We explain how a version of Floer homology can be used as an invariant of symplectic manifolds with b_1>0. As a concrete example, we look at four-manifolds produced from braids by a surgery construction. The outcome shows that the invariant is nontrivial; however, it is an open question whether it is stronger than the known ones.
SEIDEL, PAUL (2002) "Braids and symplectic four-manifolds with abelian fundamental group," Turkish Journal of Mathematics: Vol. 26: No. 1, Article 7. Available at: https://journals.tubitak.gov.tr/math/vol26/iss1/7