we-solve (beta)

Math-3027 · mathematics · Topology

Singer vanishing in dimension four

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Does every finite connected aspherical integral Poincaré-duality CW complex of formal dimension four have all L$^{2}$-Betti numbers zero outside degree two? This includes every closed aspherical topological four-manifold.

Short English statement

Does every finite connected aspherical integral Poincaré-duality CW complex of formal dimension four have all L$^{2}$-Betti numbers zero outside degree two? This includes every closed aspherical topological four-manifold.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-3027-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-3027

Manuscript 1 (family 315, 2026-09-25): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.

Singer vanishing in dimension four

OpenAI's claim

The Singer vanishing assertion is claimed in dimension four, including finite aspherical integral Poincare complexes and nonorientable cases: universal-cover $L^2$-Betti numbers vanish in every degree except two. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $Q$ be an integral Poincaré complex that is aspherical, connected and finite, with formal dimension four, allowing an arbitrary orientation character. OpenAI claims \[ b_j^{(2)}(\widetilde Q)=0\ (j\ne2),\qquad \chi(Q)=b_2^{(2)}(\widetilde Q)\ge0. \] Here $b_j^{(2)}$ is the von Neumann dimension of the homology of $\mathcal N\pi_1(Q)\otimes_{\mathbb Z\pi_1(Q)}C_*(\widetilde Q)$. The claims include closed connected aspherical topological four-manifolds; zero Euler characteristic is equivalent to vanishing of every $L^2$-Betti number. For oriented such manifolds $M$, the manuscript additionally claims $\chi(M)\ge|\sigma(M)|$, where $\sigma$ is the real intersection-form signature. These are dimension-four conclusions.

Outline of the argument

Manuscript 1 (family 315, 2026-09-25): Positive first $L^2$ cohomology would supply a bounded finite-energy function on a Cayley graph. Its translates define a boundary with controlled approach paths, and annular separation arguments organize cut points into dendritic blocks. Poincare duality imposes dimension and cohomology constraints excluding circles, so a nondegenerate block would be a dendrite with a forbidden cut point.

Formal verification

Family 315 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

Go to contribution →