Let $X$ be a topological space.
Let $\rho = \sum_j \sigma_j$, where the sum is finite, be a chain of singular $k$-simplices in $X$ (in the $k$th singular chain group of $X$, with $\mathbb{Z}$ coefficients).
For a $(k+1)$-dimensional pseudomanifold $K$ let $\partial K$ denote its boundary which should be a $k$-dimensional pseudomanifold.
If I'm not mistaken, we have the following:
If $\rho$ is a $k$-boundary, then there exists an "orientable" $(k+1)$-dimensional pseudomanifold $W$, and a continuous map $f : W \rightarrow X$, such that the boundary restriction $\partial f : \partial W \rightarrow X$ carries the triangulation on $\partial W$ to the chain $\rho$ in $X$, with signs assigned as per some "outward" orientation on $\partial W$ from $W$.
Now we assume further that $X$ is a smooth manifold.
I was wondering, in this case, can we find $W,f$ satisfying the following additional conditions?
- $W$ has a smooth structure "smooth structure with corners" on each $n$-simplex;
- $f : W \rightarrow X$ is smooth on $W\setminus U$ for some collar neighborhood $U$ of $\partial W$ which can be made "arbitrarily thin"; and
- if each $\sigma_j$ is smooth, then $f$ can be assumed smooth on the whole of $W$.
(I think the 3rd bullet point would follow from the other 2, using e.g. Whitney approximation -type theorems.)
Edit: added the last condition, for the case of $\sigma_j$'s already smooth.
Edit: per the suggestion in the comment below, allowed $W$ to be a "pseudomanifold with boundary" instead of a manifold or arbitrary finite simplicial complex.