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Version 7 |
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There are many definitions of an orientation of a manifold. The most general, in the sense that it doesn't require any extra \PMlinkescapetext{structure} on the manifold, is based on
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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
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| (co-)homology theory. For this article manifold means a connected, {\em topological} manifold possibly |
(co-)homology theory. For this article manifold means a connected, {\em topological} manifold possibly |
| with boundary. |
with boundary. |
| \begin{thm} Let $M$ be a closed, $n$--dimensional manifold. Then $H_n(M\,;\mathbb{Z})$ the |
\begin{thm} Let $M$ be a closed, $n$--dimensional manifold. Then $H_n(M\,;\mathbb{Z})$ the |
| top dimensional homology group of $M$, is either trivial |
top dimensional homology group of $M$, is either trivial |
| ($\{0\}$) or isomorphic to $ \mathbb{Z}$. |
($\{0\}$) or isomorphic to $ \mathbb{Z}$. |
| \end{thm} |
\end{thm} |
| \begin{defn} A closed $n$--manifold is called orientable if its top |
\begin{defn} A closed $n$--manifold is called orientable if its top |
| homology group is isomorphic to the integers.\newline |
homology group is isomorphic to the integers.\newline |
| An orientation of $M$ is a choice of a particular isomorphism |
An orientation of $M$ is a choice of a particular isomorphism |
| $$\mathfrak{o}\co \mathbb{Z} \to H_n(M\,;\mathbb{Z}).$$ |
$$\mathfrak{o}\co \mathbb{Z} \to H_n(M\,;\mathbb{Z}).$$ |
| An oriented manifold is a (necessarily orientable) manifold $M$ endowed with |
An oriented manifold is a (necessarily orientable) manifold $M$ endowed with |
| an orientation. \newline |
an orientation. \newline |
| If $(M,\mathfrak{o})$ is an oriented manifold then $\mathfrak{o}(1)$ is called |
If $(M,\mathfrak{o})$ is an oriented manifold then $\mathfrak{o}(1)$ is called |
| the fundamental class of $M$ , or the orientation class of $M$, and is denoted |
the fundamental class of $M$ , or the orientation class of $M$, and is denoted |
| by $[M]$. |
by $[M]$. |
| \end{defn} |
\end{defn} |
| \begin{rem} Notice that since $\mathbb{Z}$ has exactly two automorphisms an orientable manifold admits two possible orientations. |
\begin{rem} Notice that since $\mathbb{Z}$ has exactly two automorphisms an orientable manifold admits two possible orientations. |
| \end{rem} |
\end{rem} |
| \begin{rem} The above definition could be given using cohomology instead of homology. |
\begin{rem} The above definition could be given using cohomology instead of homology. |
| \end{rem} |
\end{rem} |
| 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 |
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 |
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 |
(global) orientation as compatible fitting together of local orientations. We start with |
| manifolds without boundary. |
manifolds without boundary. |
| \begin{thm}Let $M$ be an $n$-manifold without boundary and $x\in M$. Then the relative homology |
\begin{thm}Let $M$ be an $n$-manifold without boundary and $x\in M$. Then the relative homology |
| group |
group |
| $$H_n(M,M\setminus x\,;\mathbb{Z}) \cong \mathbb{Z}$$ |
$$H_n(M,M\setminus x\,;\mathbb{Z}) \cong \mathbb{Z}$$ |
| \end{thm} |
\end{thm} |
| \begin{defn} Let $M$ be an $n$-manifold and $x\in M$. An {\em orientation |
\begin{defn} Let $M$ be an $n$-manifold and $x\in M$. An {\em orientation |
| of} $M$ {\em at} $x$ is a choice of an isomorphism |
of} $M$ {\em at} $x$ is a choice of an isomorphism |
| $$\mathfrak{o}_x \co \mathbb{Z} \to H_n(M,M\setminus x\,;\mathbb{Z}).$$ |
$$\mathfrak{o}_x \co \mathbb{Z} \to H_n(M,M\setminus x\,;\mathbb{Z}).$$ |
| \end{defn} |
\end{defn} |
|
\PMlinkescapetext{One way} to make precise the notion of nicely fitting together of orientations at points, is to
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One way to make precise the notion of nicely fitting together of orientations at points, is to
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require that for nearby points the orientations are defined in a \PMlinkescapetext{uniform} way.
