two well known topological conditions on the spacetime, implying that not only regular diamonds but any reasonable set of indices for a net of local algebras is not directed: this happens when the spacetime is either nonsimply connected or has compact Cauchy surfaces (Corollary \[Ada:7\] and Lemma \[Ba:2\]). There arises, therefore, the necessity to understand the connection between net-cohomology and topology of the underlying spacetime. Progress in this direction has been achieved in [@Rob3]. The homotopy of paths, the net-cohomology under a change of the index set are issues developed in that work that will turn out to be fundamental for our aim. Moreover, it has been shown that the statistics of sectors can be classified provided that the net satisfies punctured Haag duality (see Section \[C\]). However, no result concerning the conjugation has been achieved.\
To see what is the main drawback caused by the non directness of the poset ${\mathcal{K}_{\diamond}}$, we have to describe more in detail ${\mathcal{Z}^1_t({\mathscr{A}}_{{\mathcal{K}_{\diamond}}})}$. The elements $z$ of ${\mathcal{Z}^1_t({\mathscr{A}}_{{\mathcal{K}_{\diamond}}})}$ are 1-cocycles trivial in ${\mathfrak{B}(\mathcal{H}_o)}$ or, equivalently, path-independent on ${\mathcal{K}_{\diamond}}$. The latter means that the evaluation of $z$ on a path of ${\mathcal{K}_{\diamond}}$ depends only on the endpoints of the path. When ${\mathcal{K}_{\diamond}}$ is directed 