Editing Geometric topology (section)

===High-dimensional geometric topology===
In high-dimensional topology, [[characteristic classes]] are a basic invariant, and [[surgery theory]] is a key theory.

A '''[[characteristic class]]''' is a way of associating to each [[principal bundle]] on a [[topological space]] ''X'' a [[cohomology]] class of ''X''. The cohomology class measures the extent to which the bundle is "twisted" &mdash; particularly, whether it possesses [[Section (fiber bundle)|sections]] or not. In other words, characteristic classes are global [[topological invariant|invariant]]s which measure the deviation of a local product structure from a global product structure. They are one of the unifying geometric concepts in [[algebraic topology]], [[differential geometry]] and [[algebraic geometry]].

'''[[Surgery theory]]''' is a collection of techniques used to produce one [[manifold]] from another in a 'controlled' way, introduced by {{harvs|txt|last=[[John Milnor|Milnor]]|year=1961}}. Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along the cut or boundary. This is closely related to, but not identical with, [[handle decomposition|handlebody decomposition]]s. It is a major tool in the study and classification of manifolds of dimension greater than 3.

More technically, the idea is to start with a well-understood manifold ''M'' and perform surgery on it to produce a manifold ''M ''′ having some desired property, in such a way that the effects on the [[homology (mathematics)|homology]], [[homotopy group]]s, or other interesting invariants of the manifold are known.

The classification of [[exotic sphere]]s by {{harvs|txt|author1-link=Michel Kervaire|last=Kervaire|author2-link=John Milnor|last2=Milnor|year=1963}} led to the emergence of surgery theory as a major tool in high-dimensional topology.