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Math-2413 · mathematics · Topology

Injectivity of the Baum–Connes Assembly Map

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 every discrete group $\Gamma$ and $i\in\{0,1\}$, is the Baum–Connes assembly map $\mu:K_i^\Gamma(E\Gamma)\to K_i(C_r^*(\Gamma))$ injective, where $E\Gamma$ is the universal space for proper $\Gamma$-actions?

Formulation references: [S1], [S2].

Short English statement

For every discrete group $\Gamma$ and $i\in\{0,1\}$, is the Baum–Connes assembly map $\mu:K_i^\Gamma(E\Gamma)\to K_i(C_r^*(\Gamma))$ injective, where $E\Gamma$ is the universal space for proper $\Gamma$-actions?

Sources

Catalogue entry

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

Contributions 1

Math-2413-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2413

Manuscript 1 (family 285, 2026-09-23): The infinite-order degree-zero assembly-kernel class for a finitely generated torsion-free discrete group directly refutes the universal coefficient-free reduced injectivity and isomorphism assertions, with no coefficient or maximal-completion substitution.

Injectivity of the Baum–Connes Assembly Map

OpenAI's claim

For one finitely generated group without torsion, the coefficient-free reduced assembly map annihilates an infinite-order degree-zero K-homology class; this is claimed to refute rational injectivity. 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 a discrete group, torsion-free and finitely generated, $G$, an injection $i:\mathbb Z^2\hookrightarrow G$, and $c\in H^2(BG;\mathbb Z)$ such that \[ \langle(Bi)^*c,[T^2]\rangle=1,\qquad i_*\beta=0\text{ in }K_0(C_r^*(G)), \] here $\beta\in K_0(C_r^*(\mathbb Z^2))$ is the Bott generator chosen to have rank zero. The class $h=(Bi)_*[D_{T^2}]\in K_0(BG)$ has infinite order but satisfies $\mu_G^r(h)=0$. Here $\mu_G^r:K_*(BG)\to K_*(C_r^*(G))$ is reduced coefficient-free analytic assembly and $[D_{T^2}]$ is the oriented torus spin-Dirac class. This asserts failure of reduced assembly injectivity without coefficients, even rationally.

Outline of the argument

Manuscript 1 (family 285, 2026-09-23): The construction embeds a torus subgroup with an integral cohomology detector while arranging disappearance of its Bott class analytically. Operators assembled on almost disjoint Cayley-graph sheets retain controlled spectral supports; estimates for the leftover forest permit the required homotopies in the reduced completion. A finite witness then restricts the construction to a finitely generated subgroup.

Formal verification

Family 285 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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