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require that for nearby points the orientations are defined in a uniform way.
|
| \begin{thm} |
\begin{thm} |
| Let $U$ be an open subset of $M$ that is homeomorphic to $\BR^n$ (e.g. the domain of a chart). |
Let $U$ be an open subset of $M$ that is homeomorphic to $\BR^n$ (e.g. the domain of a chart). |
| Then, |
Then, |
| $$H_n(M,M\setminus U \,; \BZ)\cong \BZ.$$ |
$$H_n(M,M\setminus U \,; \BZ)\cong \BZ.$$ |
| \end{thm} |
\end{thm} |
| \begin{defn} Let $U$ be an open subset of $M$ that is homeomorphic to $\BR^n$. A local orientation |
\begin{defn} Let $U$ be an open subset of $M$ that is homeomorphic to $\BR^n$. A local orientation |
| of $M$ on $U$ is a choice of an isomorphism |
of $M$ on $U$ is a choice of an isomorphism |
| $$\mathfrak{o}_U \co H_n(M,M\setminus U \,; \BZ) \to \BZ.$$ |
$$\mathfrak{o}_U \co H_n(M,M\setminus U \,; \BZ) \to \BZ.$$ |
| \end{defn} |
\end{defn} |
| Now notice that with $U$ as above and $x\in U$ the inclusion |
Now notice that with $U$ as above and $x\in U$ the inclusion |
| $$\imath^U_x\co M\setminus U \hookrightarrow M\setminus x$$ |
$$\imath^U_x\co M\setminus U \hookrightarrow M\setminus x$$ |
|
\PMlinkescapetext{induces} a map (actually isomorphism)
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induces a map (actually isomorphism)
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| $$\imath^U_{x\,\,*}\co H_n(M,M\setminus U \,; \BZ) \to H_n(M,M\setminus x\,;\mathbb{Z})$$ |
$$\imath^U_{x\,\,*}\co H_n(M,M\setminus U \,; \BZ) \to H_n(M,M\setminus x\,;\mathbb{Z})$$ |
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and therefore a local orientation at $U$ \PMlinkescapetext{induces} (by composing with the above isomorphism)
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and therefore a local orientation at $U$ induces (by composing with the above isomorphism)
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an orientation at each point $x\in U$. It is \PMlinkescapetext{natural} to declare that all these orientations
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an orientation at each point $x\in U$. It is natural to declare that all these orientations
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| fit nicely together. |
fit nicely together. |
| \begin{defn} |
\begin{defn} |
| Let $M$ be a manifold without boundary. An orientation of $M$ is a choice of an orientation |
Let $M$ be a manifold without boundary. An orientation of $M$ is a choice of an orientation |
| $\mathfrak{o}_x$ for each point $x\in M$, with the property that |
$\mathfrak{o}_x$ for each point $x\in M$, with the property that |
| \begin{itemize} |
\begin{itemize} |
| \item[] each point $x$ has a coordinate neighborhood $U$ so that for each $y\in U$, the orientation |
\item[] each point $x$ has a coordinate neighborhood $U$ so that for each $y\in U$, the orientation |
| $\mathfrak{o}_y$ is induced by a local orientation on $U$. |
$\mathfrak{o}_y$ is induced by a local orientation on $U$. |
| \end{itemize} |
\end{itemize} |
| A manifold is called orientable if it admits an orientation. |
A manifold is called orientable if it admits an orientation. |
| \end{defn} |
\end{defn} |
| \begin{rem} |
\begin{rem} |
| Although we avoided using this terminology what we did was to indicate how a sheaf of orientations |
Although we avoided using this terminology what we did was to indicate how a sheaf of orientations |
| could be defined, and then we defined an orientation to be a global section of that sheaf. |
could be defined, and then we defined an orientation to be a global section of that sheaf. |
| \end{rem} |
\end{rem} |
| \begin{defn} |
\begin{defn} |
| Let $M$ be a manifold with non-empty boundary, $\partial M\neq \emptyset$. $M$ is called |
Let $M$ be a manifold with non-empty boundary, $\partial M\neq \emptyset$. $M$ is called |
| orientable if its double |
orientable if its double |
| $$\hat{M}:=M\bigcup_{\partial M}M$$ |
$$\hat{M}:=M\bigcup_{\partial M}M$$ |
| is orientable, where $\bigcup_{\partial M}$ denotes gluing along the boundary.\n |
is orientable, where $\bigcup_{\partial M}$ denotes gluing along the boundary.\n |
| An orientation of $M$ is determined by an orientation of $\hat{M}$. |
An orientation of $M$ is determined by an orientation of $\hat{M}$. |
| \end{defn} |
\end{defn} |
