orientation

There are many definitions of an orientation of a manifold. The mostgeneral, in the sense that it doesn’t 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, $n$–dimensionalmanifold. Then $H_n\mathit{}\mathrm{(}M\mathrm{;}\mathrm{Z}\mathrm{)}$ the top dimensionalhomology group of $M$, is either trivial ($\mathrm{\{}\mathrm{0}\mathrm{\}}$) or isomorphicto $\mathrm{Z}$.

Definition 2.

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

$$𝔬:\mathbb{Z}\to H_n(M;\mathbb{Z}).$$

An oriented manifold is a (necessarily orientable) manifold $M$ endowed withan orientation.If $(M,𝔬)$ is an oriented manifold then $𝔬(1)$ is calledthe fundamental class of $M$ , or the orientation class of $M$, and is denotedby $[M]$.

Remark 3.

Notice that since $\mathbb{Z}$ has exactly twoautomorphisms an orientable manifold admits two possibleorientations.

Remark 4.

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

The top dimensional homology of a non-closed manifold is alwaystrivial, so it is trickier to define orientation for thosebeasts. One approach (which we will not follow) is to use specialkind of homology (for example relative to the boundary for compactmanifolds with boundary). The approach we follow defines (global)orientation as compatible fitting together of local orientations. Westart with manifolds without boundary.

Theorem 5.

Let $M$ be an $n$-manifold without boundary and $x\mathrm{\in }M$. Then the relative homology group

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

Definition 6.

Let $M$ be an $n$-manifold and $x\in M$. An orientationof $M$ at $x$ is a choice of an isomorphism

$${𝔬}_x:\mathbb{Z}\to H_n(M,M\setminus x;\mathbb{Z}).$$

to make precise the notion of nicelyfitting together of orientations at points, is to require that fornearby 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,

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

Definition 8.

Let $U$ be an open subset of $M$ that is homeomorphicto ${ℝ}^n$. A local orientation of $M$ on $U$ is a choiceof an isomorphism

$${𝔬}_U:H_n(M,M\setminus U;\mathbb{Z})\to \mathbb{Z}.$$

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

$${ı}_x^U:M\setminus U\hookrightarrow M\setminus x$$

a map (actually isomorphism)

$${ı}_{x*}^U:H_n(M,M\setminus U;\mathbb{Z})\to H_n(M,M\setminus x;\mathbb{Z})$$

and therefore a local orientation at $U$ (by composing with the aboveisomorphism) an orientation at each point $x\in U$. It is to declare that all theseorientations fit nicely together.

Definition 9.

Let $M$ be a manifold with non-empty boundary, $\partial M\ne \mathrm{\varnothing }$. $M$ is called orientable if its double

$$\widehat{M}:=M\bigcup _{\partial M}M$$

is orientable, where ${\bigcup }_{\partial M}$ denotes gluing along the boundary.
An orientation of $M$ is determined by an orientation of $\widehat{M}$.