# Homological properties of parafree Lie algebras

## TL;DR

This paper constructs specific parafree Lie algebras with nontrivial second homology and explores their cohomological dimensions, revealing new properties of these algebraic structures.

## Contribution

It provides explicit examples of parafree Lie algebras with nonzero second homology and analyzes the cohomological dimension of their pronilpotent completions.

## Key findings

- Constructed a countable parafree Lie algebra over Z/2 with nonzero second homology.
- Showed the cohomological dimension of the pronilpotent completion exceeds two.
- Proved the existence of a countable parafree group with nontrivial H2.

## Abstract

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 $H_2$.