I am working on the following problem in p.83 of Guillemin-Pollack:
Prove that intersection theory is vacuous in contractible manifolds: if $Y$ is contractible and $\dim Y > 0$, then $I_2 (f, Z) = 0$ for every $f: X \to Y$, $X$ compact and $Z$ closed, $\dim X + \dim Z = \dim Y$.
Here $I_2 (f, Z)$ denotes the $\mod 2$ intersection number, i.e. the size of the finite set (a $0$-dimensional compact manifold) $f^{-1}(Z)$ modulo $2$, assuming that $f$ is transverse to $Z$.
I am facing the contradiction below when $\dim X = 0$, but I don't see where my argument goes wrong.
Suppose $X$ is a single point, and $Z = Y$. Then any map $f: X \to Y$ is transverse to $Z$ and $f^{-1}(Z) = X$ has a single point, hence $I_2 (f, Z) = 1$.
On the other hand, we can interpret $I_2 (f, Z) = I_2 (f \times \mathrm{id}_Y, \Delta)$ where $\Delta \subset Y \times Y$ is the diagonal submanifold. Since $Y$ is contractible we have $\mathrm{id}_Y \simeq *$ (a constant map) and hence $I_2 (f \times \mathrm{id}_Y , \Delta ) = I_2 (f \times * , \Delta)$ by homotopy invariance of intersection numbers. This value is zero if we choose the image of the constant map $*$ to be disjoint from $f(X)$, which is always possible since $\dim Y > 0$ and $\dim X = 0$. So $I_2 (f, Z) = 0$ in this case.
Obviously something is wrong here. If anyone could identify the problem here it would be very helpful.