Constructions of stable generalized complex 6-manifolds and their fundamental groups

Torus surgeries in dimension four (or called $C^\infty$-log transformations) have been widely employed to construct a stable generalized complex 4-manifold with nonempty type change locus. We find a torus surgery in dimension six which can be applied to a stable generalized 6-complex manifold to yield a new stable generalized complex 6-manifold. Each torus surgery has an effect of increasing the number of path-connected components on the type change locus by one as in dimension four. Using this torus surgery, we prove that there exist countably infinite stable generalized complex 6-manifolds with nonempty type change locus that are not homologically equivalent to a product of lower dimensional manifolds. Also, it is shown that any finitely presented group is the fundamental group of a stable generalized complex 6-manifold with nonempty type change locus on which each path-connected component is diffeomorphic to $T^4$.