High Quality Content by WIKIPEDIA articles! In mathematics, the structure set mathcal{S} (X) is the basic object in the study of manifolds which are homotopy equivalent to a closed manifold X. It is a concept which helps to answer the question whether two homotopy equivalent manifolds are diffeomorphic (or PL-homeomorphic or homeomorphic). There are different versions of the structure set depending on the category (DIFF, PL or TOP) and whether Whitehead torsion is taken into account or not.Topological Spheres: The generalized Poincare conjecture in the topological category says that mathcal{S}^s (S^n) only consists of the base point. This conjecture was proved by Smale (n > 4), Freedman (n = 4) and Perelman (n = 3).Exotic Spheres: The classification of exotic spheres by Kervaire and Milnor gives mathcal{S}^s (S^n) = theta_n = pi_n(PL/O) for n > 4 (smooth category).

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