Intersection number of immersions

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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


\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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