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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This page is based on \cite{Ranicki2002}, see also \cite[Excercise 14.9.6]{Broecker&Jaenich1982}.
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</wikitex>
== Definition ==
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== Statement ==
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<wikitex>;
Let $M$ be an oriented $m$-dimensional manifold. The '''homology intersection pairing''' of $M$ $$\lambda: H_n(M)\times H_{m-n}(M) \to \Z; (x,y) \mapsto \lambda(x,y)$$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 duals of $x$, $y$ and $[M]$ is the [[fundamental class]].
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Let
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$$I_M=\lambda_M=\lambda: H_{n_1}(M)\times H_{n_2}(M) \to \Z$$
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be the [[Intersection_form#Definition|''homology intersection pairing (or product)'']] of $M$.
As a consequence of the properties of the cup product the homology intersection pairing is bilinear and satisfies
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The ''double point set'' of $f_1$ and $f_2$ is defined by
$$\lambda(y,x) = (-1)^{n(m-n)}\lambda(x,y)$$
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$$S_2(f_1,f_2) = \{(x_1,x_2)\in N_1\times N_2 | f_1(x_1) = f_2(x_2)\in M\} =
for all $x\in H_n(M)$, $y\in H_{m-n}(M)$.
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(f_1\times f_2)^{-1}(\Delta(M)),$$
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where $\Delta(M) = \{(x,x)\in M\times M | x\in M\}$ the diagonal.
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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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
</wikitex>
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$$df(x) = df_1(x_1)\oplus df_2(x_2): \tau_{N_1}(x_1)\oplus \tau_{N_2}(x_2) \to \tau_M(f(x))$$
==Alternative description==
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is an isomorphism.
<wikitex>;
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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 '''double point set''' of maps $f_i:N_i\to M$ $(i=1,2)$ is defined by
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$$S_2(f_1,f_2) = \{(x_1,x_2)\in N_1\times N_2 | f_1(x_1) = f_2(x_2)\in M\} = (f_1\times f_2)^{-1}(\Delta(M))$$
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with $\Delta(M) = \{(x,x) | x\in M\}\subset M\times M$ the diagonal subspace.
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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 ''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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$$I(x) = \left\{ \begin{array}{cc} +1, & \mathrm{if}\; df(x)\; \mathrm{preserves}\; \mathrm{orientations}\\ -1, & \mathrm{otherwise}.\end{array}\right.$$
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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{{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}}
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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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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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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$\lambda^{\mathrm{geo}}(N_1,N_2)$ 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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