# Intersection number of immersions

## 1 Introduction


## 2 Statement

Let

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

be the homology intersection pairing (or product) of $M$$M$.

The double point set of $f_1$$f_1$ and $f_2$$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\}$$\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)$$x=(x_1,x_2)\in S_2(f_1,f_2)$ of $f_1$$f_1$ and $f_2$$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$$f_1$ and $f_2$$f_2$ are transverse (or have transverse intersection) if $S_2(f_1,f_2)$$S_2(f_1,f_2)$ is finite and every double point is transverse.

The index, or the sign $I(x)\in\Z$$I(x)\in\Z$ of a transverse double point $x=(x_1,x_2)\in S_2(f_1,f_2)$$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$$f_1$ and $f_2$$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$$f_1$ and $f_2$$f_2$.