Homological properties of parafree Lie algebras

In this paper, an explicit construction of a countable parafree Lie algebra over $\mathbb Z/2$ with nonzero second homology is given. It is also shown that the cohomological dimension of the pronilpotent completion of a free noncyclic finitely generated Lie algebra over $\mathbb Z$ is greater than two. Moreover, it is proven that there exists a countable parafree group with nontrivial $H_2$.