I want to prove that Euler characteristic of the product of two compact oriented manifolds is the product of their Euler characteristics.
As always I do, I'm considering Guillemin-Pollack definitions, i.e., the Euler characteristic of M, compact and oriented, $\chi(M) = I(\Delta,\Delta)$ where $\Delta$ is the diagonal of $M\times M$ and $I(\Delta,\Delta) = I(i,\Delta) =$ sum of orientation numbers of each $p\in i^{-1}(\Delta)$ using pre image orientation. Here $i:\Delta \to M \times M$ is the inclusion.
Help!
Best Answer
With the notation $\Delta_{xy}:=\{(z,z)~|~z\in X\times Y\}$, we have to show that \begin{equation*} I(\Delta_{xy},\Delta_{xy}) =\chi(X\times Y)=\chi(X)\cdot\chi(Y) = I(\Delta_x,\Delta_x) \cdot I(\Delta_y,\Delta_y). \end{equation*} Now $\Delta_{xy}$ is also equal to $\{(x,y,x,y)~|~x\in X,~y\in Y\}$.
As a result, if we fix one $y_i$ for which $\Delta_y$ intersects with itself, then $I(\Delta_{xy_i},\Delta_{xy_i})=I(\Delta_x,\Delta_x)$. As we can repeat this for all $y_i$ for which $\Delta_y$ intersects with itself, there are then $I(\Delta_x,\Delta_x) \cdot I(\Delta_y,\Delta_y)$ points of intersection in $I(\Delta_{xy},\Delta_{xy})$.
The orientation also agrees because a point of the intersection is included with a plus sign if the orientation at the intersection adds up to the orientation of the ambient space (as explained on p. 112 of Guillemin and Pollack's book) which is the product space and thus carries the product orientation as explained on p. 97 of the (same) book.