The classifying space for commutativity was defined by Adem, Cohen and Torres-Giese back to 2012. This space is a modification of the usual classifying space that encodes the commutative structure of the group, and it comes with a comparison map to the usual clasifiying space. In this talk I will discuss recent work on the homotopy type of the homotopy fibre of the comparison map in the case that the initial group is the fundamental group of a 3-manifold. In order to understand the homotopy type we use the relations of the fundamental group in some algebraic configurations and some techniques of geometric group theory applied to these configurations and to a specific model of the fiber that involves the structure of abelian subgroups of the fundamental group of the 3-manifold. This is a joint work with O. Antolín and L. J. Saldaña.