Kirby Problem 4.50

Definitions and mathematical statement

A closed topological $4$-manifold is a compact boundaryless Hausdorff second-countable space locally homeomorphic to $\mathbb R^4$. Consider connected such manifolds $X$ with finite fundamental group $\pi_1(X)$. Both orientable and nonorientable manifolds are included, and equivalence means homeomorphism of the underlying manifolds.

Classify these manifolds up to homeomorphism by algebraic invariants. In particular, identify a collection of algebraic data associated to $X$ and the precise equivalence on that data which detects when two manifolds in the specified class are homeomorphic. Merely identifying their homotopy types or their homology groups would not by itself complete the requested classification.

The source emphasizes three families of possible fundamental groups: dihedral groups, quaternionic groups, and finite abelian groups requiring at most two or three generators. For clarity, the dihedral family has presentation $\langle r,s\mid r^m=s^2=1,\ srs^{-1}=r^{-1}\rangle$. The quaternionic family includes the generalized quaternion groups, such as $Q_8=\{\pm1,\pm i,\pm j,\pm k\}$; its $2$-power-order members have presentation $\langle a,b\mid a^{2^{r-1}}=1,\ b^2=a^{2^{r-2}},\ bab^{-1}=a^{-1}\rangle$ for $r\ge3$.

An initial concrete target is the full classification when $|\pi_1(X)|\le8$. These highlighted subfamilies do not restrict the main question, which includes every finite fundamental group. No smooth structure, triangulation, or simple-connectivity assumption is imposed. The source regards reduction to complete algebraic invariants as the intended form of a solution.

Short English statement

Classify closed topological four-manifolds with finite fundamental group, including the dihedral, quaternionic and small-generator abelian families.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2926 (KP-4.50), 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 4.50. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.