Editing Geometric topology (section)

== Important tools in geometric topology ==
{{Main|List of geometric topology topics}}

===Fundamental group===
{{main|Fundamental group}}
In all dimensions, the [[fundamental group]] of a manifold is a very important invariant, and determines much of the structure; in dimensions 1, 2 and 3, the possible fundamental groups are restricted, while in dimension 4 and above every [[finitely presented group]] is the fundamental group of a manifold (note that it is sufficient to show this for 4- and 5-dimensional manifolds, and then to take products with spheres to get higher ones).

===Orientability===
{{main|Orientability}}
A manifold is orientable if it has a consistent choice of [[orientation (mathematics)|orientation]], and a [[connected space|connected]] orientable manifold has exactly two different possible orientations.  In this setting, various equivalent formulations of orientability can be given, depending on the desired application and level of generality.  Formulations applicable to general topological manifolds often employ methods of [[homology theory]], whereas for [[differentiable manifolds]] more structure is present, allowing a formulation in terms of [[differential form]]s.  An important generalization of the notion of orientability of a space is that of orientability of a family of spaces parameterized by some other space (a [[fiber bundle]]) for which an orientation must be selected in each of the spaces which varies continuously with respect to changes in the parameter values.

===Handle decompositions===
{{main|Handle decomposition}}
[[Image:Sphere with three handles.png|right|thumb|A 3-ball with three 1-handles attached.]]
A [[handle decomposition]] of an ''m''-[[manifold]] ''M'' is a union

:<math>\emptyset = M_{-1} \subset M_0 \subset M_1 \subset M_2 \subset \dots \subset M_{m-1} \subset M_m = M</math>

where each <math>M_i</math> is obtained from <math>M_{i-1}</math>
by the attaching of <math>i</math>-'''handles'''.   A handle decomposition is to a manifold what a [[CW complex|CW-decomposition]] is to a topological space—in many regards the purpose of a handle decomposition is to have a language analogous to CW-complexes, but adapted to the world of [[smooth manifold]]s.  Thus an ''i''-handle is the smooth analogue of an ''i''-cell.  Handle decompositions of manifolds arise naturally via [[Morse theory]]. The modification of handle structures is closely linked to [[Cerf theory]].

===Local flatness===
{{main|Local flatness}}
[[Local flatness]] is a property of a [[submanifold]] in a [[topological manifold]] of larger [[dimension]]. In the [[Category (mathematics)|category]] of topological manifolds, locally flat submanifolds play a role similar to that of [[Submanifold#Embedded submanifolds|embedded submanifolds]] in the category of [[smooth manifolds]].

Suppose a ''d'' dimensional manifold ''N'' is embedded into an ''n'' dimensional manifold ''M'' (where ''d'' &lt; ''n'').  If <math>x \in N,</math> we say ''N'' is '''locally flat''' at ''x'' if there is a neighborhood <math> U \subset M</math> of ''x'' such that the [[topological pair]] <math>(U, U\cap N)</math> is [[homeomorphic]] to the pair <math>(\mathbb{R}^n,\mathbb{R}^d)</math>, with a standard inclusion of <math>\mathbb{R}^d</math> as a subspace of <math>\mathbb{R}^n</math>. That is, there exists a homeomorphism <math>U\to R^n</math> such that the [[image (mathematics)|image]] of <math>U\cap N</math> coincides with <math>\mathbb{R}^d</math>.

===Schönflies theorems===
{{main|Jordan-Schönflies theorem}}
The generalized [[Schoenflies theorem]] states that, if an (''n''&nbsp;&minus;&nbsp;1)-dimensional [[sphere]] ''S'' is embedded into the ''n''-dimensional sphere ''S<sup>n</sup>'' in a [[locally flat]] way (that is, the embedding extends to that of a thickened sphere), then the pair (''S<sup>n</sup>'',&nbsp;''S'') is homeomorphic to the pair (''S<sup>n</sup>'', ''S''<sup>''n''&minus;1</sup>), where ''S''<sup>''n''&minus;1</sup> is the equator of the ''n''-sphere.  Brown and Mazur received the [[Veblen Prize]] for their independent proofs<ref>[[Morton Brown|Brown, Morton]] (1960), A proof of the generalized Schoenflies theorem. ''Bull. Amer. Math. Soc.'', vol. 66, pp.&nbsp;74&ndash;76. {{MR|0117695}}</ref><ref>Mazur, Barry, On embeddings of spheres., ''Bull. Amer. Math. Soc.'' 65 1959 59&ndash;65. {{MR|0117693}}
</ref> of this theorem.