Orientation covering

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Let $M$${{Authors|Matthias Kreck}} <wikitex>; Let $M$ be a $n$-dimensional topological manifold. We construct an oriented manifold $\hat M$ and a $-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 $. 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 $-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. Thus we have constructed a $-fold covering of $M$ by an oriented manifold $\hat M$, which is smooth, if $M$ is smooth. This covering is called the '''orientation covering'''. 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. For more information and a discussion placing the orientation covering in a wider setting, see \cite{Dold1995|VIII § 2}. </wikitex> ==Characterization of the orientation covering== <wikitex>; One can easily characterize the orientation covering: {{beginthm|Proposition}} If $N $ is an oriented manifold and $p: N \to M$ is a $-fold covering with orientation reversing non-trivial deck transformation, then it is isomorphic to the orientation covering. {{endthm}} {{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. {{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> == Examples == <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 \mathbb Z/2 \to M$. # 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 [[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 open [[Wikipedia:Mobius_strip|Möbius strip]] $Mö$ is the canonical projection from the cylinder; $p \colon 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:Definitions]] [[Category:Definitions]]M$ be a $n$$n$-dimensional topological manifold. We construct an oriented manifold $\hat M$$\hat M$ and a $2$$2$-fold covering $p : \hat M \to M$$p : \hat M \to M$ called the orientation covering. The non-trivial deck transformation of this covering is orientation-reversing. As a set $\hat M$$\hat M$ is the set of pairs $(x, o_x)$$(x, o_x)$, where $o_x$$o_x$ is a local orientation of $M$$M$ at $x$$x$ given by a generator of the infinite cyclic group $H_n(M, M-x;\mathbb Z)$$H_n(M, M-x;\mathbb Z)$. The map $p$$p$ assignes $x$$x$ to $(x,o_x)$$(x,o_x)$. Since there are precisely two local orientations, the fibres of this map have cardinality $2$$2$.

Next we define a topology on this set. Let $\varphi : U \to V\subset \mathbb R^n$$\varphi : U \to V\subset \mathbb R^n$ be a chart of $M$$M$. We orient $\mathbb R^n$$\mathbb R^n$ by the standard orientation given by the standard basis $e_1$$e_1$, $e_2$$e_2$, ..., $e_n$$e_n$, from which we define a a continuous local orientation by identifying the tangent space with $\mathbb R^n$$\mathbb R^n$. Since for a smooth manifold a tangential orientation defines a homological orientation, this also gives a homological orientation: see [Kreck2013, §3]. We call the standard local orientation at $x \in \mathbb R^n$$x \in \mathbb R^n$ by $sto_x$$sto_x$. Using the chart we transport this standard orientation to $U$$U$ by the induced map on homology. The local orientations given by this orientation of $U$$U$ determine a subset of $\hat M$$\hat M$, which we require to be open. Doing the same starting with the non-standard orientation of $\mathbb R^n$$\mathbb R^n$ we obtain another subset, which we also call open. We give $\hat M$$\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$$\mathbb R^n$, and so we obtain an atlas of $\hat M$$\hat M$. The map $p$$p$ is by construction a $2$$2$-fold covering. By construction $\hat M$$\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$$2$-fold covering of $M$$M$ by an oriented manifold $\hat M$$\hat M$, which is smooth, if $M$$M$ is smooth. This covering is called the orientation covering.

If $M$$M$ is smooth one can use the local tangential orientation of $T_xM$$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$$\hat M$ is a smooth manifold and $p$$p$ a local diffeomorphism.

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

1 Characterization of the orientation covering

One can easily characterize the orientation covering:

Proposition 1.1. If $N$$N$ is an oriented manifold and $p: N \to M$$p: N \to M$ is a $2$$2$-fold covering with orientation reversing non-trivial deck transformation, then it is isomorphic to the orientation covering.

Proof. We have a map $N \to \hat M$$N \to \hat M$ by mapping $y \in N$$y \in N$ to $(p(y), orientation \,\, induced \,\, by \,\, p)$$(p(y), orientation \,\, induced \,\, by \,\, p)$. This is an isomorphism of these two coverings.

$\square$$\square$

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

Proposition 1.2. $M$$M$ is orientable if and only if the orientation covering is trivial. If $M$$M$ is connected, $M$$M$ is non-orientable if and only if $\hat M$$\hat M$ is connected. In particular, any simply-connected manifold is orientable.

2 Relation to the orientation character

We assume now that $M$$M$ is connected. The orientation character is a homomorphism $w: \pi_1(M) \to \{ \pm 1\}$$w: \pi_1(M) \to \{ \pm 1\}$, which attaches $+1$$+1$ to a loop $S^1 \to M$$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$$w$ is trivial if and only if $M$$M$ is orientable.

3 Examples

Here are some examples of orientation coverings.

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