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Let $X^n$ be a compact $n$-manifold with boundary. I am interested in when $X$ has the following property: For all compact $n$-manifolds $Y^n$ with $\partial Y \cong \partial X$ and any homeomorphisms $f_1,f_2 : \partial Y \to \partial X$ the resulting closed manifolds $X \cup_{f_1} Y$ and $X \cup_{f_2} Y$ are homeomorphic. I am also (in fact more) interested in the smooth case where all maps/manifolds are smooth and all homeomorphisms are diffeomorphisms.

For example $B^n$ has this property (by Alexander's trick - although this fails in the smooth case). I believe that $\natural^k S^1 \times B^3$ also has this property (Laudenbach, François, and Poénaru, Valentin. "A note on 4-dimensional handle bodies."). Certainly $S^1 \times D^2$ does not since Lens spaces are genus $1$ and there are lots of them.

Is there a nice classification of such manifolds $X$ - perhaps in dimensions $3$ and/or $4$? Does anyone know any other examples or have references?

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  • $\begingroup$ it is definitely not true that the boundary sum has that property. There are a great many manifolds without 2-cells, and that's the same as being a gluing of the handlebody to itself. The paper says that if you glue a 1-handlebody to a 2-handlebody you get a unique result. $\endgroup$ – user98602 Oct 26 '16 at 18:36
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Dimension 4 is almost certainly unreasonable unless you make eg a simple connectivity assumption. In dimension two it is true iff $Y$ is orientable and there's only one boundary component or $Y$ is non-orientable.

Dimension 3 is interesting. Suppose $Y$ is oriented and has connected boundary. Of course, if the boundary is a sphere, this is asking whether or not $Y$ has an orientation-reversing homeomorphism; this is true for some spaces and false for others.

It is false for any higher genus boundary. Adding 2-cells along the boundary you can reduce to the case when the boundary is a torus. I claim that any 3-manifold with torus boundary has infinitely many ways to fill it with a handle $S^1 \times D^2$. Note that, equivalently, after choosing a filling, I'm saying that any knot in a 3-manifold has infinitely many non-homeomorphic surgeries. This can actually be checked by homology alone; infinitely many different homology groups can arise. One could try and make the restriction along the lines of "well, suppose we fix the homology of the surgery...", but I imagine the answer is the same then too.

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