Intersection number of immersions

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	== Definition ==		== Definition ==
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−	Let $M$ be an oriented $m$-dimensional manifold. The '''homology intersection pairing''' of $M$ $$\lambda: H_n(M)\times H_{m-n}(M) \to \Z; (x,y) \mapsto \lambda(x,y)$$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]].	+	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 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]].
	As a consequence of the properties of the cup product the homology intersection pairing is bilinear and satisfies		As a consequence of the properties of the cup product the homology intersection pairing is bilinear and satisfies
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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 $(f_1)_*[N_1]\in H_{n_1}(M)$, $(f_2)_*[N_2]\in H_{n_2}(M)$: $$\lambda^{\mathrm{alg}}(N_1,N_2) := \lambda((f_1)_*[N_1],(f_2)_*[N_2]).$$		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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	==Alternative description==		==Alternative description==
	<wikitex>;		<wikitex>;

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Revision as of 17:43, 16 June 2014

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Contents 1 Introduction 2 Definition 3 Alternative description 4 Equivalence of definitions 5 References

1 Introduction

This page is based on [Ranicki2002]. Let ${{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 k$-dimensional manifold and in turn its [[Signature|signature]]: 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 $M$ $$\lambda: H_n(M)\times H_{m-n}(M) \to \Z; (x,y) \mapsto \lambda(x,y)$$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]]. As a consequence of the properties of the cup product the homology intersection pairing is bilinear and satisfies $$\lambda(y,x) = (-1)^{n(m-n)}\lambda(x,y)$$ for all $x\in H_n(M)$, $y\in H_{m-n}(M)$. 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 $(f_1)_*[N_1]\in H_{n_1}(M)$, $(f_2)_*[N_2]\in H_{n_2}(M)$: $$\lambda^{\mathrm{alg}}(N_1,N_2) := \lambda((f_1)_*[N_1],(f_2)_*[N_2]).$$ </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 see \cite{Scorpan2005|Section 3.2} or \cite{Ranicki2002|Proposition 7.22}. </wikitex> ==References== {{#RefList:}} [[Category:Definitions]]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 $4k$-dimensional manifold and in turn its signature: important invariants used in the classification of manifolds.

2 Definition

Let $M$ be an oriented $m$-dimensional manifold. The homology intersection pairing of $M$, is defined by 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.

As a consequence of the properties of the cup product the homology intersection pairing is bilinear and satisfies

for all $x\in H_n(M)$, $y\in H_{m-n}(M)$.

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 $(f_1)_*[N_1]\in H_{n_1}(M)$, $(f_2)_*[N_2]\in H_{n_2}(M)$:

3 Alternative description

The double point set of maps $f_i:N_i\to M$ $(i=1,2)$ is defined by

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 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

The geometric intersection number of transverse immersions $f_i:N_i^{n_i} \looparrowright M^{n_1+n_2}$ $(i=1,2)$ is

4 Equivalence of definitions

The algebraic and geometric intersection numbers agree, For a proof of this see [Scorpan2005, Section 3.2] or [Ranicki2002, Proposition 7.22].

5 References

• [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