Intersection number of immersions

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	== Introduction ==		== Introduction ==
	<wikitex>;		<wikitex>;
−	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]].	+	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 [[Signature|signature]] of a $4k$-dimensional manifold) and to [[Stiefel-Whitney_characteristic_classes|characteristic classes]].	+
−	These are important invariants used in the classification of [[Manifold|manifolds]].	+	This page is based on \cite{Ranicki2002}, see also \cite[Excercise 14.9.6]{Broecker&Jaenich1982}.
	</wikitex>		</wikitex>
−	== Definition ==	+	== Statement ==
	<wikitex>;		<wikitex>;
−	Let $M$ be an oriented $m$-dimensional manifold. The '''homology intersection pairing''' of $M$, $$\lambda: H_n(M)\times H_{m-n}(M) \to \Z; \quad (x,y) \mapsto \lambda(x,y),$$	+	Let
−	is defined analogously to the case $m=2n$, when it is called the [[Intersection_form|intersection form]].	+	$$I_M=\lambda_M=\lambda: H_{n_1}(M)\times H_{n_2}(M) \to \Z$$
−	Besides generalizing that simple direct definition, we can use the concept of the ''cup product'' and define the pairing by	+	be the [[Intersection_form#Definition|''homology intersection pairing (or product)'']] of $M$.
−	$$\lambda(x,y) = \langle x^*\cup y^*,[M]\rangle \in \Z,$$	+
−	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]].	+
−	The homology intersection pairing is bilinear and satisfies	+	The ''double point set'' of $f_1$ and $f_2$ is defined by
−	$$\lambda(y,x) = (-1)^{n(m-n)}\lambda(x,y)$$	+	$$S_2(f_1,f_2) = \{(x_1,x_2)\in N_1\times N_2 | f_1(x_1) = f_2(x_2)\in M\} =
−	for all $x\in H_n(M)$, $y\in H_{m-n}(M)$.	+	(f_1\times f_2)^{-1}(\Delta(M)),$$
−	These properties are easy to check using the simple direct definition; they also follow from simple properties of the cup product.	+	where $\Delta(M) = \{(x,x)\in M\times M | x\in M\}$ the diagonal.
−	The '''algebraic intersection number''' $\lambda^{\mathrm{alg}}(f_1,f_2)\in\Z$ 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, is the homology intersection of the homology classes $(f_1)_*[N_1]\in H_{n_1}(M)$, $(f_2)_*[N_2]\in H_{n_2}(M)$:	+	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
−	$$\lambda^{\mathrm{alg}}(N_1,N_2) := \lambda((f_1)_*[N_1],(f_2)_*[N_2]).$$	+	$$df(x) = df_1(x_1)\oplus df_2(x_2): \tau_{N_1}(x_1)\oplus \tau_{N_2}(x_2) \to \tau_M(f(x))$$
−	</wikitex>	+	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.
−	==Alternative description==	+	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
−	<wikitex>;	+	$$I(x) = \left\{ \begin{array}{cc} +1, & \mathrm{if}\; df(x)\; \mathrm{preserves}\; \mathrm{orientations}\\ -1, & \mathrm{otherwise}.\end{array}\right.$$
−	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.	+	{{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}}
−	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.	+	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 $f_2$.
−		+	<!--
−	The '''index''' $I(x)\in\Z$ of a transverse double point $x=(x_1,x_2)\in S_2(f_1,f_2)$ is $$I(x) = \left\{ \begin{array}{cc} +1, & \mathrm{if}\; df(x)\; \mathrm{preserves}\; \mathrm{orientations}\\ -1, & \mathrm{otherwise}.\end{array}\right.$$	+	$\lambda^{\mathrm{geo}}(N_1,N_2)$ 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 '''geometric intersection number''' of transverse immersions $f_i:N_i^{n_i} \looparrowright M^{n_1+n_2}$ $(i=1,2)$ is	+	the homology intersection of the homology classes $(f_1)_*[N_1]\in H_{n_1}(M)$, $(f_2)_*[N_2]\in H_{n_2}(M)$:
		+	==Alternative description==
		+	The ''geometric'' intersection number of transverse immersions $f_i:N_i\looparrowright M$ $(i=1,2)$ is defined as
	$$\lambda^{\mathrm{geo}}(N_1,N_2) = \sum_{x\in S_2(f_1,f_2)}{I(x)}\in \Z.$$		$$\lambda^{\mathrm{geo}}(N_1,N_2) = \sum_{x\in S_2(f_1,f_2)}{I(x)}\in \Z.$$
−	</wikitex>
−
−	== Equivalence of definitions ==
−	<wikitex>;
	The algebraic and geometric intersection numbers agree,		The algebraic and geometric intersection numbers agree,
−	$$\lambda^{\mathrm{alg}}(N_1,N_2)=\lambda^{\mathrm{geo}}(N_1,N_2).$$	+	$$\lambda^{\mathrm{alg}}(N_1,N_2)=\lambda^{\mathrm{geo}}(N_1,N_2).$$ -->
−	For a proof of this clasical fact see \cite{Scorpan2005|Section 3.2} or \cite{Ranicki2002|Proposition 7.22}.	+
−	</wikitex>	+
−		+
	==References==		==References==
	{{#RefList:}}		{{#RefList:}}

