Four-dimensional disk embedding and Wall's conjecture

The unrestricted four-dimensional disk-embedding conjecture fails: framed algebraic dual spheres do not suffice to obtain disjoint locally flat spanning disks. In particular, the free group F₂ is not good in the sense of Freedman–Quinn. Also constructs a finitely presented integral Poincaré duality group of dimension four with a finite classifying space but no realization as the fundamental group of a closed aspherical topological four-manifold, disproving Wall's conjecture.

Manuscripts (3)

A boundary-only obstruction to four-dimensional disk embedding

September 24, 2026 · source · cite as \cite{OAI:A-boundary-only-obstruction-to-four-dimensional-disk-embedding-September-24-2026}

We disprove the four-dimensional disc embedding conjecture without a fundamental-group hypothesis, even when no homotopy classes or output framings are prescribed. We construct a compact oriented smooth four-manifold containing finitely many disc maps with framed algebraic dual spheres whose boundary circles bound no disjoint locally flat discs. Consequently, the free group on two generators is not good in the sense of Freedman–Quinn, and neither is any group containing it as a subgroup.

A marked tensor obstruction to four-dimensional disk embedding

September 24, 2026 · source · cite as \cite{OAI:A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026}

We disprove the unrestricted four-dimensional disk-embedding conjecture. We construct immersed disks in a compact oriented smooth four-manifold with framed algebraic dual spheres satisfying the usual equivariant intersection and reduced self-intersection conditions, but with no pairwise disjoint locally flat replacements that preserve the boundary maps and induced normal framings. The obstruction holds even when the replacement disks' relative homotopy classes are not prescribed.

A PD4 group without an aspherical manifold model

September 24, 2026 · source · cite as \cite{OAI:A-PD4-group-without-an-aspherical-manifold-model-September-24-2026}

We construct a finitely presented integral Poincaré duality group of dimension four that has a finite classifying space but is not the fundamental group of any closed aspherical topological four-manifold. This gives a negative answer to Wall's manifold-realization question in dimension four.

Result 305 · Topology. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.