The paper constructs completions of the free associative algebra Afr_n with norms ||·||σ to form Banach algebras Afrσ and their inverse-limit Fréchet algebra Afr_. Corresponding completed free Lie algebras fr_σ and fr_∞ (denoted here by fr_) are defined, along with the groups Fr_σ = Fr_n ∩ Afr_σ and Fr_ = Fr_n ∩ Afr_. The groups Fr_σ are shown to be Banach-Lie groups, Polish, and contractible, with neighborhoods of the identity generated by the exponential map.
For Fr_ (the intersection over all σ>0), the paper proves Polishness, contractibility, and density of the subgroup generated by exponentials, but the author explicitly states uncertainty about whether Fr_ carries any standard infinite-dimensional manifold structure. Resolving this would clarify whether Fr_ fits within established frameworks of infinite-dimensional manifolds (e.g., Banach or Fréchet manifolds) used in infinite-dimensional Lie theory.