Orientation covering
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− | + | {{Authors|Matthias Kreck}} | |
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− | <wikitex | + | 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 $2$. |
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− | + | 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. | |
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− | + | 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'''. | |
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− | + | 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. | |
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+ | For more information and a discussion placing the orientation covering in a wider setting, see \cite{Dold1995|VIII § 2}. | ||
</wikitex> | </wikitex> | ||
− | == | + | ==Characterization of the orientation covering== |
− | <wikitex | + | <wikitex>; |
− | + | One can easily characterize the orientation covering: | |
− | + | {{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. | |
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− | {{beginthm|Proposition}} | + | |
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{{endthm}} | {{endthm}} | ||
{{beginproof}} | {{beginproof}} | ||
− | + | 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. | |
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{{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>; |
− | + | 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, $\ | + | # 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 $S^n$. |
− | # The orientation cover of the Klein bottle $K^2$ is the projection from the [[2-manifolds#Orientable_surfaces|2-torus]]; $T^2 \to K^2$. | + | # 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 of the open [[Wikipedia:Mobius_strip|Möbius strip]] $Mö$ is the cylinder; $S^1 \times \Rr \to Mö$. | + | # The orientation cover of the open [[Wikipedia:Mobius_strip|Möbius strip]] $Mö$ is the canonical projection from the cylinder; $p \colon S^1 \times \Rr \to Mö$. |
</wikitex> | </wikitex> | ||
− | + | == References == | |
− | ==References== | + | |
{{#RefList:}} | {{#RefList:}} | ||
== External links == | == External links == | ||
* 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:Definitions]] | ||
[[Category:Definitions]] | [[Category:Definitions]] |
Latest revision as of 20:40, 7 March 2014
An earlier version of this page was published in the Definitions section of the Bulletin of the Manifold Atlas: screen, print. You may view the version used for publication as of 18:40, 7 March 2014 and the covering&diff=cur&oldid=11510 changes since publication. |
The user responsible for this page is Matthias Kreck. No other user may edit this page at present. |
Tex syntax errorbe a -dimensional topological manifold. We construct an oriented manifold and a -fold covering called the orientation covering. The non-trivial deck transformation of this covering is orientation-reversing. As a set is the set of pairs , where is a local orientation of
Tex syntax errorat given by a generator of the infinite cyclic group . The map assignes to . Since there are precisely two local orientations, the fibres of this map have cardinality . Next we define a topology on this set. Let be a chart of
Tex syntax error. We orient by the standard orientation given by the standard basis , , ..., , from which we define a a continuous local orientation by identifying the tangent space with . 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 by . Using the chart we transport this standard orientation to by the induced map on homology. The local orientations given by this orientation of determine a subset of , which we require to be open. Doing the same starting with the non-standard orientation of we obtain another subset, which we also call open. We give 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 , and so we obtain an atlas of . The map is by construction a -fold covering. By construction is oriented in a tautological way and the non-trivial deck transformation of the covering is orientation reversing. Thus we have constructed a -fold covering of
Tex syntax errorby an oriented manifold , which is smooth, if
Tex syntax erroris smooth. This covering is called the orientation covering. If
Tex syntax erroris smooth one can use the local tangential orientation of 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 is a smooth manifold and 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
One can easily characterize the orientation covering:
Proposition 1.1. If is an oriented manifold and is a -fold covering with orientation reversing non-trivial deck transformation, then it is isomorphic to the orientation covering.
Proof. We have a map by mapping to . This is an isomorphism of these two coverings.
Tex syntax erroris orientable, we pick an orientation and see that is the disjoint union of and its complement, so it is isomorphic to the trivial covering . In turn if the orientation covering is trivial it decomposes into two open (and thus oriented) subsets homeomorphic to
Tex syntax errorand so
Tex syntax erroris orientable. Thus we have shown:
Proposition 1.2.
Tex syntax erroris orientable if and only if the orientation covering is trivial. If
Tex syntax erroris connected,
Tex syntax erroris non-orientable if and only if is connected. In particular, any simply-connected manifold is orientable.
2 Relation to the orientation character
Tex syntax erroris connected. The orientation character is a homomorphism , which attaches to a loop if and only if the pull back of the orientation covering is trivial. By the classification of coverings this implies that is trivial if and only if
Tex syntax erroris orientable.
3 Examples
Here are some examples of orientation coverings.
- If
Tex syntax error
is orientable then is isomorphic to the projection . - If is even, is non-orientable and the orientation cover is the canonical projection . The deck transformation of the orientation covering is the antipodal map on .
- The orientation cover of the Klein bottle is the canonical projection from the 2-torus; .
- The orientation cover of the open Möbius strip is the canonical projection from the cylinder; .
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.