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We demonstrate the existence of numerous non-spin 4-manifolds for which thesmooth Nielsen realization problem fails; namely, there exist finite subgroupsof their mapping class groups that cannot be realized by any group ofdiffeomorphisms. This extends and complements recent results for spin4-manifolds. Our examples span virtually all possible intersection forms, botheven and odd, indefinite and definite, and include many irreducible4-manifolds. To derive these examples, we study multi-twists, projectivetwists, and multi-reflections, which are all mapping classes supported aroundcollections of embedded spheres and projective planes. Our obstructions toNielsen realization are based on the work of Konno. We investigate projectivetwists in further detail, and notably, employ them to show that, for manyclosed symplectic 4-manifolds, the symplectic Torelli group is not generated bysquared Dehn twists.

Nielsen realization in dimension four and projective twists