Abstract
The aim of this part is to show the reader the classification of analytic structures on topological manifolds. This discussion should show that passing from an analytic structure \(\mathcal {C}\) to a more regular one \(\mathcal {C}^{\prime }\) implies two things: (1) the passage is not ever possible, i.e. there are obstructions, (2) even if the passage is possible, the resulting structures \(\mathcal {C}^{\prime }\)might be not\(\mathcal {C}^{\prime }\)-equivalent. The reader is invited to appreciate Sullivan result (Sullivan and Sullivan, Geometric topology. Localisation. Periodicity and Galois Symmetry. The 1970 MIT Notes. Ed. A. Ranicki) which states that in dimension ≠ 4 the passage from the category of topological manifolds to quasi-conformal/Lipschitz manifolds is always possible, in a unique way.
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