# Determine the homomorphism induced on the fundamental group

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.

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?

• I don't understand the second hint. can you explain better to me? – Giusy Apr 12 '19 at 16:48
• what means to determine the homomorphism induced on the fundamental group? – Giusy Apr 12 '19 at 16:52
• To determine a function means to give a formula for a function. – Lee Mosher Apr 12 '19 at 17:25
• Can you do an example of how to determine this homomorphism induced? – Giusy Apr 12 '19 at 18:20
• I think it is necessary to do the "Second" step first. – Lee Mosher Apr 12 '19 at 18:26