I believe this may be a standard algebraic topology problem, so I apologize in advance if this belongs in stackexchange (it's not a homework problem, however, and came about in a research context). I've got a continuous map $f$ from the $n$-simplex to itself, such that the image of every strict sub-simplex is itself. So, each vertex gets mapped to itself, as does each edge, and so on and so forth. Does it follow that $f$ must be surjective?


Thank you!



Map from simplex to itself that preserves sub-simplices
Given such a map $f:\Delta\_n\to\Delta\_n$, put $f\_t(x)=(1-t)x+t f(x)$. This gives a homotopy between $f$ and the identity, and each map $f\_t$ also sends every subsimplex to itself. In particular, each $f\_t$ preserves $\partial(\Delta\_n)$ and so induces a self-map $\overline{f}\_t$ of the space $\Delta\_n/\partial(\Delta\_n)\simeq S^n$. It follows that $\overline{f}$ is homotopic to the identity, and so is not homotopic to a constant map. This means that $\overline{f}$ must be surjective (because any non-surjective map factors through a space $S^n\setminus\{a\}$, which is homeomorphic to $\mathbb{R}^n$ and so contractible). As $\Delta\_n\setminus\partial(\Delta\_n)$ is dense in $\Delta\_n$, it follows that $f$ itself is also surjective. 


