PlanetMath: orientation

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orientation

(Definition)

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

Theorem 1 Let be a closed, -dimensional manifold. Then $H_n(M\,;\mathbb{Z})$ the top dimensional homology group of , is either trivial ( $\{0\}$ ) or isomorphic to $\mathbb{Z}$ .

Definition 2 A closed

-manifold is called orientable if its top homology group is isomorphic to the integers. An orientation of

is a choice of a particular isomorphism

$\displaystyle \mathfrak{o}\colon\thinspace \mathbb{Z} \to H_n(M\,;\mathbb{Z}).$

An oriented manifold is a (necessarily orientable) manifold

endowed with an orientation. If $(M,\mathfrak{o})$ is an oriented manifold then $\mathfrak{o}(1)$ is called the fundamental class of

, or the orientation class of

, and is denoted by

Remark 3 Notice that since $\mathbb{Z}$ 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 be an -manifold without boundary and $x\in M$ . Then the relative homology group

$\displaystyle H_n(M,M\setminus x\,;\mathbb{Z}) \cong \mathbb{Z}$

Definition 6 Let

be an

-manifold and $x\in M$ . An orientation of

is a choice of an isomorphism

$\displaystyle \mathfrak{o}_x \colon\thinspace \mathbb{Z} \to H_n(M,M\setminus x\,;\mathbb{Z}).$

One way 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 uniform way.

Theorem 7 Let be an open subset of that is homeomorphic to $\mathbb{R}^n$ (e.g. the domain of a chart). Then,

$\displaystyle H_n(M,M\setminus U \,; \mathbb{Z})\cong \mathbb{Z}.$

Definition 8 Let

be an open subset of

that is homeomorphic to $\mathbb{R}^n$ . A local orientation of

is a choice of an isomorphism

$\displaystyle \mathfrak{o}_U \colon\thinspace H_n(M,M\setminus U \,; \mathbb{Z}) \to \mathbb{Z}.$

Now notice that with as above and $x\in U$ the inclusion

$\displaystyle \imath^U_x\colon\thinspace M\setminus U \hookrightarrow M\setminus x$

induces a map (actually isomorphism)

$\displaystyle \imath^U_{x\,\,*}\colon\thinspace H_n(M,M\setminus U \,; \mathbb{Z}) \to H_n(M,M\setminus x\,;\mathbb{Z})$

and therefore a local orientation at

induces (by composing with the above isomorphism) an orientation at each point $x\in U$ . It is natural to declare that all these orientations fit nicely together.

Definition 9 Let

be a manifold with non-empty boundary, $\partial M\neq \emptyset$ .

is called orientable if its double

$\displaystyle \hat{M}:=M\bigcup_{\partial M}M$

is orientable, where $\bigcup_{\partial M}$ denotes gluing along the boundary.
An orientation of

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

"orientation" is owned by PrimeFan. [ full author list (3) | owner history (2) ]

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