In mathematics, the surgery structure set is the basic object in the study of manifoldss 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

) equivalent if there exists a cobordism are homotopy equivalences. The structure set.

to be simple homotopy equivalences then we obtain the simple structure set.

Remarks

is an h-cobordism resp. an s-cobordism. Using the s-cobordism theorem.

with a preferred structure of an abelian group (see chapter 18 in the book of Ranicki).

.

The basic tool to compute the simple structure set is the surgery exact sequence.

Examples

Topological Spheres: The generalized Poincaré conjecture 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 for n > 4 (smooth category).

References

External links

• Andrew Ranicki's homepage

• Shmuel Weinberger's homepage