# Orientation covering

## 1 Construction

The orientation covering of a topological manifold $M$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}M$ is a canonical two-fold covering $p \colon \hat M \to M$$p \colon \hat M \to M$ is called the For more information and a discussion placing the orientation cover in a broader setting, see [Dold1995, VIII § 2].

Theorem 1.1 c.f. [Dold1995, VIII 2.11]. Let $M$$M$ be a $n$$n$-dimensional topological manifold. There is an oriented manifold $\hat M$$\hat M$ and a $2$$2$-fold covering $p \colon \hat M \to M$$p \colon \hat M \to M$ called the orientation covering. If $M$$M$ is a smooth, resp. piecewise linear, manifold then $\hat M$$\hat M$ and the covering map $p$$p$ are smooth, resp. piecewise linear.

Proof. 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$$x$, either given by a generator of $H_n(M, M-x;\mathbb Z)$$H_n(M, M-x;\mathbb Z)$ or by an orientation of $T_xM$$T_xM$ in the smooth case (for the equivalence of these data see the atlas page on orientation of manifold). The map $p$$p$ assigns $x$$x$ to $(x,o_x)$$(x,o_x)$. Since there are precisely two local orientations, the fibers 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$ (smooth, if $M$$M$ is smooth). We orient $\mathbb R^n$$\mathbb R^n$ by the standard orientation given by the standard basis $e_1$$e_1$, $e_2$$e_2$, ..., $e_n$$e_n$, from which we define a continuous local orientation by identifying the tangent space with $\mathbb R^n$$\mathbb R^n$. Since for a smooth manifold a tangential orientation defines a homological orientation, this also gives a homological orientation (see atlas page on orientation of manifolds). 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$$U$ by the induced map on homology or the differential in the case of tangential orientations. The local orientations given by this orientation of $U$$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$$\hat M$ the topology generated by these open subsets, where we vary over all charts of $M$$M$ (smooth charts, if $M$$M$ is smooth). By construction these open subsets are homeomorphic to an open subset of $\mathbb R^n$$\mathbb R^n$, and so we obtain an atlas of $\hat M$$\hat M$. In the smooth case this is a smooth atlas making $\hat M$$\hat M$ a smooth manifold. The map $p$$p$ is by construction a $2$$2$-fold covering, smooth, if $M$$M$ is smooth. By construction $\hat M$$\hat M$ is oriented in a tautological way. 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.

$\square$$\square$

## 2 Properties

In this section we record the key properties of the orientation cover are given in Proposition \ref{prop:properties} below. The orientation covering of a manifold $M$$M$ is very closely related to the orientation character of $M$$M$. This is a homomorphism

$\displaystyle w \colon \pi_1(M) \to \Z/2$

which may be defined as follows. Fix a base-point $x \in M$$x \in M$ with lifts $\tilde x_1$$\tilde x_1$ and $\tilde x_{-1}$$\tilde x_{-1}$ in $\hat M$$\hat M$. For a loop $\gamma \colon ([0, 1], \{0, 1\}) \to (M, x)$$\gamma \colon ([0, 1], \{0, 1\}) \to (M, x)$ based at $x$$x$, let $\tilde \gamma \colon [0, 1] \to M$$\tilde \gamma \colon [0, 1] \to M$ be the lift of $\gamma$$\gamma$ with $\tilde{\gamma}(0) = 1$$\tilde{\gamma}(0) = 1$ and define $w$$w$ on the homotopy class of $\gamma$$\gamma$ by

$\displaystyle w([\gamma]) := \left\{ \begin{array}{rc} 1 & \text{if \tilde{\gamma}(1) = x_1} \\ -1 & \text{if \tilde{\gamma}(1) = x_{-1}} \end{array} \right.$

Proposition 2.1. Let $p \colon \hat M \to M$$p \colon \hat M \to M$ be the orientation covering of a topological manifold $M$$M$.

1. $M$$M$ is orientable if and only if $\hat M = M \times \Z/2$$\hat M = M \times \Z/2$ and $p$$p$ is the projection to $M$$M$.
2. Converely, if $M$$M$ is connected then $M$$M$ is non-orientable if and only if $\hat M$$\hat M$ is connected.
3. $M$$M$ is orientable if and only if $w \colon \pi_1(M) \to \Z/2$$w \colon \pi_1(M) \to \Z/2$ is the zero homomorphism.
4. By construction, the deck transformation of orientation covering is orientation reversing.
5. If $N$$N$ is an oriented manifold and $p: N \to M$$p: N \to M$ is a $2$$2$-fold covering with orientation reversing deck transformation, then $p \colon N \to M$$p \colon N \to M$ is isomorphic to the orientation covering.

Proof. (1.) 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$$\hat M$ into to open (and thus oriented) subsets homeomorphic to $M$$M$ and so $M$$M$ is orientable.

(2.) Follows immediately from (1.) since a two-fold cover of a connected space is non-trivial if and only if the total space of the covering is disconnected.

(3.) Follows from (1.) since $w$$w$ classifies the orientation cover: see the page orientation character.

(4.) Is true by construciton as stated.

(5.) 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)$. It is easily checked that his is an isomorphism of these two coverings.

$\square$$\square$

## 3 Examples

We give a list of basic in interesting orientation double 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 \Z/2 \to M$$M \times \Z/2 \to M$.
2. If $n$$n$ is even, $\Rr P^n$$\Rr P^n$ is non-orienable and with orientation cover $S^n \to \Rr P^n$$S^n \to \Rr 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 projection from the 2-torus; $T^2 \to K^2$$T^2 \to K^2$.
4. The orientation of the open Möbius strip $Mö$$Mö$ is the cylinder; $S^1 \times \Rr \to Mö$$S^1 \times \Rr \to Mö$.