In a recent paper accepted for publication in the Annals of Mathematics, Davis, Hayden, Huang, Ruberman, and Sunukjian proved the existence of exotic pairs of smooth, closed, aspherical -manifolds (Davis et al. 2024).
Let and be topological spaces. Two continuous maps are said to be homotopic if there exists a continuous map such that and for every . A continuous map is a homotopy equivalence if there is a continuous map such that is homotopic to the identity map on , and is homotopic to the identity map on . In this case, is called a homotopy inverse of .
A path-connected topological -manifold is called aspherical if all of its higher homotopy groups vanish. This means that for every integer , every continuous map from the -sphere to is homotopic to a constant map.
A topological manifold can sometimes support more than one smooth structure. We shall call two smooth manifolds and an exotic pair if they are homeomorphic but not diffeomorphic. Equivalently, after identifying and by a homeomorphism, one obtains two distinct smooth structures on the same underlying topological manifold, distinct up to diffeomorphism.
The Borel conjecture predicts that closed aspherical topological manifolds are topologically rigid: if and are closed aspherical topological manifolds and is a homotopy equivalence, then should be homotopic to a homeomorphism. A natural smooth analogue asks whether two closed smooth aspherical -manifolds that are homotopy equivalent must be diffeomorphic.
This smooth analogue is classical in dimensions . In dimension , it is known for orientable manifolds as a consequence of Perelman’s proof of geometrization. In contrast, counterexamples had long been known in every dimension . Thus dimension was the remaining case.
Davis, Hayden, Huang, Ruberman, and Sunukjian resolved this case by proving the following theorem.
Theorem shows that the topological rigidity suggested by asphericity does not extend to the smooth category in dimension . In particular, some closed aspherical topological -manifolds carry more than one smooth structure, up to diffeomorphism.
The word “closed” is important here. Many exotic phenomena in dimension were already known for non-closed manifolds, but Theorem produces exotic smooth structures in the compact, boundaryless setting. Moreover, the construction in (Davis et al. 2024) yields infinitely many such exotic pairs.