In the proof of no retraction theorem for $C^{2}$ function I have the necessity to justify why can't exist a homeomorphism between open subsets of $\mathbb{S}^{n}$ and $B(0,1,\mathbb{R}^{n})$. This doubt arises from the structure of the proof directly : we suppose that such restraction exists and create and homotopy $g_{t} : B \longmapsto B$ between the identity, time $t=0$ and the retraction, time $t=1$. We now consider the function $t \in [0,1] \longmapsto \phi(t) = \int_{B}\mbox{det} Dg_{t}(x) dx$, where $Dg_{t}$ denotes the differential of $g_{t}$ in the point $x \in B$. We then notice that $\phi(t)' = 0$ which implies that $\phi(1) = \phi(0)$. But $\phi(0) = \int_{B} \mbox{det}DId = \lvert B \rvert > 0$ and $\phi(1) = 0$ because if it were $\mbox{det} D r(x) \ne 0$ by inversion local theorem we would have in particular an homeomorphism between an open subset of $\mathbb{S}^{n}$ and an open subset of $B(0,1)$ which can't occur since the first has empty interior while the second one don't.
There are any simple reason to affirm as stated in the question ? I don't know anything about algebraic topology so I'd like to keep the proof as simple as possible using just general topology.
Any help or hint would be appreciated.