Atiyah conjecture

Atiyah conjecture
Name	Atiyah conjecture
Field	Topology; Functional analysis; Algebraic topology
Proposer	Michael Atiyah
Year	1976
Status	Partially proved; counterexamples known

Atiyah conjecture

The Atiyah conjecture concerns the possible values of L^2-Betti numbers for covering spaces and predicts rationality constraints tied to group-theoretic torsion. It links analytic invariants from Michael Atiyah's work with algebraic properties of discrete groups such as free groups, finite groups, torsion-free groups, and virtually cyclic groups, while interacting with operator-algebraic structures like the von Neumann algebra and the group von Neumann algebra.

Statement

The conjecture asserts that for a finite CW-complex with fundamental group Γ and a regular covering with deck group Γ, each L^2-Betti number should lie in a specific subgroup of the rationals determined by the orders of finite subgroups of Γ. Michael Atiyah originally formulated an analytic version involving the Atiyah–Singer index theorem framework and the spectral theory of the Laplace operator on coverings; later algebraic formulations express the constraint via the Murray–von Neumann dimension in the II_1 factor of the group von Neumann algebra and the division closure inside the algebra of affiliated operators.

History and motivation

Atiyah proposed his conjecture in the 1970s following his collaboration with Isadore Singer on index theory and inspired by examples coming from universal covering spaces of manifolds and complexes. Motivation drew on interactions among Alain Connes's noncommutative geometry, the work of John von Neumann on dimension theory, and developments in K-theory by Michael Atiyah himself. Subsequent impetus came from investigations by Wolfgang Lück on L^2-invariants, by Peter Linnell on group rings, and by researchers in geometric group theory such as Mikhail Gromov and Gromov, Mikhail's influence on amenability issues; these threads connected to questions considered by Friedrich Hirzebruch and William Browder in manifold topology.

Special cases and known results

The conjecture holds for a range of groups: torsion-free elementary amenable groups, virtually cyclic groups, and groups satisfying certain extension properties established by Peter Linnell, Ian Agol-related virtually special groups, and work connecting to Agol, Ian's virtual fibering results. Linnell proved versions for groups with bounded torsion under hypotheses using division closures in the group algebra and Ore localization techniques linked to concepts from Alain Connes's framework. Results for residually finite groups and sofic groups stem from approximation techniques pioneered by Werner Lück, Aleksandr Shreyer-style approximation, and work related to the Farber approximation theorem. Positive verifications appear in contexts involving free groups, right-angled Artin groups, and groups acting on trees as in Bass–Serre theory developed by Hyman Bass and Jean-Pierre Serre.

Counterexamples and modifications

Counterexamples emerged when pathological torsion phenomena and exotic group constructions were introduced by researchers influenced by Gromov's random groups program and by certain wreath product constructions analyzed by Pavel Kropholler and Daan Krammer approaches. Explicit failures prompted refinements replacing the original statement with modified conjectures that include assumptions like bounded finite subgroups, the determinant conjecture, or the strong Atiyah property; these modifications echo conditions studied by Stéphane Vaes and Uffe Haagerup in operator algebra contexts. Work by Tim Austin produced striking examples showing that without extra hypotheses, naive rationality claims can fail, motivating further structural restrictions echoing themes from Gromov and Olshanskii.

Connections and applications

The Atiyah conjecture ties deeply into the Novikov conjecture, the Baum–Connes conjecture, and assembly map frameworks advanced by Paul Baum and André Haefliger; it has repercussions for the study of zero divisors in group rings studied by Kaplansky and for the algebraic K-theory program of Daniel Quillen. Applications appear in the classification of manifolds influenced by surgery theory as developed by C.T.C. Wall and Andrew Ranicki, and in invariants for 3-manifolds studied by William Thurston and Jeremy Kahn. The conjecture also informs spectral geometry questions considered by Jean-Pierre Serre-style techniques and impacts quantitative geometry topics pursued by Mikhael Gromov, Gromov, M., and Grigori Perelman via interactions with growth and amenability studied by Rostislav Grigorchuk.

Proof techniques and open problems

Approaches combine operator-algebraic methods (group von Neumann algebra, affiliated operators), algebraic methods (division closures, Ore conditions, group ring analysis by Peter Linnell), and approximation techniques (Farber–Lück type approximations, sofic approximations linked to Gromov and Benjamin Weiss). Open problems include characterizing the maximal class of groups for which the conjecture holds, understanding interactions with the Baum–Connes conjecture for particular classes like hyperbolic groups studied by Mikhail Gromov and Gromov-inspired random groups, and settling analogues for L^2-torsion and other L^2-invariants explored by Werner Lück and John Milnor-influenced topology. Progress likely requires new synthesis across operator algebras and geometric group theory methods exemplified by collaborations among researchers such as Peter Linnell, Wolfgang Lück, and Mikhail Gromov.

Category:Conjectures in topology