Since the 1980s, the study of invariants of 3-dimensional manifolds has benefited from the connections between topology, physics and number theory. Recently, a new topological invariant has been discovered: the homological block (also known as the half-index of certain 3d N=2 theories). When the 3-manifold is a Seifert manifold given by a negative-definite plumbing the homological block turned out to be related to false theta functions and characters of logarithmic VOA's. In this talk I describe the role of quantum modular forms, false and mock theta functions in the study of the topology of 3-manifolds. The talk is based on the article 1809.10148 and work in progress with Cheng, Chun, Feigin, Gukov, and Harrison.