_i\to M$ $(i=1,2)$ is defined by
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Let
$$S_2(f_1,f_2) = \{(x_1,x_2)\in N_1\times N_2 | f_1(x_1) = f_2(x_2)\in M\} = (f_1\times f_2)^{-1}(\Delta(M))$$
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$$I_M=\lambda_M=\lambda: H_{n_1}(M)\times H_{n_2}(M) \to \Z$$
with $\Delta(M) = \{(x,x) | x\in M\}\subset M\times M$ the diagonal subspace.
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be the [[Intersection_form#Definition|''homology intersection pairing (or product)'']] of $M$.
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.
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The ''double point set'' of $f_1$ and $f_2$ is defined by
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$$S_2(f_1,f_2) = \{(x_1,x_2)\in N_1\times N_2 | f_1(x_1) = f_2(x_2)\in M\} =
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(f_1\times f_2)^{-1}(\Delta(M)),$$
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where $\Delta(M) = \{(x,x)\in M\times M | x\in M\}$ the diagonal.
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.
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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
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$$df(x) = df_1(x_1)\oplus df_2(x_2): \tau_{N_1}(x_1)\oplus \tau_{N_2}(x_2) \to \tau_M(f(x))$$
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is an isomorphism.
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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''' $I(x)\in\Z$ of a transverse double point $x=(x_1,x_2)\in S_2(f_1,f_2)$ is $$I(x) = \left\{ \begin{array}{cc} +1, & \mathrm{if}\; df(x)\; \mathrm{preserves}\; \mathrm{orientations}\\ -1, & \mathrm{otherwise}.\end{array}\right.$$
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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
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$$I(x) = \left\{ \begin{array}{cc} +1, & \mathrm{if}\; df(x)\; \mathrm{preserves}\; \mathrm{orientations}\\ -1, & \mathrm{otherwise}.\end{array}\right.$$
The '''geometric intersection number''' of transverse immersions $f_i:N_i^{n_i} \looparrowright M^{n_1+n_2}$ $(i=1,2)$ is
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{{beginthm|Theorem}}\label{t:algeo} If $f_1$ and $f_2$ are transverse, then
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$$\lambda(f_{1*}[N_1],f_{2*}[N_2])=\sum_{x\in S_2(f_1,f_2)}{I(x)}.$$
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{{endthm}}
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This clasical fact is either a theorem or a definition depending on which definition of [[Intersection_form#Definition|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 $f_2$.
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<!--
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$\lambda^{\mathrm{geo}}(N_1,N_2)$ defined below.
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$\lambda^{\mathrm{alg}}(f_1,f_2)\in\Z$ $$\lambda^{\mathrm{alg}}(N_1,N_2) := \lambda((f_1)_*[N_1],(f_2)_*[N_2]).$$
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the homology intersection of the homology classes $(f_1)_*[N_1]\in H_{n_1}(M)$, $(f_2)_*[N_2]\in H_{n_2}(M)$:
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==Alternative description==
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The ''geometric'' intersection number of transverse immersions $f_i:N_i\looparrowright M$ $(i=1,2)$ is defined as
$$\lambda^{\mathrm{geo}}(N_1,N_2) = \sum_{x\in S_2(f_1,f_2)}{I(x)}\in \Z.$$
$$\lambda^{\mathrm{geo}}(N_1,N_2) = \sum_{x\in S_2(f_1,f_2)}{I(x)}\in \Z.$$
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The algebraic and geometric intersection numbers agree,
== Equivalence of definitions ==
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$$\lambda^{\mathrm{alg}}(N_1,N_2)=\lambda^{\mathrm{geo}}(N_1,N_2).$$ -->
<wikitex>;
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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 see \cite{Scorpan2005|Section 3.2} or \cite{Ranicki2002|Proposition 7.22}.
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</wikitex>
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==References==
==References==
{{#RefList:}}
{{#RefList:}}
[[Category:Definitions]]
[[Category:Definitions]]
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[[Category:Forgotten in Textbooks]]

Latest revision as of 14:17, 2 April 2019

This page has not been refereed. The information given here might be incomplete or provisional.

[edit] 1 Introduction

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 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

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