The mapping class group of a K3 surface splits, yet an order-two subgroup lifts to homeomorphisms but not to diffeomorphisms, and the fundamental group of the diffeomorphism group does not map onto that of the homeomorphism group.
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A note on the Nielsen realization problem for K3 surfaces
The mapping class group of a K3 surface splits, yet an order-two subgroup lifts to homeomorphisms but not to diffeomorphisms, and the fundamental group of the diffeomorphism group does not map onto that of the homeomorphism group.