Do exotic symmetries of 4-manifolds survive stabilisation?

In 4-manifold topology, differences between the smooth and topological categories often “dissolve” after stabilisation by connected sum with enough copies of S²xS². I will discuss recent joint work exploring whether this holds for the mapping class group of a 4-manifold: if two self-diffeomorphisms are topologically isotopic, are they always smoothly isotopic, after stabilisation? We produce general conditions on the fundamental group that guarantee the answer is indeed “yes”. On the other hand, by weakening the initial hypothesis to topologically pseudo-isotopic, we produce examples where the answer is “no”. This is joint with Mark Powell and Oscar Randal-Williams.