we-solve (beta)

Math-2365 · mathematics · Group theory

Algebraic K-Theory and Rigidity Conjectures for Discrete Groups

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
For a torsion-free group $G$ and an integral domain $R$, are $0$ and $1$ the only idempotents of $RG$? Do the negative K-groups of $\mathbb{Z}G$, the reduced group $K_0(\mathbb{Z}G)$, and $\operatorname{Wh}(G)$ vanish? Are higher signatures homotopy invariant, closed aspherical manifolds topologically rigid, and groups of type FP of type FF? Is every finitely presented Poincaré-duality group of dimension at least five the fundamental group of an aspherical homology ANR-manifold? Is every hyperbolic group with boundary $S^n$ the fundamental group of a closed aspherical manifold? Are the Farrell–Jones assembly maps in algebraic K- and L-theory bijective?

Formulation references: [S1], [S2].

Short English statement

For a torsion-free group $G$ and an integral domain $R$, are $0$ and $1$ the only idempotents of $RG$? Do the negative K-groups of $\mathbb{Z}G$, the reduced group $K_0(\mathbb{Z}G)$, and $\operatorname{Wh}(G)$ vanish? Are higher signatures homotopy invariant, closed aspherical manifolds topologically rigid, and groups of type FP of type FF? Is every finitely presented Poincaré-duality group of dimension at least five the fundamental group of an aspherical homology ANR-manifold? Is every hyperbolic group with boundary $S^n$ the fundamental group of a closed aspherical manifold? Are the Farrell–Jones assembly maps in algebraic K- and L-theory bijective?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2365-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2365

Manuscript 1 (family 320, 2026-10-04): The nonhomeomorphic aspherical pair refutes the Borel-rigidity clause of this broad bundle, but does not decide its idempotent, K-group, signature, FP/FF or higher-dimensional realization clauses.

Algebraic K-Theory and Rigidity Conjectures for Discrete Groups

OpenAI's claim

Two closed aspherical topological four-manifolds are claimed homotopy equivalent but not homeomorphic, sharing a word-hyperbolic fundamental group; one also admits a self-equivalence outside every homeomorphism homotopy class. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
OpenAI claims closed connected aspherical topological four-manifolds $M,N$ with a common word-hyperbolic fundamental group such that \[ M\simeq N\quad\text{but}\quad M\not\cong N\text{ by any homeomorphism}. \] Aspherical means that the universal cover is contractible; $\simeq$ denotes homotopy equivalence. A second theorem claims a topological four-manifold that is aspherical, closed and connected $M$ and a self homotopy equivalence $f:M\to M$ for which no homeomorphism is homotopic to $f$. These are four-dimensional topological counterexamples; the first excludes homeomorphism existence in any homotopy class.

Outline of the argument

Manuscript 1 (family 320, 2026-10-04): Reflected marked chambers retain an odd-cover filling obstruction, and carefully chosen binary label relations eliminate unwanted chamber symmetries. Hyperbolicity and Poincare duality make the outer automorphism group finite, with a controlled normalizer of the realized chamber action. A hypothetical homeomorphism would transport another finite action into that realized subgroup and force an excluded conjugacy.

Formal verification

Family 320 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 →