Linear topological classification of certain function spaces on manifolds and CW complexes

For a Tychonoff space X, Cp(X) denotes the space of all continuous real-valued functions on X with the pointwise convergent topology. Two Tychonoff spaces X and Y are said to be ℓ-equivalent if Cp(X) and Cp(Y) are linearly homeomorphic. We prove that any compact topological n-manifold is ℓ-equivalent to the n-disk. This complements a result of V. Valov on noncompact manifolds. Further the classification of CW complexes up to ℓ-equivalence, originated by D. Pavlovskiĭ, is completed by classifying noncompact CW complexes up to the above equivalence.