The Borel conjecture for higher graph manifolds in dimensions below six and for non-trivial bundle pieces

The Borel conjecture asserts that every homotopy equivalence between closed aspherical manifolds is homotopic to a homeomorphism. Bárcenas, Juan-Pineda and Suárez-Serrato (2017) announced it for a broad family of aspherical manifolds that includes higher graph manifolds — manifolds glued from bundles whose fibers are compact quotients of aspherical simply connected Lie groups over truncated finite-volume pinched-negatively-curved bases — but with two restrictions: their theorems are stated for dimensions $n \ge 6$ , and their complete decomposition-based proof is given only when every piece is a trivial bundle. The open cases are: (a) the conjecture for this family in dimensions $n=4$ and $n=5$ (dimension 3 is settled by 3-manifold geometrization), and (b) a complete L-theoretic verification for pieces with non-trivial bundle structure for $n \ge 6$ , since the published argument for the general family establishes only an algebraic K-theory assembly isomorphism, whose passage to topological rigidity requires an input that is not supplied. Davis and Hillman (2025) prove rel-boundary topological rigidity for aspherical 4-manifolds with elementary amenable fundamental group, and Contreras Peruyero and Suárez-Serrato (2025) prove the Novikov conjecture for geometrizable 4-manifolds, but neither addresses the glued manifolds of this family.

Background

An aspherical manifold is a manifold whose universal cover is contractible; for closed manifolds this makes the homotopy type an invariant of the fundamental group $\pi_1(M)$ alone. The Borel conjecture asserts that a closed aspherical manifold is topologically rigid: every homotopy equivalence $f: M \to M'$ to another closed manifold is homotopic to a homeomorphism. The conjecture is open in general. For manifolds of dimension at least 5 the standard attack is surgery theory, the classification machinery of high-dimensional topology that converts a homotopy equivalence into a homeomorphism once certain obstruction groups vanish: rigidity of $M$ follows once the assembly map in quadratic L-theory (L-theory being the obstruction-group framework underlying surgery) is an isomorphism for $\Gamma = \pi_1(M)$ — an instance of the Farrell–Jones conjecture, a broad framework describing the algebraic K- and L-theory of group rings — together with vanishing of the Whitehead group $\mathrm{Wh}(\Gamma)$ — the group of obstruction classes governing the $s$ -cobordism theorem — which itself follows from the K-theoretic assembly isomorphism. For groups of finite asymptotic dimension (a coarse-geometric finiteness invariant introduced by Gromov: the coarse analogue of covering dimension), the Novikov conjecture (homotopy invariance of the higher rational signatures) and the integral K-theoretic assembly isomorphism are known (Yu; Carlsson–Goldfarb; Goldfarb), but the L-theoretic isomorphism needed for topological rigidity is not automatic from these. Higher-dimensional graph manifolds generalize the JSJ (Jaco–Shalen–Johannson) decomposition of 3-manifolds, the canonical splitting of a 3-manifold along embedded tori into geometric pieces: Frigerio, Lafont and Sisto glue pieces of the form (truncated finite-volume hyperbolic manifold) $\times$ torus along boundary tori via affine maps, and prove asphericity, vanishing of lower algebraic K-theory, and the Borel conjecture for this class in dimensions $n \ge 6$ ; Nguyen Phan's cusp-decomposable manifolds and the higher graph manifolds of Connell and Suárez-Serrato (pieces are infra-nilmanifold bundles — an infra-nilmanifold being a compact manifold finitely covered by a quotient of a nilpotent Lie group by a lattice — over truncated negatively curved bases) are adjacent classes. The source of the present question, a note by Bárcenas, Juan-Pineda and Suárez-Serrato, treats a more general family (their Definition 1): pieces are fiber bundles $Z_i \to M_i$ where $M_i$ is obtained by truncating the cusps of a complete finite-volume manifold $V_i$ whose sectional curvature is pinched negative (bounded between two fixed negative constants) and the fiber $N_i$ is a compact quotient of an aspherical simply connected Lie group by a uniform lattice, i.e. a discrete subgroup acting with compact quotient (for example an infra-nilmanifold), and the pieces are glued along a complete pairing of boundary components by arbitrary diffeomorphisms. The glued hypersurfaces are the internal walls; concretely they arise as extensions of the boundary groups of the truncated bases by the uniform lattices acting on the fibres. Two dimension thresholds shape what is known. In dimension 4, topological category results come not from surgery but from Freedman's disk embedding theorem — the technical device that upgrades immersed disks to embedded ones and thereby supports topological homeomorphism constructions in dimension 4 — together with the class of so-called good groups (those for which the disk embedding theorem holds), known to include the elementary amenable ones (the smallest class of groups containing all finite and all abelian groups and closed under extensions and directed unions); whether groups containing nonabelian free subgroups are good is open (the disk embedding conjecture). In dimension 5 surgery applies to the total manifold, but any inductive strategy that deduces rigidity of $M$ from rigidity of its walls and pieces needs rigidity of the 4-dimensional walls. The note's theorems stop at $n \ge 6$ — the range in which every auxiliary manifold entering such an induction has dimension at least 5 — and its fully verified decomposition proof (an extension of the Frigerio–Lafont–Sisto criterion) additionally assumes that all pieces are trivial bundles $Z_i \cong N_i \times M_i$ .

