Surgery structure set

In mathematics, the surgery structure set formula_1 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 formula_2 from closed manifolds formula_3 of dimension formula_4 to formula_5 (formula_6) equivalent if there exists a cobordism formula_7 together with a map formula_8 such that formula_9, formula_10 and formula_11 are homotopy equivalences.
The structure set formula_12 is the set of equivalence classes of homotopy equivalences formula_13 from closed manifolds of dimension n to X.
This set has a preferred base point: formula_14.
There is also a version which takes Whitehead torsion into account. If we require in the definition above the homotopy equivalences F, formula_10 and formula_11 to be simple homotopy equivalences then we obtain the simple structure set formula_17.
Remarks.
Notice that formula_18 in the definition of formula_12 resp. formula_17 is an h-cobordism resp. an s-cobordism. Using the s-cobordism theorem we obtain another description for the simple structure set formula_17, provided that n>4: The simple structure set formula_17 is the set of equivalence classes of homotopy equivalences formula_13 from closed manifolds formula_24 of dimension n to X with respect to the following equivalence relation. Two homotopy equivalences formula_2 (i=0,1) are equivalent if there exists a
diffeomorphism (or PL-homeomorphism or homeomorphism) formula_26 such that formula_27 is homotopic to formula_10.
As long as we are dealing with differential manifolds, there is in general no canonical group structure on formula_17. If we deal with topological manifolds, it is possible to endow formula_17 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 formula_31 whose equivalence class is the base point in formula_17. Some care is necessary because it may be possible that a given simple homotopy equivalence formula_31 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 formula_17.
The basic tool to compute the simple structure set is the surgery exact sequence.
Examples.
Topological Spheres: The generalized Poincaré conjecture in the topological category says that formula_35 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 formula_36 for n > 4 (smooth category).