It is known that a manifold $B$ bounds a compact contractible topological manifold if and only if $B$ is a homology sphere. The "only if" direction follows by excising a small ball in the interior of the contractible manifold, and noting that its boundary sphere has the same homology as $B$ by Poincare duality. The "if" direction is due to Freedman in dimension $3$ and to Kervaire in dimensions $>3$.

** Question.** Is there a characterization of boundaries of
*noncompact* contractible manifolds?

Note that if $B$ is a boundary component of a contractible $n$-manifold $W$, then the following holds.

If $B$ is compact, then $W$ is compact and $\partial W=B$.

(If $V$ denotes the union of $B$ and the interior of $W$, then the long exact sequence of the pair gives $H_{n-1}(B)\cong H_{n}(V,B)$ and Poincare duality gives implies that $H_{n-1}(B)$ is nontrivial, and hence so is $H_{n}(V,B)\cong H^0_c(V)$ which is only possible if $V$ is compact.)$B$ is stably parallelizable (because $W$ is parallelizable).

If $U$ is a proper open subset of $B$, then $U$ bounds a noncompact contractible manifold, namely, $U\cup\mathrm{Int}(W)$.

The product of $B$ with any open contractible manifold bounds a noncompact contractible manifold.

If $B'$ bounds a contractible manifold $W'$, then the connected sum $B\# B'$ bounds a contractible manifold, namely, the boundary connected sum of $W$, $W'$.

The above question may be too hard, so I would be happy with any addition to 1--5.