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We introduce a new operation, double point surgery, on immersed surfaces in a 4‐manifold, and use it to construct knotted configurations of surfaces in many 4‐manifolds. Taking branched covers, we produce smoothly exotic actions of ℤ  m   ⊕ ℤ  n   on simply connected 4‐manifolds with complicated fixed‐point sets.

Double point surgery and configurations of surfaces