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Latest revision as of 15:17, 2 April 2019

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

[edit] 1 Introduction

Let

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${{Stub}} == Introduction == <wikitex>; 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 manifold (and so to the [[Signature|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> == Definition == <wikitex>; Let $M$ be an oriented $m$-dimensional manifold. The '''homology intersection pairing''' of $M$, $$\lambda: H_n(M)\times H_{m-n}(M) \to \Z; \quad (x,y) \mapsto \lambda(x,y),$$ is defined analogously to the case $m=2n$, when it is called the [[Intersection_form|intersection form]]. Besides generalizing that simple direct definition, we can use the concept of the ''cup product'' and define the pairing 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]]. The homology intersection pairing is bilinear and satisfies $$\lambda(y,x) = (-1)^{n(m-n)}\lambda(x,y)$$ for all $x\in H_n(M)$, $y\in H_{m-n}(M)$. These properties are easy to check using the simple direct definition; they also follow from simple properties of the cup product. The '''algebraic intersection number''' $\lambda^{\mathrm{alg}}(f_1,f_2)\in\Z$ 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, is the homology intersection of the homology classes $(f_1)_*[N_1]\in H_{n_1}(M)$, $(f_2)_*[N_2]\in H_{n_2}(M)$: $$\lambda^{\mathrm{alg}}(N_1,N_2) := \lambda((f_1)_*[N_1],(f_2)_*[N_2]).$$ </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 $f_i:N_i^{n_i} \looparrowright M^{n_1+n_2}$ $(i=1,2)$ have '''transverse intersections''' (or are '''transverse''') if each double point is transverse and $S_2(f_1,f_2)$ is finite. The '''index''' $I(x)\in\Z$ of a transverse double point $x=(x_1,x_2)\in S_2(f_1,f_2)$ is $$I(x) = \left\{ \begin{array}{cc} +1, & \mathrm{if}\; df(x)\; \mathrm{preserves}\; \mathrm{orientations}\ -1, & \mathrm{otherwise}.\end{array}\right.$$ The '''geometric intersection number''' of transverse immersions $f_i:N_i^{n_i} \looparrowright M^{n_1+n_2}$ $(i=1,2)$ is $$\lambda^{\mathrm{geo}}(N_1,N_2) = \sum_{x\in S_2(f_1,f_2)}{I(x)}\in \Z.$$ </wikitex> == Equivalence of definitions == <wikitex>; 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 clasical fact see \cite{Scorpan2005|Section 3.2} or \cite{Ranicki2002|Proposition 7.22}. </wikitex> ==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

be the homology intersection pairing (or product) of

Tex syntax error

$M$.

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

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

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

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