Orientation covering

Latest revision as of 20:40, 7 March 2014

Let ${{Stub}}{{Authors|Matthias Kreck}} == Construction == <wikitex include="TeXInclude:">; Let $M$ be a $n$-dimensional topological (or smooth) 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 $x$, 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 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 $M$ (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 about all charts (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. This covering is called the '''orientation covering'''. For more information, see \cite{Dold1995|VIII § 2}. </wikitex> == Some remarks== <wikitex include="TeXInclude:">; 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. By construction of $\hat M$ the deck transformation of the 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 it is isomorphic to the orientation covering. Namely 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. 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. </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 $M$ 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 $M$. 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 $M$ 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; [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].

1 Characterization of the orientation covering

One can easily characterize the orientation covering:

Proof. 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.

$\square$

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:

2 Relation to the orientation character

We assume now that $M$ 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 $M$ is orientable.

3 Examples

Here are some examples of orientation coverings.

1. If $M$ 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.