Orientation covering

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−	{{Stub}}{{Authors|Matthias Kreck}}	+	{{Authors|Matthias Kreck}}
−	== Construction ==	+	<wikitex>;
−	<wikitex include="TeXInclude:">;	+	Let $M$ be a $n$-dimensional topological manifold. We construct an oriented manifold $\hat M$ and a $2$-fold covering $p : \hat M \to M$ called the orientation covering. The non-trivial deck transformation of this covering is orientation-reversing. As a set $\hat M$ is the set of pairs $(x, o_x)$, where $o_x$ is a local orientation of $M$ at $x$ given by a generator of the infinite cyclic group $H_n(M, M-x;\mathbb Z)$. The map $p$ assignes $x$ to $(x,o_x)$. Since there are precisely two local orientations, the fibres of this map have cardinality $2$.
−	The orientation covering of a topological manifold $M$ is a canonical two-fold covering	+
−	of $M$.	+
−	{{beginthm|Theorem|c.f. \cite{Dold1995|VIII 2.11}}}	+
−	Let $M$ be a $n$-dimensional topological manifold.	+
−	There is an oriented manifold $\hat M$ and a $2$-fold covering $p \colon \hat M \to M$ called the orientation covering.	+
−	If $M$ is a smooth, resp. piecewise linear, manifold then $\hat M$ and the covering map $p$ are smooth, resp. piecewise linear.	+
−	{{endthm}}	+
−	{{beginrem|Remark}}	+	Next we define a topology on this set. Let $\varphi : U \to V\subset \mathbb R^n$ be a chart of $M$. We orient $\mathbb R^n$ by the standard orientation given by the standard basis $e_1$, $e_2$, ..., $e_n$, from which we define a a continuous local orientation by identifying the tangent space with $\mathbb R^n$. Since for a smooth manifold a tangential orientation defines a homological orientation, this also gives a [[Orientation_of_manifolds#Reformulation_in_terms_of_local_homological_orientations|homological orientation]]: see \cite[§3]{Kreck2013}. We call the standard local orientation at $x \in \mathbb R^n$ by $sto_x$. Using the chart we transport this standard orientation to $U$ by the induced map on homology. The local orientations given by this orientation of $U$ determine a subset of $\hat M$, which we require to be open. Doing the same starting with the non-standard orientation of $\mathbb R^n$ we obtain another subset, which we also call open. We give $\hat M$ the topology generated by these open subsets, where we vary about all charts. By construction each of these open subsets is are homeomorphic to an open subset of $\mathbb R^n$, and so we obtain an atlas of $\hat M$. The map $p$ is by construction a $2$-fold covering. By construction $\hat M$ is oriented in a tautological way and the non-trivial [[Wikipedia:Covering_space#Deck_transformation_group.2C_regular_covers|deck transformation]] of the covering is orientation reversing.
−	The covering $p \colon \hat M \to M$ is called the '''orientation covering'''. For more information, see \cite{Dold1995|VIII § 2}.	+
−	{{endrem}}	+
−	{{beginproof}}	+	Thus we have constructed a $2$-fold covering of $M$ by an oriented manifold $\hat M$, which is smooth, if $M$ is smooth. This covering is called the '''orientation covering'''.
−	As a set $\hat M$ is the set of pairs $(x, o_x)$, where $o_x$ is a local orientation of $M$ at $x$, either given by a generator of $H_n(M, M-x;\mathbb Z)$ or by an orientation of $T_xM$ in the smooth case (for the equivalence of these data see the atlas page on orientation of manifold). The map $p$ assigns $x$ to $(x,o_x)$. Since there are precisely two local orientations, the fibers of this map have cardinality $2$.	+
−	Next we define a topology on this set. Let $\varphi : U \to V\subset \mathbb R^n$ be a chart of $M$ (smooth, if $M$ is smooth). We orient $\mathbb R^n$ by the standard orientation given by the standard basis $e_1$, $e_2$, ..., $e_n$, from which we define a continuous local orientation by identifying the tangent space with $\mathbb R^n$. Since for a smooth manifold a tangential orientation defines a homological orientation, this also gives a homological orientation (see atlas page on orientation of manifolds). We call the standard local orientation at $x \in \mathbb R^n$ by $sto_x$. Using the chart we transport this standard orientation to $U$ by the induced map