Simple connectivity at infinity and the smooth type of the universal Davis covers of the exotic aspherical 4-manifolds

The first pairs of closed, aspherical, smooth 4-manifolds that are homeomorphic but not diffeomorphic — counterexamples to the smooth analogue of the Borel conjecture in dimension four — were constructed by Davis, Hayden, Huang, Ruberman, and Sunukjian by applying the reflection group trick to an exotic pair of compact 4-manifolds $X$ and $X'$ . Their paper leaves two questions open about the universal Davis covers $D(\widetilde{X})$ and $D(\widetilde{X}')$ , where $\widetilde{X}$ and $\widetilde{X}'$ are the universal covers of $X$ and $X'$ : the authors conjecture that both open contractible 4-manifolds fail to be simply connected at infinity and hence are not homeomorphic to $\mathbb{R}^4$ (in particular, neither is an exotic $\mathbb{R}^4$ ), and they ask whether $D(\widetilde{X})$ and $D(\widetilde{X}')$ are diffeomorphic to each other — that is, whether the exotic smooth structure distinguishing the closed quotients persists or disappears after passing to the universal covers. This problem asks for a resolution of both questions.

Background

A closed manifold is called aspherical when its universal cover is contractible; its homotopy type is then determined by its fundamental group. The Borel conjecture predicts that closed aspherical manifolds are topologically rigid — every homotopy equivalence between them is homotopic to a homeomorphism — and its smooth analogue asks whether homotopy-equivalent closed aspherical smooth manifolds must be diffeomorphic. The smooth analogue holds in dimensions at most 2 and for orientable 3-manifolds (by geometrization), and fails in dimensions at least 5; Davis, Hayden, Huang, Ruberman, and Sunukjian settled the remaining case by constructing the first pairs of closed, aspherical, smooth 4-manifolds that are homeomorphic but not diffeomorphic.

Their construction starts from a pair of compact smooth 4-manifolds with boundary, $X$ and $X'$ , each built from the Akbulut cork (a standard contractible compact smooth 4-manifold with boundary) by attaching a thickened once-punctured torus, so that both are homotopy equivalent to the 2-torus $T^2$ ; $X'$ is obtained from $X$ by removing the interior of the cork and regluing it by a boundary involution that extends to a homeomorphism but not to a diffeomorphism (a cork twist), following Hayden and Piccirillo's trace-embedding examples. Freedman's classification theory gives a homeomorphism $X\cong X'$ , while adjunction inequalities from Stein (complex-geometric) structures show that $H_2(X')$ is generated by a smoothly embedded torus whereas every homologically essential smoothly embedded surface in $X$ has genus at least 2.

To produce closed examples, the authors apply Davis's reflection group trick. Given a flag triangulation $T$ of $\partial X$ (a simplicial complex is flag when every set of pairwise-adjacent vertices spans a simplex), the 1-skeleton of $T$ defines a right-angled Coxeter group $W(T)$ : the group generated by one order-two reflection for each vertex of $T$ , with commuting relations along edges. The associated Davis complex $D(X)$ is the noncompact manifold obtained by gluing a countable collection of copies of $X$ and $-X$ along the mirror panels of $\partial X$ according to the Coxeter relations. Quotienting $D(X)$ by a torsion-free finite-index subgroup of $W$ yields closed aspherical smooth 4-manifolds $Q(X)$ and $Q(X')$ that are homeomorphic but not diffeomorphic.

This problem concerns the universal covers of these manifolds, as posed in Remark 4.3 of that paper. Let $\widetilde{X}$ and $\widetilde{X}'$ be the universal covers of $X$ and $X'$ . Lifting the triangulation of $\partial X$ to $\partial\widetilde{X}$ defines another right-angled Coxeter group $\widetilde{W}$ and Davis complexes $D(\widetilde{X})$ and $D(\widetilde{X}')$ , and $D(\widetilde{X})$ is the universal cover of $D(X)$ , hence of the closed aspherical manifold $Q(X)$ ; similarly for $X'$ . Both are open contractible smooth 4-manifolds. A noncompact space is simply connected at infinity when loops can be killed off outside sufficiently large compact subsets: for every compact $K$ there is a larger compact $L$ such that the induced map $\pi_1(M\setminus L)\to\pi_1(M\setminus K)$ is trivial. For open contractible $n$ -manifolds with $n\ge 4$ , simple connectivity at infinity is the precise obstruction to being homeomorphic to $\mathbb{R}^n$ (Stallings for $n\ge 5$ ; Freedman's classification in dimension 4). An open 4-manifold that is homeomorphic but not diffeomorphic to the standard $\mathbb{R}^4$ is called an exotic $\mathbb{R}^4$ .

