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Given a symplectic manifold ( M , ω ) and a Lagrangian submanifold L , we construct versions of the symplectic blow-up and blow-down which are defined relative to L . We further show that if M admits an anti-symplectic involution ϕ , i.e., a diffeomorphism such that ϕ 2 = I d and ϕ ω = ω , and we blow-up an appropriately symmetric embedding of symplectic balls, then there exists an antisymplectic involution on the blow-up ˜ M as well. We then derive a homological condition for real Lagrangian surfaces L = F i x ( ϕ ) which determines when the topology of L changes after a blowdown, and we use these constructions to study the relative packing numbers and packing stability for real symplectic four manifolds which are non-Seiberg-Witten simple.

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