Orientation covering

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1 Construction

Let $M$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}M$ be a $n$$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 $M$$M$ at $x$$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$$\varphi : U \to V\subset \mathbb R^n$ be a chart of $M$$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$$U$ is a subset of $\hat M$$\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$$2$-fold covering of $M$$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].

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 $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 2.2. $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.

3 Relation to the orientation character

We assume now that $M$$M$ is connected. The orientation character is a homomorphism $w: \pi_1(M) \to \mathbb Z/2= \{ \pm 1\}$$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$$w$ is trivial if and only if $M$$M$ is orientable.

4 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$$\mathbb R P^n$ is non-orienable and the orientation cover is the canonical projection $S^n \to \mathbb R P^n$$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.