Smooth singular simplex

Let $M$ be a smooth manifold, possibly with boundary. For $k\ge 0$, put $A^k=\{(t_0,\dots ,t_k)\in {\mathbb{R}}^{k+1}:{\sum }_i{t}_i=1\}\mathrm{.}$ A smooth singular $k$-simplex in $M$ is a map $\sigma :{\mathrm{\Delta }}^k\to M$ for which there are an open set $O\subseteq A^k$ containing ${\mathrm{\Delta }}^k$ and a smooth map $\bar{\sigma }:O\to M$ with $\bar{\sigma }{\mathrm{\mid }}_{{\mathrm{\Delta }}^k}=\sigma$. The simplex and its face maps are The standard topological simplex and its affine face maps. Smoothness into a boundary target is that of Smooth maps between manifolds with boundary.

The extension takes values in $M$ on all of $O$, including points outside the simplex. Merely extending its chart coordinates to a Euclidean space with values outside $M$ does not suffice. Nor is separate smoothness of the face restrictions the definition. For a boundaryless target this is the usual neighbourhood-extension convention.

Write $S_k^{\mathrm{\infty }}(M)$ for this set of maps; the extension itself is not additional simplex data. Every such map is continuous. At $k=0$, $A^0={\mathrm{\Delta }}^0$ is a point, so every point of $M$ is a smooth zero-simplex. Constant maps are smooth in every degree, including maps to a boundary point, and degenerate parametrizations are allowed. The empty manifold has no simplices. No simultaneous choice of extensions is part of this definition.