There are many definitions of an orientation of a manifoldMathworldPlanetmath. The most general, in the sense that it doesn’t require any extra on the manifold, is based on (co-)homologyMathworldPlanetmathPlanetmath theory. For this article manifold means a connected, topological manifold possibly with boundary.

Theorem 1.

Let M be a closed, n–dimensional manifold. Then Hn⁒(M;Z) the top dimensional homology group of M, is either trivial ({0}) or isomorphic to Z.

Definition 2.

A closed n–manifold is called orientable if its top homology group is isomorphic to the integers. An orientation of M is a choice of a particular isomorphism


An oriented manifold is a (necessarily orientable) manifold M endowed with an orientation. If (M,𝔬) is an oriented manifold then 𝔬⁒(1) is called the fundamental classMathworldPlanetmath of M , or the orientation class of M, and is denoted by [M].

Remark 3.

Notice that since β„€ has exactly two automorphismsPlanetmathPlanetmathPlanetmath an orientable manifold admits two possible orientations.

Remark 4.

The above definition could be given using cohomologyMathworldPlanetmath instead of homology.

The top dimensional homology of a non-closed manifold is always trivial, so it is trickier to define orientation for those beasts. One approach (which we will not follow) is to use special kind of homology (for example relative to the boundary for compact manifolds with boundary). The approach we follow defines (global) orientation as compatible fitting together of local orientations. We start with manifolds without boundary.

Theorem 5.

Let M be an n-manifold without boundary and x∈M. Then the relative homology group

Definition 6.

Let M be an n-manifold and x∈M. An orientation of M at x is a choice of an isomorphism


to make precise the notion of nicely fitting together of orientations at points, is to require that for nearby points the orientations are defined in a way.

Theorem 7.

Let U be an open subset of M that is homeomorphicMathworldPlanetmath to Rn (e.g. the domain of a chart). Then,

Definition 8.

Let U be an open subset of M that is homeomorphic to ℝn. A local orientation of M on U is a choice of an isomorphism


Now notice that with U as above and x∈U the inclusion


a map (actually isomorphism)


and therefore a local orientation at U (by composing with the above isomorphism) an orientation at each point x∈U. It is to declare that all these orientations fit nicely together.

Definition 9.

Let M be a manifold with non-empty boundary, βˆ‚β‘Mβ‰ βˆ…. M is called orientable if its double


is orientable, where β‹ƒβˆ‚β‘M denotes gluing along the boundary.
An orientation of M is determined by an orientation of M^.

Title orientation
Canonical name Orientation1
Date of creation 2013-03-22 12:56:24
Last modified on 2013-03-22 12:56:24
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 15
Author PrimeFan (13766)
Entry type Definition
Classification msc 57N99
Related topic ThomClass
Defines orientable
Defines oriented
Defines orientable manifold
Defines oriented manifold
Defines fundamental class
Defines orientation class
Defines local orientation