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Math-1608 · mathematics · Topology

Kirby Problem 4.19

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

A connected manifold is aspherical if its universal cover is contractible, equivalently if all its homotopy groups in degrees at least two vanish. For a closed four-manifold $X$, define its Euler characteristic by $\chi(X)=\sum_{i=0}^4(-1)^i\dim H_i(X;\mathbb Q)$. If $X$ is oriented, its intersection form on $H_2(X;\mathbb R)$ is nonsingular and symmetric; its signature $\sigma(X)$ is the number of positive eigenvalues minus the number of negative ones.

Determine the possible Euler characteristics of closed aspherical four-manifolds. In particular, for every oriented member of this class, is \[ \chi(X)\ge |\sigma(X)|? \] The aspherical question can be posed in the topological category as well as the smooth category; existence of a smoothing is not part of asphericity. Reversing orientation changes the sign of the signature and leaves this inequality unchanged.

Separately, what is the smallest Euler characteristic of a closed hyperbolic four-manifold? Hyperbolic means a smooth manifold with a complete Riemannian metric of constant sectional curvature $-1$. The unqualified minimum includes nonorientable examples; the minimum restricted to orientable examples is a useful separate version.

This second question concerns closed manifolds, not cusped finite-volume ones. An exact minimum requires both a lower bound applying to every eligible manifold and an example attaining it. The two questions are related through the fact that hyperbolic manifolds are aspherical, but their hypotheses are different.

Short English statement

Determine Euler characteristics of closed aspherical four-manifolds and the least one in the hyperbolic class.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-1608-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-1608

Manuscript 1 (family 315, 2026-09-25): The Euler and signature corollaries prove chi>=|signature| and nonnegativity for aspherical four-manifolds, but do not determine every possible Euler characteristic or the least one in the hyperbolic class.

Kirby Problem 4.19

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

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