Let us consider the closed disk $\overline{\mathbb{B}(0,1)} \subsetneqq \mathbb{R}^2$. Let moreover $\mathbb{RP}^2:=\overline{\mathbb{B}(0,1)}/\sim$, where the equivalence relation $\sim$ identifies two antipodal points. Given $f:(x,y) \in \overline{\mathbb{B}(0,1)} \to (-x,-y) \in \overline{\mathbb{B}(0,1)}$, it induces a continuous map $\tilde{f}: \mathbb{RP}^2 \to \mathbb{RP}^2$. In fact, denoted by $\pi$ the classical projection onto the quotient, we may define $\tilde{f}([(x,y)])=[f(x,y)]$. As $\pi \circ \tilde{f}=f \circ \pi$, $\tilde{f}$ has to be continuous. At this point, I consider the homomorphism induced on the fundamental group of $\mathbb{RP}^2$, i.e. $\tilde{f}_*: \pi_1(\mathbb{RP}^2)=\mathbb{Z}/2\mathbb{Z} \to \pi_1(\mathbb{RP}^2)=\mathbb{Z}/2\mathbb{Z}$? What is such a homomorphism?
Induced homomorphism between fundamental groups
algebraic-topologygeneral-topologyprojective-space
Best Answer
The map $f$ is a homeomorphism. The maps $\tilde{f}$ and $\tilde{f}_*$ are then induced by functoriality, hence $\tilde{f}_*$ is an isomorphism. The only self isomorphism of $\mathbb{Z} / \mathbb{2Z}$ is the identity, so $\tilde{f}_*$ has to be the identity.
Edit: I fill in some details as requested.
The map $\tilde{f}$ is induced by the universal property of the quotient applied to $f$ followed by the quotient map $\pi$. It is an isomorphism because the quotient is unique and $\pi\circ f=\pi$ ($\tilde{f}$ is in fact the identity).
The map $\tilde{f}_*$ is then induced by the functor $\pi_1$. This being a functor means that it preserves isomorphisms (because it preserves composition and identities). Hence $\tilde{f}_*$ is a self isomorphism of $\mathbb{Z}/2\mathbb{Z}$, which must be the identity (and in fact if you believed before that $\tilde{f}$ is the identity then you don't need to argue that much, the functor just sends identity to identity).