Kirby Problem 4.54

Definitions and mathematical statement

For an oriented connected four-manifold $M$, let $\pi=\pi_1(M)$ and $A=\pi_2(M)$, viewed as a left $\mathbb Z[\pi]$-module by deck transformations. The homotopy two-type has a Postnikov class $\kappa\in H^3(\pi;A)$, specifying its $K(A,2)$ fibration over $K(\pi,1)$.

The equivariant intersection form is the sesquilinear pairing $\lambda_M:A\times A\to\mathbb Z[\pi]$ obtained by intersecting lifts of sphere representatives with translates in the universal cover. With compatible conventions, \[ \lambda_M(a,b)=\sum_{g\in\pi}(a\cdot gb)g, \qquad \lambda_M(b,a)=\overline{\lambda_M(a,b)}, \] where the involution sends $g$ to $g^{-1}$. Compact representatives make the sum finite. The source's quadratic two-type is the quadruple $(\pi,A,\lambda_M,\kappa)$; no additional unspecified quadratic refinement is included in this terminology.

Which algebraic quadruples occur for closed oriented topological four-manifolds? Which occur for closed oriented smooth four-manifolds? Seek intrinsic necessary and sufficient realization conditions, including results for prescribed families of fundamental groups such as finite noncyclic groups.

Two quadruples are identified by a group isomorphism and a compatible module isomorphism preserving the form and Postnikov class. Realizing the module and class by a CW complex alone is insufficient: the requested realization is a genuine closed manifold with the specified intersection pairing. Smooth and topological realizability are distinct questions, and stabilization by extra hyperbolic summands does not by itself realize the original quadruple.

Short English statement

Which fundamental-group, second-homotopy, intersection-form, and Postnikov data arise from closed topological or smooth four-manifolds?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2930 (KP-4.54), 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 4.54. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.