on homology or the differential in the case of tangential orientations. The local orientations given by this orientation of $U$ is a subset of $\hat M$, which we require to be open. Doing the same starting with the non-standard orientation of $\mathbb R^n$ we obtain another subset, which we also call open. We give $\hat M$ the topology generated by these open subsets, where we vary over all [[charts]] of $M$ (smooth charts, if $M$ is smooth). By construction these open subsets are homeomorphic to an open subset of $\mathbb R^n$, and so we obtain an atlas of $\hat M$. In the smooth case this is a smooth atlas making $\hat M$ a smooth manifold. The map $p$ is by construction a $2$-fold covering, smooth, if $M$ is smooth. By construction $\hat M$ is oriented in a tautological way. Thus we have constructed a $2$-fold covering of $M$ by an oriented manifold $\hat M$, which is smooth, if $M$ is smooth.	+	If $M$ is smooth one can use the local tangential orientation of $T_xM$ instead of the homological orientation to construct the orientation covering (for the equivalence of these data see the Manifold Atlas page [[Orientation_of_manifolds#Reformulations_of_orientation_for_smooth_manifolds|Orientation of manifolds]]; \cite[§6]{Kreck2013}). Since a countable covering of a smooth manifold has a unique smooth structure such that the projection map is a local diffeomorphism, in the smooth case $\hat M$ is a smooth manifold and $p$ a local diffeomorphism.
−	{{endproof}}	+
		+	For more information and a discussion placing the orientation covering in a wider setting, see \cite{Dold1995|VIII § 2}.
	</wikitex>		</wikitex>
−	== Properties ==	+	==Characterization of the orientation covering==
−	<wikitex include="TeXInclude:">;	+	<wikitex>;
−	In this section we record the key properties of the orientation cover are given in Proposition \ref{prop:properties} below.	+	One can easily characterize the orientation covering:
−	The orientation covering of a manifold $M$ is very closely related to the [[orientation character]] of $M$. This is a homomorphism	+	{{beginthm|Proposition}} If $N $ is an oriented manifold and $p: N \to M$ is a $2$-fold covering with orientation reversing non-trivial deck transformation, then it is isomorphic to the orientation covering.
−	$$ w \colon \pi_1(M) \to \Z/2 $$	+
−	which may be defined as follows. Fix a base-point $x \in M$ with lifts $\tilde x_1$ and $\tilde x_{-1}$ in $\hat M$. For a loop	+
−	$\gamma \colon ([0, 1], \{0, 1\}) \to (M, x)$ based at $x$, let $\tilde \gamma \colon [0, 1] \to M$ be the lift of $\gamma$	+
−	with $\tilde{\gamma}(0) = 1$ and define $w$ on the homotopy class of $\gamma$ by	+
−	$$	+
−	w([\gamma]) := \left\{ \begin{array}{rc} 1 & \text{if $\tilde{\gamma}(1) = x_1$} \\ -1 & \text{if $\tilde{\gamma}(1) = x_{-1}$} \end{array} \right. $$	+
−	{{beginthm|Proposition}} \label{properties}	+
−	Let $p \colon \hat M \to M$ be the orientation covering of a topological manifold $M$.	+
−	# $M$ is orientable if and only if $\hat M = M \times \Z/2$ and $p$ is the projection to $M$.	+
−	# Converely, if $M$ is connected then $M$ is non-orientable if and only if $\hat M$ is connected.	+
−	# $M$ is orientable if and only if $w \colon \pi_1(M) \to \Z/2$ is the zero homomorphism.	+
−	# By construction, the [[Wikipedia:Covering_space#Deck_transformation_group.2C_regular_covers|deck transformation]] of orientation covering is orientation reversing.	+
−	# If $N $ is an oriented manifold and $p: N \to M$ is a $2$-fold covering with orientation reversing deck transformation, then $p \colon N \to M$ is isomorphic to the orientation covering.	+
	{{endthm}}		{{endthm}}
	{{beginproof}}		{{beginproof}}
−	(1.) If $M$ is orientable, we pick an orientation and see that $\hat M$ is the disjoint union of $\{(x,o_x)| \,\, o_x \,\, is \,\, the \,\, local \,\, orientation \,\, given \,\, by \,\, the \,\, orientation \,\, of \,\, M\}$ and its complement, so it is isomorphic to the trivial covering $ M \times \mathbb Z/2$. In turn if the orientation covering is trivial it decomposes $\hat M$ into to open (and thus oriented) subsets homeomorphic to $M$ and so $M$ is orientable.	+	We have a map $N \to \hat M$ by mapping $y \in N$ to $(p(y), orientation \,\, induced \,\, by \,\, p)$. This is an isomorphism of these two coverings.
