Let $B^3$ denote the closed unit ball in $\Bbb R^3$ and $S^2$ be the unit sphere. Does there exist a retraction $r$ from $B^3$ onto $S^2$?
I cannot argue it using fundamental group since both have the trivial fundamental group. Please help me in this regard.
Thank you very much.
A standard way of doing this is via homology. Namely, if there exists a retract $r: B^3 \to S^2$, then $r \circ i = \text{Id}$ (with $i: S^2 \to B^3$ inclusion). In particular, one looks at: $$\mathbb{Z} = H_2(S^2) \xrightarrow{i_*} H_2(B^2) \xrightarrow{r_*} H_2(S^2) = \mathbb{Z}$$ and notes that the composition should be the identity map, but $H_2(B^2) = 0$, a contradiction. This generalizes for any $n$-ball and its boundary sphere.