Universality of infinite exotic smooth structures in dimension four

Determine whether every smoothable topological 4-manifold admits infinitely many pairwise non-diffeomorphic smooth structures, thereby establishing the universality of the phenomenon of infinite exotic smooth structures in dimension four.

Background

In dimension four, unlike other dimensions, a single topological manifold can admit infinitely many non-diffeomorphic smooth structures. A large body of work constructs infinite families of exotic smooth structures on various closed, oriented 4-manifolds, often with constraints on the fundamental group, Euler characteristic, or parity of b2^+.

The present paper extends general existence results to non-spin, closed, oriented 4-manifolds with fundamental group Z_{4k} (k≥1) and to manifolds with fundamental group Z_2 × G for any finite group G, producing infinitely many pairwise non-diffeomorphic smooth structures in these settings. Despite such advances, it remains unknown whether every smoothable topological 4-manifold admits infinitely many smooth structures.