Kirby Problem 4.119

Definitions and mathematical statement

A topological $4$-manifold is a Hausdorff second-countable space locally homeomorphic to $\mathbb R^4$ or, at boundary points, to a closed half-space. A CW complex is built inductively from points by attaching disks along continuous maps of their boundary spheres to the preceding skeleton. It has the closure-finite weak topology: each cell closure meets finitely many cells, and closedness is tested on cell closures.

Is every topological $4$-manifold homeomorphic to a CW complex? Explicitly, for each such manifold $M$, the question asks for some CW complex $K$ and a continuous bijection $|K|\to M$ with continuous inverse. The source's genuinely difficult case includes closed manifolds without a smooth structure; no smoothability hypothesis is part of the question.

The requested map is a homeomorphism, not just a homotopy equivalence or a simple homotopy equivalence. Thus constructing a CW model with the same algebraic topology would not suffice. The cells need not form a regular CW structure: an attaching map need not be an embedding, and a cell closure need not itself be an embedded disk.

Likewise, a CW complex need not be a simplicial complex. The question is therefore distinct from triangulability, which would require a homeomorphism to the realization of a simplicial complex. No preferred cell decomposition of the given manifold is supplied; the task is to find some CW structure realizing its actual topology.

Short English statement

Is every topological four-manifold homeomorphic to a CW complex, rather than merely homotopy equivalent to one?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2995 (KP-4.119), 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 4.119. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.