  In this article, we give new means of constructing and distinguishing closed
exotic four-manifolds. Using Heegaard Floer homology, we define new closed
four-manifold invariants that are distinct from the Seiberg--Witten and
Bauer--Furuta invariants and can remain distinct in covers. Our constructions
include exotic definite manifolds with fundamental group $\mathbb Z/2$,
infinite families of exotic manifolds that are related by knot surgeries on
Alexander polynomial 1 knots, and exotic manifolds that contain square-zero
spheres.
