In mathematics, the surgery structure set 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.
Let X be a closed smooth (or PL- or topological) manifold of dimension n. We call two homotopy equivalences f i : M i → X from closed manifolds M i of dimension n to X ( i = 0 , 1 ) equivalent if there exists a cobordism ( W ; M 0 , M 1 ) together with a map ( F ; f 0 , f 1 ) : ( W ; M 0 , M 1 ) → ( X × [ 0 , 1 ] ; X × { 0 } , X × { 1 } ) such that F , f 0 and f 1 are homotopy equivalences. The structure set S h ( X ) is the set of equivalence classes of homotopy equivalences f : M → X from closed manifolds of dimension n to X. This set has a preferred base point: i d : X → X .
There is also a version which takes Whitehead torsion into account. If we require in the definition above the homotopy equivalences F, f 0 and f 1 to be simple homotopy equivalences then we obtain the simple structure set S s ( X ) .
Notice that ( W ; M 0 , M 1 ) in the definition of S h ( X ) resp. S s ( X ) is an h-cobordism resp. an s-cobordism. Using the s-cobordism theorem we obtain another description for the simple structure set S s ( X ) , provided that n>4: The simple structure set S s ( X ) is the set of equivalence classes of homotopy equivalences f : M → X from closed manifolds M of dimension n to X with respect to the following equivalence relation. Two homotopy equivalences f i : M i → X (i=0,1) are equivalent if there exists a diffeomorphism (or PL-homeomorphism or homeomorphism) g : M 0 → M 1 such that f 1 ∘ g is homotopic to f 0 .
As long as we are dealing with differential manifolds, there is in general no canonical group structure on S s ( X ) . If we deal with topological manifolds, it is possible to endow S s ( X ) with a preferred structure of an abelian group (see chapter 18 in the book of Ranicki).
Notice that a manifold M is diffeomorphic (or PL-homeomorphic or homeomorphic) to a closed manifold X if and only if there exists a simple homotopy equivalence ϕ : M → X whose equivalence class is the base point in S s ( X ) . Some care is necessary because it may be possible that a given simple homotopy equivalence ϕ : M → X is not homotopic to a diffeomorphism (or PL-homeomorphism or homeomorphism) although M and X are diffeomorphic (or PL-homeomorphic or homeomorphic). Therefore, it is also necessary to study the operation of the group of homotopy classes of simple self-equivalences of X on S s ( X ) .
The basic tool to compute the simple structure set is the surgery exact sequence.
Topological Spheres: The generalized Poincaré conjecture in the topological category says that 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 S s ( S n ) = θ n = π n ( P L / O ) for n > 4 (smooth category).
