Intersection number of immersions

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== Introduction ==
== Introduction ==
<wikitex>;
<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]].
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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]].
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 $4k$-dimensional manifold) and to [[Stiefel-Whitney_characteristic_classes|characteristic classes]].
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These are important invariants used in the classification of [[Manifold|manifolds]].
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This page is based on \cite{Ranicki2002}, see also \cite[Excercise 14.9.6]{Broecker&Jaenich1982}.
</wikitex>
</wikitex>
== Statement ==
== Statement ==
<wikitex>;
<wikitex>;
Let $M$ be a connected oriented manifold of dimension $m=n_1+n_2$ and
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Let
$$I_M=\lambda_M=\lambda: H_{n_1}(M)\times H_{n_2}(M) \to \Z$$
$$I_M=\lambda_M=\lambda: H_{n_1}(M)\times H_{n_2}(M) \to \Z$$
the [[Intersection_form#Definition|''homology intersection pairing (or product)'']] of $M$.
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be the [[Intersection_form#Definition|''homology intersection pairing (or product)'']] of $M$.
Let $N_i$ be oriented $n_i$-manifolds $(i=1,2)$.
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The ''double point set'' of $f_1$ and $f_2$ is defined by
The ''double point set'' of maps $f_i:N_i\to M$ $(i=1,2)$ is defined by
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$$S_2(f_1,f_2) = \{(x_1,x_2)\in N_1\times N_2 | f_1(x_1) = f_2(x_2)\in M\} =
$$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)),$$
(f_1\times f_2)^{-1}(\Delta(M)),$$
where $\Delta(M) = \{(x,x)\in M\times M | x\in M\}$ the diagonal.
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
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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
$$df(x) = df_1(x_1)\oplus df_2(x_2): \tau_{N_1}(x_1)\oplus \tau_{N_2}(x_2) \to \tau_M(f(x))$$
$$df(x) = df_1(x_1)\oplus df_2(x_2): \tau_{N_1}(x_1)\oplus \tau_{N_2}(x_2) \to \tau_M(f(x))$$
is an isomorphism.
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.
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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'', 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
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
$$I(x) = \left\{ \begin{array}{cc} +1, & \mathrm{if}\; df(x)\; \mathrm{preserves}\; \mathrm{orientations}\\ -1, & \mathrm{otherwise}.\end{array}\right.$$
$$I(x) = \left\{ \begin{array}{cc} +1, & \mathrm{if}\; df(x)\; \mathrm{preserves}\; \mathrm{orientations}\\ -1, & \mathrm{otherwise}.\end{array}\right.$$
{{beginthm|Theorem}}\label{t:algeo}
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{{beginthm|Theorem}}\label{t:algeo} If $f_1$ and $f_2$ are transverse, then
For any transverse immersions $f_1:N_1\looparrowright M$, $f_2:N_2\looparrowright M$ of oriented $n_1$- and $n_2$-manifolds
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$$\lambda(f_{1*}[N_1],f_{2*}[N_2])=\sum_{x\in S_2(f_1,f_2)}{I(x)}.$$
$$\lambda(f_{1*}[N_1],f_{2*}[N_2])=\sum_{x\in S_2(f_1,f_2)}{I(x)}.$$
{{endthm}}
{{endthm}}
This this clasical fact is either a theorem or a definition depending on which definition of homology intersection pairing one accepts.
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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 $f_2$.
For a proof see \cite{Scorpan2005|Section 3.2} or \cite{Ranicki2002|Proposition 7.22}.
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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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<!--
<!--
$\lambda^{\mathrm{geo}}(N_1,N_2)$ defined below.
$\lambda^{\mathrm{geo}}(N_1,N_2)$ defined below.

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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