Orientation covering

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.

This page has not been refereed. The information given here might be incomplete or provisional.

1 Construction

Let

Tex syntax error

${{Authors|Matthias Kreck}}{{Stub}} ==Construction== <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. 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 $H_n(M, M-x;\mathbb Z)$ 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 $U$ 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 about all charts. By construction these open subsets 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. {{endthm}} </wikitex> ==Relation to the orientation character== <wikitex>; We assume now that $M$ is connected. The '''orientation character''' is a homomorphims $w: \pi_1(M) \to \mathbb Z/2= \{ \pm 1\}$, 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-orienable 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. 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

Tex syntax error

$M$ at $x$$x$ given by a generator of $H_n(M, M-x;\mathbb Z)$$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

Tex syntax error

$M$. 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$$U$ is a subset of $\hat M$$\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 these open subsets 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

Tex syntax error

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

Tex syntax error

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

Tex syntax error

$M$ 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].

2 Characterization of the orientation covering

One can easily characterize the orientation covering:

Proposition 2.1. 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

Tex syntax error

$M$ 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

Tex syntax error

$M$ and so

Tex syntax error

$M$ is orientable. Thus we have shown:

Proposition 2.2.

Tex syntax error

$M$ is orientable if and only if the orientation covering is trivial. If

Tex syntax error

$M$ is connected,

Tex syntax error

$M$ is non-orientable if and only if $\hat M$$\hat M$ is connected.

3 Relation to the orientation character

We assume now that

Tex syntax error

$M$ is connected. The orientation character is a homomorphims $w: \pi_1(M) \to \mathbb Z/2= \{ \pm 1\}$$w: \pi_1(M) \to \mathbb Z/2= \{ \pm 1\}$, which attaches $+1$$+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

Tex syntax error

$M$ is orientable.

4 Examples

Here are some examples of orientation coverings.

1. If

   Tex syntax error

   $M$ 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-orienable 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$.

5 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).

6 External links

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