Embedded surfaces and the intersection forms of non-simply connected 4-manifolds (Q2707099)

From the introduction and summary: When the celebrated work of Freedman at the beginning of the eighties showed that simply-connected smooth 4-manifolds are classified up to homeomorphism by their intersection forms, the problem to determine all bilinear forms which can be realized as the intersection forms of smooth 4-manifolds became one of the central topics in 4-dimensional topology.NEWLINENEWLINENEWLINEIt had been understood early on that this problem is intimately related to another classical question in the theory of 4-manifolds, namely to the question of determining the minimal genus of an embedded surface representing a given homology class.NEWLINENEWLINENEWLINEThis thesis is devoted to some aspects of these two fundamental problems and their relation, in particular to the sometimes neglected rôle of the fundamental group. On the one hand, we will see that the presence of certain fundamental groups imposes additional constraints on the intersection form, such as inequalities between the signature and the Betti numbers which are considerably stronger than in the simply-connected case. By the correspondence indicated above, this in turn can be used to study embeddings of surfaces with additional assumptions on the fundamental group of the complement. On the other hand, there are bilinear forms which cannot be realized as the intersection forms of simply-connected smooth 4-manifolds, but appear as the intersection forms of 4-manifolds with non-trivial fundamental group, a fact which is again related to the existence of embedded or immersed surfaces with special properties. The main objective of this thesis is to investigate these connections between the fundamental group of a 4-manifold, its intersection form and embeddings of surfaces.