In a paper (Kent and Leininger 2024) recently accepted for publication in the Annals of Mathematics, Kent and Leininger constructed the first examples of compact atoroidal surface bundles over surfaces.
One natural way to build higher-dimensional manifolds from surfaces is to let one surface vary continuously over another. Let and be closed surfaces. A surface bundle over with fiber is a surjective continuous map where is a topological -manifold, such that every point has an open neighbourhood for which there is a homeomorphism satisfying on . Here has the product topology, and is the projection . Thus, although the total space may be globally complicated, it looks locally like a product of an open set in with the surface .
For , the set is called the fiber over , and every fiber is homeomorphic to . The manifold is called the total space, and is called the base. Such a bundle is often denoted by The simplest example is the product bundle , with projection
Let denote the group with componentwise addition. If is a path-connected manifold, let denote its fundamental group. Since the fundamental group of the torus is isomorphic to , the presence of a subgroup isomorphic to in may be viewed as an algebraic trace of a torus. In the present context, a manifold is called atoroidal if its fundamental group contains no subgroup isomorphic to .
Many familiar surface bundles over surfaces are not atoroidal. For instance, if both and have positive genus, then the product bundle has fundamental group which contains a copy of . Thus it is natural, but far from obvious, to ask whether atoroidal surface bundles exist in the genuinely surface-by-surface case, where both and are closed orientable surfaces of genus at least .
Kent and Leininger answered this question affirmatively in their paper (Kent and Leininger 2024), constructing the first such examples.
Theorem shows that surface bundles over surfaces can have much subtler global topology than product bundles. Locally, the total space still looks like a product of two surfaces, but globally its fundamental group avoids even the basic rank-two abelian subgroup . In this sense, Kent and Leininger’s examples demonstrate that the local product structure of a surface bundle imposes far fewer restrictions on the large-scale algebraic topology of the total space than one might first expect.