1 · The essentials
In four-dimensional spaces, a central question asks when circles can be filled by disks that do not intersect one another. The manuscripts describe examples where certain mathematical checks are satisfied, yet the desired disks cannot exist. The main manuscript makes this failure depend only on the boundary circles: allowing more freedom in the choice of disks does not remove it. It claims that a restriction on the surrounding space, omitted from the unrestricted conjecture, cannot simply be dropped. A companion manuscript addresses a different realization question, describing a group with four-dimensional duality properties that cannot come from the kind of four-dimensional space requested by that question. Together, these manuscripts claim negative answers to the unrestricted disk-embedding conjecture and to Wall's manifold-realization question in dimension four. The main result is not formalized in Lean and has not been formally verified.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine several wire loops that you want to fill with soap films without any films touching. Passing a checklist about how the proposed fillings interact need not guarantee that separated fillings exist, even if you are free to redesign them. The manuscripts concern mathematical disks in four dimensions, not physical soap films.
Why it matters
The claimed disk examples distinguish algebraic conditions on proposed disks from the existence of actual disjoint fillings. The main manuscript also states a consequence for which fundamental groups are good in the sense of Freedman–Quinn. The group-theoretic companion separates four-dimensional Poincaré duality and a finite classifying space from realization by a closed aspherical topological four-manifold.
2 · Going further
The disk-embedding problem here concerns disk maps in a compact oriented smooth four-manifold, equipped with framed algebraic dual spheres. The unrestricted conjecture asks for disk embedding without a hypothesis on the fundamental group. Manuscript 1 claims an example whose boundary circles bound no pairwise disjoint locally flat disks, even when neither the relative homotopy classes nor the output framings of those disks are prescribed. Its stated consequence is that 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.
Manuscript 2 describes immersed disks with framed algebraic dual spheres satisfying the usual equivariant intersection and reduced self-intersection conditions. It claims that these disks have no pairwise disjoint locally flat replacements preserving the boundary maps and induced normal framings, even without prescribing the replacements' relative homotopy classes. The important distinction is that Manuscript 1 removes the requirement to preserve output framings: its obstruction is stated in terms of the boundary circles alone.
At the level of the supplied abstracts, the strategy is to construct examples that meet the algebraic requirements but obstruct the requested geometric fillings. The titles identify a boundary-only obstruction and a marked tensor obstruction, but the abstracts do not explain their construction or the mechanism establishing nonexistence. A more detailed account of the argument is therefore not available from this source.
Manuscript 3 claims a finitely presented integral Poincaré duality group of dimension four with a finite classifying space that is not the fundamental group of any closed aspherical topological four-manifold. It presents this as a negative answer to Wall's manifold-realization question in dimension four. The supplied abstracts do not specify how its construction depends on the disk-embedding manuscripts.
Useful notions
- Four-dimensional manifolds
- Disk maps, immersions, and embeddings
- Fundamental groups
- Intersection conditions and normal framings
- Poincaré duality and classifying spaces
What was known before / what is new
Before
The source identifies an unrestricted four-dimensional disk-embedding conjecture and Wall's manifold-realization question in dimension four as the questions being addressed. It also refers to the usual equivariant intersection and reduced self-intersection conditions, and to goodness in the sense of Freedman–Quinn. It supplies no further historical account or description of previously established cases.
New
Manuscript 1 claims a boundary-only obstruction: framed algebraic dual spheres do not suffice for disjoint locally flat spanning disks, even without prescribed homotopy classes or output framings. Manuscript 2 gives a related claimed obstruction for replacements required to preserve boundary maps and induced normal framings. Manuscript 3 claims a group with the stated duality and finiteness properties but no closed aspherical topological four-manifold realization.
Limits and assumptions
The disk constructions are stated for compact oriented smooth four-manifolds, while the forbidden fillings are disjoint locally flat disks. Manuscript 2 preserves boundary maps and induced normal framings; Manuscript 1 claims nonexistence without prescribing output framings or homotopy classes. These claims concern the unrestricted conjecture, not the failure of every version with a fundamental-group hypothesis. The group-realization claim concerns dimension four and closed aspherical topological four-manifolds. The main result is not formalized in Lean and has not been formally verified; the supplied abstracts do not provide enough argument to independently assess the claims. They also do not list which restricted cases remain open.
Mini glossary
- Disk embedding —
- A placement of a disk into a space without identifying distinct points of that disk.
- Locally flat disk —
- An embedded disk whose local placement in the surrounding manifold looks like the standard flat model.
- Fundamental group —
- A group recording loops in a space up to continuous deformation.
- Normal framing —
- A consistent choice of coordinates in directions normal to a disk or sphere.
- Relative homotopy classes —
- Classes of maps related by continuous deformation while their boundary maps remain fixed.
Keywords : Disk embedding · Four-dimensional topology · Wall's conjecture
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
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.
To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.
Manuscripts in this family
A boundary-only obstruction to four-dimensional disk embedding
24/09/2026We 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
24/09/2026We 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
24/09/2026We 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.