A new construction of symplectic manifolds


In each even dimension $\ge 4$, families of compact, symplectic manifolds are constructed, such that all finitely presentable groups occur as fundamental groups. For each group, these manifolds can be assumed not to be homotopy equivalent to Kähler manifolds. Other examples are constructed that are homeomorphic, but not diffeomorphic, to simply connected Kähler surfaces. The geography of compact, symplectic $4$-manifolds is studied, with a result that for any fixed fundamental group it is possible to realize any value of the first Chern number $\mathrm{c}_1^2$ or the signature. Explicit results are obtained about simultaneously realizing both Chern numbers (or equivalently, the signature and Euler characteristic), for any fixed fundamental group. Various other applications are presented.


Robert E. Gompf