I know that every smooth manifold $M$ admits a triangulation. That means, there exists some simplicial complex $K$ homeomorphic to $M$.
Does this mean that it also admits a smooth $\Delta$-complex structure? In Hatcher, this is defined as
a collection of smooth maps $\sigma_\alpha :\Delta_\alpha^n\to M$ satisfying
- The restriction of each map to each face is injective, and each point of X is in the image of exactly one such restriction.
- Each restriction to a face is one of the maps $\sigma_\beta : \Delta^{n−1}\to M$.
- A set $U\subset M$ is open iff $\sigma^{-1}(A)$ is open in $\Delta^n$ for each $\sigma$.