Surgery theory

thumb|A cobordism (W; M, N). In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French bord, giving cobordism) of a manifold. Two manifolds of the same dimension are cobordant if their disjoint union is the boundary of a compact manifold one dimension higher.

right|thumb|A genus three handlebody. In the mathematical field of geometric topology, a handlebody is a decomposition of a manifold into standard pieces. Handlebodies play an important role in Morse theory, cobordism theory and the surgery theory of high-dimensional manifolds. Handles are used to particularly study 3-manifolds.

smooth manifold that is homeomorphic but not diffeomorphic to a sphere

techniques in topology used to produce one finite-dimensional manifold from another

Mathematical theories

In geometric topology and differential topology, an (n + 1)-dimensional cobordism W between n-dimensional manifolds M and N is an '''h-cobordism' (the h'' stands for homotopy equivalence) if the inclusion maps

Cohomology class of topological structures