Intersection number of immersions

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Contents

1 Introduction

This page is based on [Ranicki2002], see also [Broecker&Jaenich1982, Excercise 14.9.6]. 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 of a manifold (and so to the signature of a 4k-dimensional manifold) and to characteristic classes. These are important invariants used in the classification of manifolds.

2 Statement

Let M be a connected oriented manifold of dimension m=n_1+n_2 and

\displaystyle I_M=\lambda_M=\lambda: H_{n_1}(M)\times H_{n_2}(M) \to \Z

the homology intersection pairing (or product) of M.

Theorem 2.1. For any immersions f_1:N_1\looparrowright M, f_2:N_2\looparrowright M of oriented n_1- and n_2-manifolds the number \lambda(f_{1*}[N_1],f_{2*}[N_2]) equals to \lambda^{\mathrm{geo}}(N_1,N_2) defined below.

This 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].

The number \lambda(f_{1*}[N_1],f_{2*}[N_2]) can be called algebraic intersection number of f_1 and f_2.


3 Alternative description

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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.

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
\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))
is an isomorphism.

Immersions f_i:N_i\looparrowright M (i=1,2) have transverse intersection (or are transverse) if each double point is transverse and S_2(f_1,f_2) is finite.

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.

The geometric intersection number of transverse immersions f_i:N_i\looparrowright M (i=1,2) is defined as

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


4 References

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