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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.

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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?

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

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