In Davis's classical construction, when the input is a compact contractible manifold whose boundary is not simply connected, the resulting contractible Davis complex fails to be simply connected at infinity — the mechanism behind the first closed aspherical manifolds not covered by Euclidean space. In the present situation the building blocks $\widetilde{X}$ are themselves simply connected and noncompact, so the classical argument does not decide the end topology. Moreover, the authors' own obstruction distinguishes only the intermediate covers $D(X)$ and $D(X')$ (via the genera of their essential surfaces) and says nothing about the simply connected covers, where all pieces are contractible. The end topology and smooth type of $D(\widetilde{X})$ and $D(\widetilde{X}')$ are therefore open.

Problem Statement

As posed in Remark 4.3 of "Exotic aspherical 4-manifolds" (arXiv:2411.19400), determine the end topology and the smooth type of the universal Davis covers $D(\widetilde{X})$ and $D(\widetilde{X}')$ , where $X$ and $X'$ are the compact smooth 4-manifolds used to construct the first exotic pair of closed aspherical 4-manifolds (each obtained from the Akbulut cork by attaching a thickened once-punctured torus, hence homotopy equivalent to $T^2$ , with $X'$ obtained from $X$ by the cork twist), $\widetilde{X}$ and $\widetilde{X}'$ are their universal covers, and $D(\widetilde{X})$ , $D(\widetilde{X}')$ are the associated Davis complexes — open contractible smooth 4-manifolds that universally cover the closed aspherical manifolds $Q(X)$ and $Q(X')$ . The problem has two linked parts:

(a) The authors conjecture that $D(\widetilde{X})$ and $D(\widetilde{X}')$ are not simply connected at infinity, and hence that neither is homeomorphic to $\mathbb{R}^4$ (so, in particular, neither is an exotic $\mathbb{R}^4$ ). Prove or refute this conjecture.

(b) Determine whether the open contractible manifolds $D(\widetilde{X})$ and $D(\widetilde{X}')$ are diffeomorphic to each other — that is, whether the exotic smooth structure distinguishing the closed quotients $Q(X)$ and $Q(X')$ persists or disappears after passing to the universal Davis covers.

The common objective is a complete determination of the topology at infinity and the diffeomorphism type of these two explicitly constructed manifolds; parts (a) and (b) are facets of that objective.

References

Michael Davis, Kyle Hayden, Jingyin Huang, Daniel Ruberman, Nathan Sunukjian, "Exotic aspherical 4-manifolds", arXiv:2411.19400 (v1 2024, v3 revised 5 May 2026; to appear in Annals of Mathematics), doi:10.48550/arxiv.2411.19400, https://arxiv.org/abs/2411.19400
Michael W. Davis, "Groups generated by reflections and aspherical manifolds not covered by Euclidean space", Annals of Mathematics (2) 117 (1983), 293–324, doi:10.2307/2007079, https://doi.org/10.2307/2007079
Michael Hartley Freedman, "The topology of four-dimensional manifolds", Journal of Differential Geometry 17 (1982), 357–453, doi:10.4310/jdg/1214437136, https://doi.org/10.4310/jdg/1214437136
John Stallings, "The Piecewise-Linear Structure of Euclidean Space", Mathematical Proceedings of the Cambridge Philosophical Society 58 (1962), 481–488, doi:10.1017/s0305004100036756, https://doi.org/10.1017/s0305004100036756
Kyle Hayden, Lisa Piccirillo, "The trace embedding lemma and spinelessness", Journal of Differential Geometry 131 (2025), 133–163, doi:10.4310/jdg/1755543268, https://doi.org/10.4310/jdg/1755543268
Selman Akbulut, Daniel Ruberman, "Absolutely exotic compact 4-manifolds", Commentarii Mathematici Helvetici 91 (2016), 1–19, doi:10.4171/CMH/375, https://doi.org/10.4171/CMH/375
Bakul Sathaye, "Link Obstruction to Riemannian smoothings of locally CAT(0) 4-manifolds", arXiv:1707.03433 (2017), doi:10.48550/arxiv.1707.03433, https://arxiv.org/abs/1707.03433