# Geometric and topological ways to define intersection number

Resently I’m reading Bott-Tu’s differential forms in algebraic topology, and comparing it with some differential topology textbook.

While proving the Poincare-Hopf theorem, it defines the intersection number $I(M,N)$, where M, N are submanifold of another smooth manifold K with condition $\dim M+\dim N=\dim K$, to be $I(M,N)=\int_K \eta_M \wedge\eta_N.$ Here $\eta_M$ is the Poincare dual of M, so $\eta_M \wedge\eta_N$ is a top form in K, assuming the dimension condition.

My question is why this definition is coincide with the definition appeared in differential topology, which using the orientable intersection number that “sum up” $\pm 1$ locally according to the orientation.

If it’s possible, I perfer an answer without using much algebraic geometry. Any links to webpage or other reference are welcomed as well. Great Thanks!

Imagine nonnegative function $f:\mathbb{R}\rightarrow \mathbb{R}$ that is supported in the interval $[-\epsilon,\epsilon]$ with If $\gamma:[0,1]\rightarrow \mathbb{R}$ is any path with $\gamma(0)$ and $\gamma(1)$ outside of $[-\epsilon,\epsilon]$ then $\int_{[0,1]}\gamma^*(fdx)$ is equal to the algebraic intersection number relative to its boundary, of $\gamma$ with $0$.

We can play the same game in $\mathbb{R}^n$, defining an $n$-form $\omega$ supported in an $\epsilon$-ball of $\vec{0}$, so that
Once again if $\gamma:B\rightarrow \mathbb{R}^n$ is a smooth map of the ball with $\gamma(\partial B)$ outside the $\epsilon$-ball where $\omega$ is supported, then

will be the intersection number of $B$ relative to its boundary, or alternately the winding number of $\partial B$ about $\vec{0}$.

The reason for this is the full change of variables formula for integrating $n$-forms of compact support on an oriented $n$-manifold. If $f:X\rightarrow Y$ is a smooth, proper map of oriented $n$-manifolds, with $Y$ connected, define the degree of $f$ as follows. Let $y\in Y$ be a regular value of $f$. Since $f$ is proper $f^{-1}(y)$ is compact. Since it is a zero manifold, it is a finite set of points. At each $x\in f^{-1}(y)$ define the sign of $x$ to be $\pm 1$ depending on whether $df_x:T_xX\rightarrow T_yY$ is orientation preserving or orientation reversing. The degree of $f$, denoted $deg(f)$ is
Standard elementary arguments show that $deg(f)$ is independent of the point $y$, and unchanged by homotopies of $f$ through smooth proper maps.

Change of variables says that if $\omega$ is an $n$-form with compact support on $Y$, then $f^*\omega$ will have compact support. Both integrals are well defined and

Here is a slightly flawed but intuitive proof. The set of singular points of $f$ has measure zero by Sard’s theorem. Remove the singular points of $f$, $S(f)\subset Y$ from $Y$ and $f^{-1}(S(f))$ from $X$. You now have a map
that is a local diffeomorphism.
Since you removed a set of measure zero,

Using the stack of records theorem,

Since the pull back of an $n$-form at points where $df$ is singular is zero, we have
Completing the proof.

If $M\subset N$ is a compact regular submanifold, you can cover $M$ with coordinate patches $(U_{\alpha},\phi_{\alpha})$ of $N$ where $M\cap U_{\alpha}$ is a slice for all $\alpha$. That is in local coordinates $\phi_{\alpha}$, $M$ is the set of points with $x_{m+1}=x_{m+2}\ldots=x_n=0$. We call $(x_{m+1},\ldots,x_n)$ the complementary coordinates to $M$.

Complete this to a cover of $N$ by adding $N-M$.
Choose a partition of unity subordinate to the open cover of $N$ so that $\rho_{\alpha}:M\rightarrow \mathbb{R}_{\geq 0}$ has support inside $U_{\alpha}$. Choose $\epsilon>0$ so that inside the support of each $\rho_{\alpha}$ if the complementary coordinates $M$ have norm less than $\epsilon$ are contained in $U_{\alpha}$. Let $\omega_{\alpha}$ be an $n-m$ form with support in the points in $U_{\alpha}$ whose complementary coordinates have norm less than $\epsilon$ and $\int_{\mathbb{R}^{n-m}}\omega_{\alpha}=1$. Finally let
The form is smooth.
With a little more care you can make sure its closed.

The point is if $K$ is a regular manifold that is transverse to $M$ of dimension $n-m$, then
the pull back under inclusion $i:K\rightarrow N$,
has its support near the points where $K$ intersects $M$, and if $\epsilon$ is small enough, the integral near each point is $\pm 1$ depending on the sign of the intersection of $K$ and $M$ at that point.

That is it.