Does every four-manifold admit either no smooth structure or infinitely many of them? I'll report on recent work, joint with A. Stipsicz and Z. Szabó, addressing this question for four-manifolds with finite cyclic fundamental groups. A bonus discussion may feature fake projective planes in the context of yet another dichotomy.
Links:
[1] https://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/de/node/3444
[3] https://www.mpim-bonn.mpg.de/de/node/12018