Kirby Problem 4.49

Definitions and mathematical statement

An open topological $4$-manifold $M$ has empty boundary and no compact component; manifolds are Hausdorff and second countable. It admits a compactification by a manifold boundary if there is a compact topological $4$-manifold $W$ with boundary and a homeomorphism $M\cong\operatorname{int}(W)$. The added set must be an actual $3$-manifold boundary, rather than arbitrary points or a singular compactifying space.

Give an effective necessary and sufficient criterion for this property. Such a criterion should identify exactly the interiors of compact $4$-manifolds using accessible data about $M$, in particular its behavior near infinity. It should supply both an obstruction when compactification fails and a sufficient condition ensuring a manifold-boundary compactification.

A neighborhood of infinity is a subset containing the complement of some compact subset of $M$. An end records a compatible choice of components of such complements as the compact subset increases. A compact manifold boundary gives collared neighborhoods of its ends, modeled on $Y\times[0,\infty)$ for closed $3$-manifolds $Y$. Recognizing when such collars exist is therefore part of the problem.

The question is in the topological category; neither a smooth compactification nor a chosen smooth structure is required. The source does not specify a finite encoding of arbitrary open manifolds, so "effective" asks for a usable criterion rather than an algorithm on an unstated class of finite inputs. Merely restating the existence of a compact filling is not such a criterion.

Short English statement

Find an effective criterion for an open four-manifold to be homeomorphic to the interior of a compact manifold with boundary.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2925 (KP-4.49), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 4.49. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.