Problem Statement

Consider the family of closed smooth $n$ -manifolds constructed as follows (Definition 1 of Bárcenas–Juan-Pineda–Suárez-Serrato): for $i = 1, \dots, r$ , take a complete finite-volume non-compact $n_i$ -manifold $V_i$ whose sectional curvature is pinched negative (bounded between two fixed negative constants), with $2 \le n_i \le n$ ; truncate the cusps of $V_i$ to obtain a compact manifold with boundary $M_i$ ; let $Z_i \to M_i$ be a fiber bundle whose fiber $N_i$ is a compact quotient of an aspherical simply connected Lie group by a uniform lattice, i.e. a discrete subgroup acting with compact quotient (for example an infra-nilmanifold, a compact manifold finitely covered by a quotient of a nilpotent Lie group by a lattice); finally pair the boundary components of the $Z_i$ completely — each component with exactly one other, belonging to a different piece — and glue each paired couple by an arbitrary diffeomorphism, obtaining a closed $n$ -manifold $M$ . The Borel conjecture for this family is claimed for $n \ge 6$ , and rigorously proved via the decomposition-based criterion only when all pieces are trivial bundles $Z_i \cong N_i \times M_i$ . Settle the remaining cases, in the affirmative or by explicit counterexample: (a) [low dimensions] does every such $M$ of dimension $n = 5$ , respectively $n = 4$ , satisfy the Borel conjecture, i.e. is every homotopy equivalence $f: M \to M'$ from $M$ to a closed aspherical manifold $M'$ homotopic to a homeomorphism (the case $n = 3$ holds by 3-manifold geometrization, so the open cases are $n = 4$ and $n = 5$ ); (b) [non-trivial bundle pieces] give a complete verification of the conjecture for $n \ge 6$ when the pieces $Z_i$ are non-trivial bundles (the piece groups $\pi_1(Z_i)$ are extensions of the hyperbolic-type group $\pi_1(M_i)$ by the lattice group $\pi_1(N_i)$ , and the total group is a graph of groups amalgamated along the internal-wall groups). The common objective of (a) and (b) is topological rigidity of the entire Definition-1 family in every dimension $n \ge 3$ . The three cases at issue are $n = 4$ , $n = 5$ , and non-trivial bundle pieces for $n \ge 6$ ; a counterexample to the Borel conjecture for one of these cases would be two manifolds of the family that are homotopy equivalent but not homeomorphic, or a self-homotopy equivalence of one of them not homotopic to a homeomorphism.

References

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Boris Goldfarb, “Weak coherence of groups and finite decomposition complexity”, preprint arXiv:1307.5345 (2013, revised 2018). https://arxiv.org/abs/1307.5345
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