It's problem 2.4.7 of Guillemin's Differential Topology. The problem asked me to prove $S^1$ is not simply connected, and it has a hint: consider the identity map. But I don't know the correlation between simply connected and intersection mod 2, which is the topic of this section.
Anyone knows how to approach? Thanks!
Since the circle is connected, all the constant maps $S^1 \to S^1$ are homotopic. Also, their graphs are clearly transverse to the diagonal, intersecting it in exactly one place (picture a horizontal line crossing a diagonal line).
On the other hand, the graph of the identity map $\varphi : S^1 \to S^1$ is not transverse to the diagonal. In fact, the graph equals the diagonal! However, it is not difficult to see how to homotope the map to make it transverse. Let $\varphi_t(x)=x+t$ (viewing the circle as $\mathbb{R}/\mathbb{Z}$). For any small positive $t$, the graph of $\varphi_t$ is even disjoint to the diagonal and, in particular, transverse to it. So, what is the mod 2 intersection number of the identity?