Determine the homomorphism induced on the fundamental group

Question

Consider the unit disk $D^2$ and consider the application f: $D^2 \to D^2$, f(x,y)=(-x,-y). Prove that by identifying the border of $D^2$ in the usual way f induces a continuous map $\bar{f}: \mathbf{RP}^2 \to \mathbf{RP}^2$, where $\mathbf{RP}^2$ indicates real projective plane. Determine the homomorphism induced on the fundamental group $\bar{f}_\ast:\pi_1(\mathbf{RP}^2,x_0) \to \pi_1(\mathbf{RP}^2,x_0) $. I showed the first part, but I don't know how to determine the homomorphism induced on the fundamental group.

Answer 1

Let me give you three hints.

First show that $\bar f$ is actually a homeomorphism. It follows that the induced homomorphism $\bar f_*$ is an automorphism of the group $\pi_1(\mathbf{RP}^2,x_0)$.

Second, do you know what familiar group the fundamental group $\pi_1(\mathbf{RP}^2,x_0)$ is isomorphic to?

Third, can you figure out how many automorphisms that group has?