Luttinger surgery is used to produce minimal symplectic 4-
manifolds with small Euler characteristics. We construct a minimal
symplectic 4-manifold which is homeomorphic but not diffeomorphic
to CP2#3CP
2
, and which contains a genus two symplectic
surface with trivial normal bundle and simply-connected complement.
We also construct a minimal symplectic 4-manifold which is
homeomorphic but not diffeomorphic to 3CP2#5CP
2
, and which
contains two disjoint essential Lagrangian tori such that the complement
of the union of the tori is simply-connected.
These examples are used to construct minimal symplectic manifolds
with Euler characteristic 6 and fundamental group Z, Z3, or
Z/p ⊕ Z/q ⊕ Z/r for integers p, q, r. Given a group G presented
with g generators and r relations, a symplectic 4-manifold with
fundamental group G and Euler characteristic 10 + 6(g + r) is
constructed.