Different ways to define intersection number?

06-05

Recently Im reading Bott-Tus "differential forms in algebraic topology", and comparing it with some differential topology textbook.
While proving the Poincare-Hopf theorem, they define 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 $I(M,N)=int_K eta_M wedgeeta_N.$ Here $eta_M$ is the Poincare dual of M, so $eta_M wedgeeta_N$ is a top form in K, assuming the dimension condition.
My question is why this definition coincides with the definition appeared in differential topology, which using the orientable intersection number that "sum up" $pm 1$ locally according to the orientation.

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

Let $M$ be a closed manifold and $X, Y$ be closed submanifolds that intersect transversally. Denote $eta_X = PD[X], eta_Y = PD[Y]$ . Then the intersection number is

$X cdot Y = int_M eta_X wedge eta_Y = int_X eta_Y|_X$ .

Note that we can choose $eta_Y$ such that it is supported on a small tubular neighbourhood $U_Y$ of $Y$ . Identify $U_Y$ with $u_{< lambda} Y$ , where $u_{< lambda}Y$ is the normal bundle of $Y$ with the norm of normal vectors restricted less than $lambda$ . Then $eta_Y$ can be viewed as a form on $u_{< lambda}Y$ the integral of $eta_Y$ on each fiber is $1$ .

In addition, we are able to fix a Riemannian structure on $M$ such that $forall, p in X cap Y$ , $T_pX , ot , T_pY$ and $X$ is totally geodesic on a neighbourhood of $p$ (This claim is left for the reader as an exercise, though it is not quite trivial). Under this circumstance, we only need to calculate the integral

$int_X eta_Y = sum_{p in X cap Y} , int_{X cap B(p, lambda)} eta_Y = sum_{p in X cap Y} ,int_{ ( u_{< lambda}Y)_p} pm eta_Y.$

Thus we only need to check the sign of each integral, since they are all $pm 1$ .

Let $omega_X$ be the volume form on $X$ and $omega_Y$ be the volume form on $Y$ . Then $eta_Y wedge omega_Y$ is a positive orientation on $T_YM = u Y oplus TY$ . Therefore if $omega_X wedge omega_Y$ is a positive (negative) orientation on $T_pM = T_pX oplus T_pY$ , then the orientation on $u_pY$ and $T_pX$ should be the same (opposite). We are through.

Since the Poincare dual of a k-submanifold $X$ in a larger n-manifold $M$ , is cohomology to $j_*(Phi)$ , the pulling back of the Thom class of the normal bundle $N_X$ , we could simply use it to replace Poincare dual in our integral. (This is implied in the first answer.)

Thom class is such a "natrual" class, that if two vector bundle $E_i$ over the same manifold $M$ , with $pi_i:E_1 oplus E_2 o E_i$ , then the Thom class of $E_1 oplus E_2$ is $Phi(E_1 oplus E_2) = pi_1^* Phi(E_1) wedgepi_2^* Phi(E_2)$ . And we have $X$ and $Y$ intersect transversally, so $N_{X cap Y} = N_X oplus N_Y$ . Therefore, $eta_{(X cap Y)} = eta_X wedge eta_Y$ .

Now coming to our calculation, that is

$int_M eta_X wedge eta_Y =int_{p in X cap Y} eta_{X cap Y}=sum_{p in X cap Y} pm 1.$

The value is determined by the orientation of $p in X cap Y$ , which is induced by $N_pX oplus N_pY = T_pY oplus T_pX$ , which is exactly the local intersection number of these two submanifold.