any 1-cocycle is trivial in ${\mathfrak{B}(\mathcal{H}_o)}$, but this might not be hold when ${\mathcal{K}_{\diamond}}$ is not directed. The consequences can be easily showed: let $\widehat{\otimes}$ be the tensor product introduced in [@GLRV]: for any $z,z_1\in{\mathcal{Z}^1_t({\mathscr{A}}_{{\mathcal{K}_{\diamond}}})}$, it turns out that $z\widehat{\otimes} z_1$ is a 1-cocycle of ${\mathcal{K}_{\diamond}}$ with values in ${\mathscr{A}}_{{\mathcal{K}_{\diamond}}}$, but it is not clear whether it is trivial in ${\mathfrak{B}(\mathcal{H}_o)}$ (we will see in Remark \[Cc:5a\] that this 1-cocycle is trivial in ${\mathfrak{B}(\mathcal{H}_o)}$). Now, we know that the nonsimply connectedness and the compactness of the Cauchy surfaces are topological obstructions to the directness of the index sets. The first aim of this paper is to understand whether these conditions are also obstructions to the triviality in ${\mathfrak{B}(\mathcal{H}_o)}$ of 1-cocycles. This problem is analyzed in great generality in Section \[A\]. We introduce the notions of the first homotopy group and fundamental group for an abstract poset ${\mathcal{P}}$ (Definition \[Ab:12\]) and prove that any 1-cocycle $z$ of ${\mathcal{P}}$, with values in a net of local algebras ${\mathscr{A}}_{\mathcal{P}}$ indexed by ${\mathcal{P}}$, defines a unitary representation of the fundamental group of ${\mathcal{P}}$ (Theorem \[Ac:4\]). In the case that ${\mathcal{P}}$ is a basis for a topological space ordered under inclusion, and whose elements are 