Given two continuous surjective functions $f$ and $g$ from the unit disk to itself and $f(z) \neq g(z)$ for all $z$ in the unit disk is it possible to construct a retraction map from the unit disk to its boundary?
[Math] Retraction map from unit disk to its boundary
algebraic-topologygeneral-topology
Best Answer
There is no retract from $D^2$ to $S^1$. If there were such a retract $r : D^2 \to S^1$ then we would have an injection
$$i_\ast : H_1(S^1) \to H_1(D^2)$$
where $i_\ast$ is the map induced from inclusion $i : S^1 \to D^2$. This is because $r \circ i =1$ and hence on homology $r_\ast \circ i_\ast = 1$, viz. $i_\ast$ has a left inverse. But this is impossible because $H_1(D^2) = \tilde{H}_1(D^2)= 0$ while $H_1(S^1) = \tilde{H}_1(S^1) = \Bbb{Z}$.