Topology of manifolds modeled on countable direct limits of Menger compacta

We construct n-dimensional counterparts of manifolds modeled on the space ℓ 2 equipped by the bounded weak topology ( μ n ∞ -manifolds). For μ n ∞ -manifolds we prove the characterization, triangulation and classification theorems. In addition, a universal map of μ n ∞ onto Q ∞ (the countable direct limit of Hilbert cubes and Z-embeddings) is constructed and characterized.