Four-dimensional disk embedding and Wall's conjecture

The explanation

Two surfaces in a four-dimensional space can cross at points. Each crossing gets a sign, +1 or −1. Suppose two crossings have opposite signs. Their algebraic count is +1 + (−1) = 0, but they still meet at 2 points. The Whitney trick slides one surface along a disc to remove such a pair. The disc is called a Whitney disc. In five or more dimensions this works. In four dimensions the Whitney disc can itself hit the surfaces.

Casson replaced the disc by an infinite tower of discs, called a Casson handle. Freedman showed it behaves like an ordinary thickened disc. This settled simply connected four-manifolds, where every loop can shrink to a point. Freedman and Quinn extended the method to good groups. The fundamental group records the loops in a space up to deformation. A group is good if certain loops in grope models can be shrunk. A grope is a tower of surfaces glued along curves. The test models have free fundamental groups, so the free group is the central caseSource 1. Count words in the free group on a and b, where no letter sits next to its inverse. There are 4 × 3⁴ = 324 such words of length 5. In the grid group ℤ², only 20 elements need exactly 5 steps. Freedman and Teichner showed that groups of slow, subexponential growth are goodSource 2. The free group grows exponentially, and its case remained openSource 3.

The disc embedding conjecture says the discs always exist, with no condition on the group. The first paper claims to disprove itSource 4. Its discs come with dual spheres, which meet each disc once in the algebraic count. Even with no framing or dual sphere asked for, no disjoint discs existSource 5. It concludes that the free group on two generators is not goodSource 6. A second paper claims a separate disproof, in which the new discs must keep the boundary framingsSource 7. It studies flat connections with values in the matrix group SL₂Source 8. These are rules for moving data around loops without twisting. An algebraic obstruction rules out the needed data when at least five pieces are activeSource 9.

The third paper takes up Wall's question. A Poincaré duality group has the algebra of a closed manifold's fundamental group. Wall asked whether each one is the fundamental group of a closed aspherical manifoldSource 10. A space is aspherical if its universal cover can shrink to a point. Davis gave examples that are not finitely presentedSource 11. For finitely presented groups, a 2026 survey records no known counterexampleSource 12. The third paper claims a finitely presented four-dimensional example with a finite classifying space. It claims no closed aspherical topological four-manifold has that groupSource 13. It builds on the second paper's obstructionSource 14. It refutes the all-dimensions question, not each dimensionSource 15.

How sure is this? There is no Lean check. A computer proof checker has not looked at any part of it. The arguments combine four-manifold geometry with long algebraic and analytic steps. No specialist review of these papers has been published.

What it doesn't show. This does not give a counterexample in every dimension, and it does not settle which other groups are good.