# Intersection number of immersions

## 1 Introduction

This page is based on [Ranicki2002]. Let $f_1:N_1^{n_1} \looparrowright M^{n_1+n_2}$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}f_1:N_1^{n_1} \looparrowright M^{n_1+n_2}$, $f_2:N_2^{n_2} \looparrowright M^{n_1+n_2}$$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$$\lambda([N_1],[N_2])\in\Z$ has both an algebraic and geometric formulation; roughly speaking it counts with sign the number of intersection points that the two immersions have. The intersection number is an obstruction to perturbing the immersions into being disjoint. When it vanishes this perturbation can often be achieved using the Whitney trick. The intersection number of immersions is closely related to the intersection form of a $4k$$4k$-dimensional manifold and in turn its signature: important invariants used in the classification of manifolds.

## 2 Definition

Let $M$$M$ be an oriented $m$$m$-dimensional manifold. The homology intersection pairing of $M$$M$,
$\displaystyle \lambda: H_n(M)\times H_{m-n}(M) \to \Z; \quad (x,y) \mapsto \lambda(x,y),$
is defined by
$\displaystyle \lambda(x,y) = \langle x^*\cup y^*,[M]\rangle \in \Z$
where $x^*\in H^{m-n}(M)$$x^*\in H^{m-n}(M)$, $y^*\in H^n(M)$$y^*\in H^n(M)$ are the PoincarĂ© duals of $x$$x$, $y$$y$ and $[M]$$[M]$ is the fundamental class.

As a consequence of the properties of the cup product 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)$$x\in H_n(M)$, $y\in H_{m-n}(M)$$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}$$f_1:N_1^{n_1} \looparrowright M^{n_1+n_2}$, $f_2:N_2^{n_2} \looparrowright M^{n_1+n_2}$$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$$\lambda^{\mathrm{alg}}(N_1,N_2)\in\Z$, is the homology intersection of the homology classes $(f_1)_*[N_1]\in H_{n_1}(M)$$(f_1)_*[N_1]\in H_{n_1}(M)$, $(f_2)_*[N_2]\in H_{n_2}(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$$f_i:N_i\to M$ $(i=1,2)$$(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$$\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)$$x=(x_1,x_2)\in S_2(f_1,f_2)$ of immersions $f_i:N_i^{n_i} \looparrowright M^{n_1+n_2}$$f_i:N_i^{n_i} \looparrowright M^{n_1+n_2}$ $(i=1,2)$$(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^{n_i} \looparrowright M^{n_1+n_2}$$f_i:N_i^{n_i} \looparrowright M^{n_1+n_2}$ $(i=1,2)$$(i=1,2)$ have transverse intersections (or are transverse) if each double point is transverse and $S_2(f_1,f_2)$$S_2(f_1,f_2)$ is finite.

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

The geometric intersection number of transverse immersions $f_i:N_i^{n_i} \looparrowright M^{n_1+n_2}$$f_i:N_i^{n_i} \looparrowright M^{n_1+n_2}$ $(i=1,2)$$(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 see [Scorpan2005, Section 3.2] or [Ranicki2002, Proposition 7.22].