arcwise and simply connected sets, then the fundamental group of ${\mathcal{P}}$ is isomorphic to the fundamental group of the underlying topological space (Theorem \[Ada:6\]). This states that *the only possible topological obstruction to the triviality in ${\mathfrak{B}(\mathcal{H}_o)}$ of 1-cocycles is the nonsimply connectedness* (Corollary \[Adb:2\]).\
Before studying superselection sectors in a globally hyperbolic spacetime ${\mathcal{M}}$, we have to point out another problem arising in [@GLRV; @Rob3]. Regular diamonds do not need to have arcwise connected causal complements. This, on the one hand creates some technical problems; on the other hand it is not clear whether it is justified to assume Haag duality on ${\mathscr{A}}_{{\mathcal{K}_{\diamond}}}$: the only known result showing that a net of local observables, in the presence of a nontrivial superselection structure, inherits Haag duality from fields makes use of the arcwise connectedness of causal complements of the elements of the index set [@GLRV Theorem 3.15]. We start to deal with this problem by showing that net-cohomology is invariant under a change of the index set (Theorem \[Adb:4\]), provided the new index set is a refinement of ${\mathcal{K}_{\diamond}}$ (see Definition \[Aca:1\] and Lemma \[Adb:3\]a). In Section \[Bb\] we introduce the set ${\mathcal{K}}$ of diamonds of ${\mathcal{M}}$. ${\mathcal{K}}$ is a 