Differentiable structure on the boundary induces differentiable structure on the entire manifold

Question

I am trying to prove the uniqueness of smoothed corners, in detail. In Milnor's notes, https://www.maths.ed.ac.uk/~v1ranick/papers/homsph.pdf, there's an explanation in p.34-37 on how to do it.

The guiding question is: Given two manifolds with boundary $W_1, W_2$, can we put a smooth structure on $W = W_1\times W_2$ so that $W_1\times W_2-\partial W_1\times\partial W_2$, $\partial W_1\times W_2$, and $W_1\times\partial W_2$ are smoothly embedded submanifolds? And if there is, is it unique?

Milnor shows there is at least one. He also uses his Lemma 8.2 $(\dagger)$ to assert that all possible structures will induce in the boundary $\partial W$ identical structures up to diffeomorphism. This is because $\partial W$ is equal to $\partial W_1\times W_2\cup W_1\times\partial W_2$ and the differentiable structures of both are fixed so the lemma applies.

Notice he didn't show all possible structures on $W$ are identical up to diffeomorphism. He only says the induced smooth structures on $\partial W$ will be identical up to diffeomorphism.

My question is: though he didn't say it, is it true that all possible structures on $W$ are equivalent as well?

$(\dagger)$ The Lemma 8.2 says that when gluing two smooth manifolds along their diffeomorphic boundaries, the smooth structures on the end manifold (compatible with the first two) are equivalent up to diffeomorphism.

What I have done: One can immediately see that this smooth structure in $\partial W$ has to extend to a collar, so there is an inclusion $\partial W\times [0,\infty)\subseteq W$ which is a smooth embeddeding, with $\partial W\times [0,\infty)$ given the product smooth structure, with the standard one in $[0,\infty)$.

The hope is that by connectedness we can claim that the differentiable structure on $W$ is uniquely determined, once we have the one in the collar.

Answer 1

Let me try to answer your question to the best of my understanding of it.

1. Your first question is: Suppose that $W$ is a topological manifold with boundary whose boundary is smoothable and extends to a smooth structure on $W$. Is this extension unique up to diffeomorphism?

The answer to this question is negative. For instance, take a closed 4-dimensional manifold $M$ which admits two nonequilavent smooth structures $s_1, s_2$. Without loss of generality, we may assume that $s_1, s_2$ agree on an open nonempty subset $U\subset M$. Let $W$ be obtained by removing from it the interior of smooth ball contained in $U$. Then restricting $s_1, s_2$ to $W$ we obtain two smooth structures on it (I retain the notation $s_1, s_2$ for these). If they were equivalent, there would be a diffeomorphism $$ f: (W,s_1)\to (W,s_2). $$ But each diffeomorphism $S^3\to S^3$ is smoothly extends to the 4-ball $B^4$. Then we would obtain a diffeomorphism $(M,s_1)\to (M,s_2)$, which is a contradiction.

2. Your second question appears to be: Suppose that in the setting of Question 1, we have two extensions $s_1, s_2$ of the given smooth structure on $\partial W$ to $W$. Suppose, in addition, that $$ (int(W), s_1), (int(W), s_2) $$ are diffeomorphic. Doe it follow that $(W,s_1), (W,s_2)$ are diffeomorphic?

This is a harder question which also has negative answer. See Benoit's answer to an MO question here.