$S^n$ is not a strong deformation retract of $B^{n+1}$

Show that $S^n$ is not a strong deformation retract of $\bB^{n+1}$.

Solution

If $\iota S^n \injects \bB^{n+1}$ then a strong deformation retract yields a $p: \bB^{n+1} \to S^n$ with $p\circ \iota = \id$. Then apply homology to factor the identity map $\ZZ \to \ZZ$ through a constant zero map $\ZZ\to 0$.