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Four-dimensional disk embedding and Wall’s conjecture

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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.

The release has no Lean statement for this family, so there is no formal statement to compare with the result described above.

The paper states

Theorem 1.1

In A PD4 group without an aspherical manifold model

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

Four other possible matches are listed below the card, among them Theorem 1.1 in A marked tensor obstruction to four-dimensional disk embedding and Theorem 1.1 in A boundary-only obstruction to four-dimensional disk embedding.

There is a finitely presented group G with a finite, oriented, four-dimensional aspherical Poincaré complex Q such that

H^i(G;\Z G)=0\quad(i\ne4),\qquad H^4(G;\Z G)\cong\Z

with trivial right G-action on the last group, but no closed aspherical topological four-manifold has fundamental group G. In particular the trivial left \Z G-module \Z has a finite resolution by finitely generated free modules.

Four more possible matches

  • Theorem 1.1 in A marked tensor obstruction to four-dimensional disk embedding.
  • Theorem 1.1 in A boundary-only obstruction to four-dimensional disk embedding.
  • Corollary 1.2 (Nongoodness of the free group) in A boundary-only obstruction to four-dimensional disk embedding.
  • Corollary 1.3 in A boundary-only obstruction to four-dimensional disk embedding.
Reported proof verification
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Papers and explanations
Three current papers, each readable here as a page beside its PDF.

Papers

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A boundary-only obstruction to four-dimensional disk embedding

Undated, current edition.

Read it herePDF

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

Undated, current edition.

Read it herePDF

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

Undated, current edition.

Read it herePDF

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.

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  • An unsigned public tier list, the OpenAI Math Tier List, classes it as a counterexample and rates its significance, if it holds up, as “Career-defining publication”. The page says that nothing on it is verified.

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