Intersection number of immersions

Revision as of 12:21, 29 March 2019

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 $M$, $$\lambda: H_n(M)\times H_{m-n}(M) \to \Z; \quad (x,y) \mapsto \lambda(x,y),$$ is defined analogously to the case $m=2n$ \cite{Intersection}. Besides generalizing the 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''' 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

$$\displaystyle \lambda: H_n(M)\times H_{m-n}(M) \to \Z; \quad (x,y) \mapsto \lambda(x,y),$$

is defined analogously to the case $m=2n$, when it is called the intersection_form. Besides generalizing that simple direct definition, we can use the concept of the cup product and define the pairing by

$$\displaystyle \lambda(x,y) = \langle x^*\cup y^*,[M]\rangle \in \Z,$$

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.

The homology intersection pairing is bilinear and satisfies

$$\displaystyle \lambda(y,x) = (-1)^{n(m-n)}\lambda(x,y)$$

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

$$\displaystyle \lambda^{\mathrm{alg}}(N_1,N_2) := \lambda((f_1)_*[N_1],(f_2)_*[N_2]).$$

3 Alternative description

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

$$\displaystyle S_2(f_1,f_2) = \{(x_1,x_2)\in N_1\times N_2 | f_1(x_1) = f_2(x_2)\in M\} = (f_1\times f_2)^{-1}(\Delta(M))$$

with $\Delta(M) = \{(x,x) | x\in M\}\subset M\times M$ the diagonal subspace.

$$\displaystyle df(x) = (df_1(x_1),df_2(x_2)): \tau_{N_1}(x_1)\oplus \tau_{N_2}(x_2) \to \tau_M(f(x))$$

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.

$$\displaystyle I(x) = \left\{ \begin{array}{cc} +1, & \mathrm{if}\; df(x)\; \mathrm{preserves}\; \mathrm{orientations}\\ -1, & \mathrm{otherwise}.\end{array}\right.$$

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

$$\displaystyle \lambda^{\mathrm{geo}}(N_1,N_2) = \sum_{x\in S_2(f_1,f_2)}{I(x)}\in \Z.$$

4 Equivalence of definitions

The algebraic and geometric intersection numbers agree,

$$\displaystyle \lambda^{\mathrm{alg}}(N_1,N_2)=\lambda^{\mathrm{geo}}(N_1,N_2).$$

For a proof of this clasical fact 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