−		+
−	(2.) Follows immediately from (1.) since a two-fold cover of a connected space is non-trivial if and only if the total space of the covering is disconnected.	+
−		+
−	(3.) Follows from (1.) since $w$ classifies the orientation cover: see the page [[orientation character]].	+
−		+
−	(4.) Is true by construciton as stated.	+
−	<!-- By construction of $\hat M$ the deck transformation of the covering is orientation reversing. -->	+
−		+
−	(5.) We have a map $N \to \hat M$ by mapping $y \in N$ to $(p(y), orientation \,\, induced \,\, by \,\, p)$. It is easily checked that his is an isomorphism of these two coverings.	+
−	<!-- By the considerations above, $M$ is orientable if and only if this covering is trivial, or $M$ is non-orientable if and only if $N$ is connected. -->	+
	{{endproof}}		{{endproof}}
		+	If $M$ is orientable, we pick an orientation and see that $\hat M$ is the disjoint union of $\{(x,o_x)| \,\, o_x \,\, is \,\, the \,\, local \,\, orientation \,\, given \,\, by \,\, the \,\, orientation \,\, of \,\, M\}$ and its complement, so it is isomorphic to the trivial covering $ M \times \mathbb Z/2$. In turn if the orientation covering is trivial it decomposes $\hat M$ into two open (and thus oriented) subsets homeomorphic to $M$ and so $M$ is orientable. Thus we have shown:
		+	{{beginthm|Proposition}} $M$ is orientable if and only if the orientation covering is trivial. If $M$ is connected, $M$ is non-orientable if and only if $\hat M$ is connected. In particular, any simply-connected manifold is orientable.
		+	{{endthm}}
		+	</wikitex>
−		+	==Relation to the orientation character==
−		+	<wikitex>;
		+	We assume now that $M$ is connected. The [[Orientation character|orientation character]] is a homomorphism $w: \pi_1(M) \to \{ \pm 1\}$, which attaches $+1$ to a loop $S^1 \to M$ if and only if the pull back of the orientation covering is trivial. By the classification of coverings this implies that $w$ is trivial if and only if $M$ is orientable.
	</wikitex>		</wikitex>
	== Examples ==		== Examples ==
−	<wikitex>;	+	<wikitex>;
−	We give a list of basic in interesting orientation double coverings.	+	Here are some examples of orientation coverings.
−	# If $M$ is orientable then $p \colon \hat M \to M$ is isomorphic to the projection $M \times \Z/2 \to M$.	+	# If $M$ is orientable then $p \colon \hat M \to M$ is isomorphic to the projection $M \times \mathbb Z/2 \to M$.
−	# If $n$ is even, $\Rr P^n$ is non-orienable and with orientation cover $S^n \to \Rr P^n$. The deck transformation	+	# If $n$ is even, $\mathbb R P^n$ is non-orientable and the orientation cover is the canonical projection $S^n \to \mathbb R P^n$. The deck transformation of the orientation covering is the [[Wikipedia:Antipodal point|antipodal map]] on $S^n$.
−	is the [[Wikipedia:Antipodal_point|antipodal map]] on $S^n$.	+	# The orientation cover of the [[Wikipedia:Klein bottle|Klein bottle]] $K^2$ is the canonical projection from the [[2-manifolds#Orientable_surfaces|2-torus]]; $p \colon T^2 \to K^2$.
−	# The orientation cover of the Klein bottle $K^2$ is the projection from the [[$2$-torus]]; $T^2 \to K^2$.	+	# The orientation cover of the open [[Wikipedia:Mobius_strip|Möbius strip]] $Mö$ is the canonical projection from the cylinder; $p \colon S^1 \times \Rr \to Mö$.
−	# The orientation of the open [[Wikipedia:Mobius_strip|Möbius strip]] $Mö$ is the cylinder; $S^1 \times \Rr \to Mö$.	+
	</wikitex>		</wikitex>
−
	== References ==		== References ==
	{{#RefList:}}		{{#RefList:}}
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	* The Encylopedia of Mathematics article on [http://www.encyclopediaofmath.org/index.php/Orientation orientation].		* The Encylopedia of Mathematics article on [http://www.encyclopediaofmath.org/index.php/Orientation orientation].
	* The Wikipedia page on the [[Wikipedia:Orientability#Orientable_double_cover|orientability]].		* The Wikipedia page on the [[Wikipedia:Orientability#Orientable_double_cover|orientability]].
−	[[Category:Theory]]	+	[[Category:Definitions]]
		+	[[Category:Definitions]]

