# orientation

There are many definitions of an orientation of a manifold^{}. The most
general, in the sense that it doesnβt require any extra
on the manifold, is based on
(co-)homology^{} theory. For this article manifold means a connected,
topological manifold possibly with boundary.

###### Theorem 1.

Let $M$ be a closed, $n$βdimensional
*manifold*. Then ${H}_{n}\mathit{\beta \x81\u2019}\mathrm{(}M\mathrm{;}\mathrm{Z}\mathrm{)}$ the top dimensional
homology group of $M$, is either trivial ($\mathrm{\{}\mathrm{0}\mathrm{\}}$) or isomorphic
to $\mathrm{Z}$.

###### Definition 2.

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

$$\mathrm{\pi \x9d\x94\neg}:\mathrm{\beta \x84\u20ac}\beta \x86\x92{H}_{n}\beta \x81\u2019(M;\mathrm{\beta \x84\u20ac}).$$ |

An oriented manifold is a (necessarily orientable) manifold $M$ endowed with
an orientation.
If $(M,\mathrm{\pi \x9d\x94\neg})$ is an oriented manifold then $\mathrm{\pi \x9d\x94\neg}\beta \x81\u2019(1)$ is called
the *fundamental class ^{}* of $M$ , or the

*orientation class*of $M$, and is denoted by $[M]$.

###### Remark 3.

Notice that since $\mathrm{\beta \x84\u20ac}$ has exactly two
automorphisms^{} an orientable manifold admits two possible
orientations.

###### Remark 4.

The above definition could be given using cohomology^{} 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 $x\mathrm{\beta \x88\x88}M$. Then the relative homology group

$${H}_{n}\beta \x81\u2019(M,M\beta \x88\x96x;\mathrm{\beta \x84\u20ac})\beta \x89\x85\mathrm{\beta \x84\u20ac}$$ |

###### Definition 6.

Let $M$ be an $n$-manifold and $x\beta \x88\x88M$. An orientation of $M$ at $x$ is a choice of an isomorphism

$${\mathrm{\pi \x9d\x94\neg}}_{x}:\mathrm{\beta \x84\u20ac}\beta \x86\x92{H}_{n}\beta \x81\u2019(M,M\beta \x88\x96x;\mathrm{\beta \x84\u20ac}).$$ |

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 homeomorphic^{} to ${\mathrm{R}}^{n}$
(e.g. the domain of a chart). Then,

$${H}_{n}\beta \x81\u2019(M,M\beta \x88\x96U;\mathrm{\beta \x84\u20ac})\beta \x89\x85\mathrm{\beta \x84\u20ac}.$$ |

###### Definition 8.

Let $U$ be an open subset of $M$ that is homeomorphic
to ${\mathrm{\beta \x84\x9d}}^{n}$. A *local orientation* of $M$ on $U$ is a choice
of an isomorphism

$${\mathrm{\pi \x9d\x94\neg}}_{U}:{H}_{n}\beta \x81\u2019(M,M\beta \x88\x96U;\mathrm{\beta \x84\u20ac})\beta \x86\x92\mathrm{\beta \x84\u20ac}.$$ |

Now notice that with $U$ as above and $x\beta \x88\x88U$ the inclusion

$${\mathrm{\Delta \pm}}_{x}^{U}:M\beta \x88\x96U\beta \x86\u037aM\beta \x88\x96x$$ |

a map (actually isomorphism)

$${\mathrm{\Delta \pm}}_{x\beta \x81\pounds *}^{U}:{H}_{n}\beta \x81\u2019(M,M\beta \x88\x96U;\mathrm{\beta \x84\u20ac})\beta \x86\x92{H}_{n}\beta \x81\u2019(M,M\beta \x88\x96x;\mathrm{\beta \x84\u20ac})$$ |

and therefore a local orientation at $U$ (by composing with the above isomorphism) an orientation at each point $x\beta \x88\x88U$. It is to declare that all these orientations fit nicely together.

###### Definition 9.

Let $M$ be a manifold with non-empty boundary, $\beta \x88\x82\beta \x81\u2018M\beta \x89\mathrm{\beta \x88\x85}$. $M$ is called *orientable* if its double

$$\widehat{M}:=M\beta \x81\u2019\underset{\beta \x88\x82\beta \x81\u2018M}{\beta \x8b\x83}M$$ |

is orientable, where ${\beta \x8b\x83}_{\beta \x88\x82\beta \x81\u2018M}$ denotes gluing along the boundary.

An orientation of $M$ is determined by an orientation of $\widehat{M}$.

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 |