# Stable smoothing of topological manifolds relative to an embedding

Let $$M$$ be a topological manifold. We know that $$M$$ is stably smoothable if and only its tangent microbundle, up to stabilization, admits a reduction to vector bundle.

Now I wonder if there is a relative version of this fact. Suppose $$M$$ and $$N$$ are topological manifolds and $$f: M \to N$$ is a locally flat topological embedding and assume it admits a normal microbundle. We also assume that the normal microbundle is equivalent to a vector bundle. Suppose $$M$$ and $$N$$ are both stably smoothable. I wonder if one can prove that there are smoothable stablizations $$M = M \times {\mathbb R}^m$$ and $$N' = N \times {\mathbb R}^{m + n}$$ such that the natural extension $$f': M' \to N'$$ (extending $$f$$ by the inclusion $${\mathbb R}^m \to {\mathbb R}^{m+n}$$) is a smooth embedding.

If the above speculation is true, I also wonder if such stable smoothings are in any sense unique.

I think the answer is yes, it follows from this: if $$M$$ is a triangulable manifold of dimension $$n$$ greater than 4, and if the total space $$E$$ of a vector bundle over $$M$$ is smoothable, then the smooth structure is concordant to one where the zero section is a smooth submanifold.
To see why it follows, stabilize $$M$$ so that it has dimension at least $$5$$. Then the normal bundle of the embedding satisfies the requirements of our theorem since the tubular neighborhood is assumed to be a vector bundle and is an open subset of a smoothable space. By the above proposition and the concordance extension theorem, we can find a concordant smooth structure on $$N$$ which agrees with the smooth structure on the tubular neighborhood of $$M$$ which we have manufactured to have $$M$$ as a submanifold.
So why is the first proposition true? Pick a triangulation of $$M$$. Over an n-simplex $$\Delta$$ we inductively apply the fact that any smooth structure on $$\Delta \times \mathbb{R}^k$$ which is a product structure when restricted to some set of faces $$I$$ is concordant relative $$I \times \mathbb{R}^k$$ to a product structure. This is a consequence of the product structure theorem. After this process is completed, the smooth structure on our vector bundle has the property that the restriction to every $$n$$-simplex in $$M$$ is a smooth submanifold. Hence, $$M$$ is a smooth submanifold. The vector bundle requirement ensures that any two trivializations over a shared face $$\sigma$$ induce the same smooth structure on $$\sigma \times \mathbb{R}^k$$, meaning the notion of a local smooth product structure on a vector bundle is well defined.