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Latest revision as of 20:40, 7 March 2014

The user responsible for this page is Matthias Kreck. No other user may edit this page at present.

Let

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${{Stub}}{{Authors|Matthias Kreck}} == Construction == <wikitex include="TeXInclude:">; The orientation covering of a topological manifold $M$ is a canonical two-fold covering of $M$. {{beginthm|Theorem|c.f. \cite{Dold1995|VIII 2.11}}} Let $M$ be a $n$-dimensional topological manifold. There is an oriented manifold $\hat M$ and a $-fold covering $p \colon \hat M \to M$ called the orientation covering. If $M$ is a smooth, resp. piecewise linear, manifold then $\hat M$ and the covering map $p$ are smooth, resp. piecewise linear. {{endthm}} {{beginrem|Remark}} The covering $p \colon \hat M \to M$ is called the '''orientation covering'''. For more information, see \cite{Dold1995|VIII § 2}. {{endrem}} {{beginproof}} As a set $\hat M$ is the set of pairs $(x, o_x)$, where $o_x$ is a local orientation of $M$ at $x$, either given by a generator of $H_n(M, M-x;\mathbb Z)$ or by an orientation of $T_xM$ in the smooth case (for the equivalence of these data see the atlas page on orientation of manifold). The map $p$ assigns $x$ to $(x,o_x)$. Since there are precisely two local orientations, the fibers of this map have cardinality $. Next we define a topology on this set. Let $\varphi : U \to V\subset \mathbb R^n$ be a chart of $M$ (smooth, if $M$ is smooth). We orient $\mathbb R^n$ by the standard orientation given by the standard basis $e_1$, $e_2$, ..., $e_n$, from which we define a continuous local orientation by identifying the tangent space with $\mathbb R^n$. Since for a smooth manifold a tangential orientation defines a homological orientation, this also gives a homological orientation (see atlas page on orientation of manifolds). We call the standard local orientation at $x \in \mathbb R^n$ by $sto_x$. Using the chart we transport this standard orientation to $U$ by the induced map on homology or the differential in the case of tangential orientations. The local orientations given by this orientation of $U$ is a subset of $\hat M$, which we require to be open. Doing the same starting with the non-standard orientation of $\mathbb R^n$ we obtain another subset, which we also call open. We give $\hat M$ the topology generated by these open subsets, where we vary over all [[charts]] of $M$ (smooth charts, if $M$ is smooth). By construction these open subsets are homeomorphic to an open subset of $\mathbb R^n$, and so we obtain an atlas of $\hat M$. In the smooth case this is a smooth atlas making $\hat M$ a smooth manifold. The map $p$ is by construction a $-fold covering, smooth, if $M$ is smooth. By construction $\hat M$ is oriented in a tautological way. Thus we have constructed a $-fold covering of $M$ by an oriented manifold $\hat M$, which is smooth, if $M$ is smooth. {{endproof}} </wikitex> == Properties == <wikitex include="TeXInclude:">; In this section we record the key properties of the orientation cover are given in Proposition \ref{prop:properties} below. The orientation covering of a manifold $M$ is very closely related to the [[orientation character]] of $M$. This is a homomorphism $$ w \colon \pi_1(M) \to \Z/2 $$ which may be defined as follows. Fix a base-point $x \in M$ with lifts $\tilde x_1$ and $\tilde x_{-1}$ in $\hat M$. For a loop $\gamma \colon ([0, 1], \{0, 1\}) \to (M, x)$ based at $x$, let $\tilde \gamma \colon [0, 1] \to M$ be the lift of $\gamma$ with $\tilde{\gamma}(0) = 1$ and define $w$ on the homotopy class of $\gamma$ by $$ w([\gamma]) := \left\{ \begin{array}{rc} 1 & \text{if $\tilde{\gamma}(1) = x_1$} \ -1 & \text{if $\tilde{\gamma}(1) = x_{-1}$} \end{array} \right. $$ {{beginthm|Proposition}} \label{properties} Let $p \colon \hat M \to M$ be the orientation covering of a topological manifold $M$. # $M$ is orientable if and only if $\hat M = M \times \Z/2$ and $p$ is the projection to $M$. # Converely, if $M$ is connected then $M$ is non-orientable if and only if $\hat M$ is connected. # $M$ is orientable if and only if $w \colon \pi_1(M) \to \Z/2$ is the zero homomorphism. # By construction, the [[Wikipedia:Covering_space#Deck_transformation_group.2C_regular_covers|deck transformation]] of orientation covering is orientation reversing. # If $N $ is an oriented manifold and $p: N \to M$ is a $-fold covering with orientation reversing deck transformation, then $p \colon N \to M$ is isomorphic to the orientation covering. {{endthm}} {{beginproof}} (1.) If $M$ is orientable, we pick an orientation and see that $\hat M$ is the disjoint union of $\{(x,o_x)| \,\, o_x \,\, is \,\, the \,\, local \,\, orientation \,\, given \,\, by \,\, the \,\, orientation \,\, of \,\, M\}$ and its complement, so it is isomorphic to the trivial covering $ M \times \mathbb Z/2$. In turn if the orientation covering is trivial it decomposes $\hat M$ into to open (and thus oriented) subsets homeomorphic to $M$ and so $M$ is orientable. (2.) Follows immediately from (1.) since a two-fold cover of a connected space is non-trivial if and only if the total space of the covering is disconnected. (3.) Follows from (1.) since $w$ classifies the orientation cover: see the page [[orientation character]]. (4.) Is true by construciton as stated. <!-- By construction of $\hat M$ the deck transformation of the covering is orientation reversing. --> (5.) We have a map $N \to \hat M$ by mapping $y \in N$ to $(p(y), orientation \,\, induced \,\, by \,\, p)$. It is easily checked that his is an isomorphism of these two coverings. <!-- By the considerations above, $M$ is orientable if and only if this covering is trivial, or $M$ is non-orientable if and only if $N$ is connected. --> {{endproof}} </wikitex> == Examples == <wikitex>; We give a list of basic in interesting orientation double coverings. # If $M$ is orientable then $p \colon \hat M \to M$ is isomorphic to the projection $M \times \Z/2 \to M$. # If $n$ is even, $\Rr P^n$ is non-orienable and with orientation cover $S^n \to \Rr P^n$. The deck transformation is the [[Wikipedia:Antipodal_point|antipodal map]] on $S^n$. # The orientation cover of the Klein bottle $K^2$ is the projection from the [[$-torus]]; $T^2 \to K^2$. # The orientation of the open [[Wikipedia:Mobius_strip|Möbius strip]] $Mö$ is the cylinder; $S^1 \times \Rr \to Mö$. </wikitex> == References == {{#RefList:}} == External links == * The Encylopedia of Mathematics article on [http://www.encyclopediaofmath.org/index.php/Orientation orientation]. * The Wikipedia page on the [[Wikipedia:Orientability#Orientable_double_cover|orientability]]. [[Category:Theory]]M$ be a $n$-dimensional topological manifold. We construct an oriented manifold $\hat M$ and a $2$-fold covering $p : \hat M \to M$ called the orientation covering. The non-trivial deck transformation of this covering is orientation-reversing. As a set $\hat M$ is the set of pairs $(x, o_x)$, where $o_x$ is a local orientation of

