Editing Geometric topology (section)

==Branches of geometric topology==

===Low-dimensional topology===
{{main|Low-dimensional topology}}
[[Low-dimensional topology]] includes:
* [[Surface (topology)|Surfaces]] (2-manifolds)
* [[3-manifold]]s
* [[4-manifold]]s
each have their own theory, where there are some connections.

Low-dimensional topology is strongly geometric, as reflected in the [[uniformization theorem]] in 2 dimensions – every surface admits a constant curvature metric; geometrically, it has one of 3 possible geometries: positive curvature/spherical, zero curvature/flat, negative curvature/hyperbolic – and the [[geometrization conjecture]] (now theorem) in 3 dimensions – every 3-manifold can be cut into pieces, each of which has one of 8 possible geometries.

2-dimensional topology can be studied as [[complex geometry]] in one variable ([[Riemann surface]]s are complex curves) – by the uniformization theorem every conformal class of metrics is equivalent to a unique complex one, and 4-dimensional topology can be studied from the point of view of complex geometry in two variables (complex surfaces), though not every 4-manifold admits a complex structure.

===Knot theory===
{{main|Knot theory}}
[[Knot theory]] is the study of [[knot (mathematics)|mathematical knot]]s. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In mathematical language, a knot is an [[embedding]] of a [[circle]] in 3-dimensional [[Euclidean space]], '''R'''<sup>3</sup> (since we're using topology, a circle isn't bound to the classical geometric concept, but to all of its [[homeomorphism]]s). Two mathematical knots are equivalent if one can be transformed into the other via a deformation of '''R'''<sup>3</sup> upon itself (known as an [[ambient isotopy]]); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.

To gain further insight, mathematicians have generalized the knot concept in several ways. Knots can be considered in other [[3-manifold|three-dimensional spaces]] and objects other than circles can be used; see ''[[knot (mathematics)]]''. Higher-dimensional knots are [[n-sphere|''n''-dimensional spheres]] in ''m''-dimensional Euclidean space.

===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.