Surgery structure set
In mathematics, the surgery structure set 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. Definition Let X be a closed smooth (or PL- or topological) manifold of dimension n. We call two homotopy equivalences from closed manifolds of dimension to () equivalent if there exists a cobordism together with a map such that , and are homotopy equivalences. The structure set is the set of equivalence classes of homotopy equivalences from closed manifolds of dimension n to X. This set has a preferred base point: . There is also a version which takes Whitehead torsion into account. If we require in the definition above the homotopy equivalences F, and to be simple homotopy equivalences then we obtain the simple structure set . Remarks Notice that in the definition of resp. is an h-cobordism resp. an s-cobordism. Using the s-cobordism theorem we obtain another