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$M$ at $x$ given by a generator of the infinite cyclic group $H_n(M, M-x;\mathbb Z)$. The map $p$ assignes $x$ to $(x,o_x)$. Since there are precisely two local orientations, the fibres of this map have cardinality $2$. Next we define a topology on this set. Let $\varphi : U \to V\subset \mathbb R^n$ be a chart of

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$M$. We orient $\mathbb R^n$ by the standard orientation given by the standard basis $e_1$, $e_2$, ..., $e_n$, from which we define a a continuous local orientation by identifying the tangent space with $\mathbb R^n$. Since for a smooth manifold a tangential orientation defines a homological orientation, this also gives a homological orientation: see [Kreck2013, §3]. We call the standard local orientation at $x \in \mathbb R^n$ by $sto_x$. Using the chart we transport this standard orientation to $U$ by the induced map on homology. The local orientations given by this orientation of $U$ determine a subset of $\hat M$, which we require to be open. Doing the same starting with the non-standard orientation of $\mathbb R^n$ we obtain another subset, which we also call open. We give $\hat M$ the topology generated by these open subsets, where we vary about all charts. By construction each of these open subsets is are homeomorphic to an open subset of $\mathbb R^n$, and so we obtain an atlas of $\hat M$. The map $p$ is by construction a $2$-fold covering. By construction $\hat M$ is oriented in a tautological way and the non-trivial deck transformation of the covering is orientation reversing. Thus we have constructed a $2$-fold covering of

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$M$ by an oriented manifold $\hat M$, which is smooth, if

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$M$ is smooth. This covering is called the orientation covering. If

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$M$ is smooth one can use the local tangential orientation of $T_xM$ instead of the homological orientation to construct the orientation covering (for the equivalence of these data see the Manifold Atlas page Orientation of manifolds; [Kreck2013, §6]). Since a countable covering of a smooth manifold has a unique smooth structure such that the projection map is a local diffeomorphism, in the smooth case $\hat M$ is a smooth manifold and $p$ a local diffeomorphism.

For more information and a discussion placing the orientation covering in a wider setting, see [Dold1995, VIII § 2].

Contents 1 Characterization of the orientation covering 2 Relation to the orientation character 3 Examples 4 References 5 External links

1 Characterization of the orientation covering

One can easily characterize the orientation covering:

If

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$M$ is orientable, we pick an orientation and see that $\hat M$ is the disjoint union of $\{(x,o_x)| \,\, o_x \,\, is \,\, the \,\, local \,\, orientation \,\, given \,\, by \,\, the \,\, orientation \,\, of \,\, M\}$ and its complement, so it is isomorphic to the trivial covering $M \times \mathbb Z/2$. In turn if the orientation covering is trivial it decomposes $\hat M$ into two open (and thus oriented) subsets homeomorphic to

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$M$ and so

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$M$ is orientable. Thus we have shown:

2 Relation to the orientation character

We assume now that

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$M$ is connected. The orientation character is a homomorphism $w: \pi_1(M) \to \{ \pm 1\}$, which attaches $+1$ to a loop $S^1 \to M$ if and only if the pull back of the orientation covering is trivial. By the classification of coverings this implies that $w$ is trivial if and only if

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$M$ is orientable.

3 Examples

Here are some examples of orientation coverings.

1. If

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   $M$ is orientable then $p \colon \hat M \to M$ is isomorphic to the projection $M \times \mathbb Z/2 \to M$.
2. If $n$ is even, $\mathbb R P^n$ is non-orientable and the orientation cover is the canonical projection $S^n \to \mathbb R P^n$. The deck transformation of the orientation covering is the antipodal map on $S^n$.
3. The orientation cover of the Klein bottle $K^2$ is the canonical projection from the 2-torus; $p \colon T^2 \to K^2$.
4. The orientation cover of the open Möbius strip $Mö$ is the canonical projection from the cylinder; $p \colon S^1 \times \Rr \to Mö$.

4 References

• [Dold1995] A. Dold, Lectures on algebraic topology, Springer-Verlag, 1995. MR1335915 (96c:55001) Zbl 0872.55001
• [Kreck2013] M. Kreck, Orientation of manifolds, Bull. Man. Atl. (2013).

5 External links

• The Encylopedia of Mathematics article on orientation.
• The Wikipedia page on the orientability.