# Orientation character

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## 1 Definition

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 connected topological manifold throughout. The following definition can be found in [Davis&Kirk2001, Section 5]. The orientation character is a homomorphism

$\displaystyle w \colon \pi_1(M) \to \{ \pm 1 \} =: C_2$

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

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

The orientation character $w\colon\pi_1(M)\to C_2$$w\colon\pi_1(M)\to C_2$ induces a map $M \to BC_2$$M \to BC_2$ from $M$$M$ to the classifying space of $C_2$$C_2$, which is unique up to homotopy. By the definition of the orientation character, this map classifies the orientation covering. So $w$$w$ is trivial if and only if $\widehat M \cong M\times C_2$$\widehat M \cong M\times C_2$, and therefore $M$$M$ is orientable.

## 2 Alternative descriptions

Let $\widetilde M$$\widetilde M$ be the universal covering of $M$$M$. Then $\pi_1(M)$$\pi_1(M)$ acts on $\widetilde M$$\widetilde M$ by deck transformations.

Proposition 2.1. Let $w\colon \pi_1(M)\to C_2$$w\colon \pi_1(M)\to C_2$ be the orientation character. Then $w([\gamma])=1$$w([\gamma])=1$ if and only if the action of $[\gamma]$$[\gamma]$ on $\widetilde M$$\widetilde M$ is orientation preserving.

Proof. If $M$$M$ is orientable, then an orientation on $M$$M$ induces an orientation on $\widetilde M$$\widetilde M$ and every deck transformation is orientation preserving. If $M$$M$ is non-orientable, then the standard orientation of the orientation covering $\widehat M$$\widehat M$ of $M$$M$ induces an orientation on $\widetilde M$$\widetilde M$. By construction the non-trivial deck transformation of $\widehat M$$\widehat M$ is orientation-reversing. Therefore, a deck transformation of $\widetilde M\to M$$\widetilde M\to M$ is orientation preserving if and only if it acts trivially on $\widehat M$$\widehat M$.

$\square$$\square$

If the manifold $M^n$$M^n$ carries a smooth structure, the orientation character can be additionally characterised in terms of tangential data. Define a map $w^\prime \colon \pi_1(M) \to C_2$$w^\prime \colon \pi_1(M) \to C_2$ in the following way: Let $x \in M$$x \in M$ be the base point. Pick a chart $\varphi \colon U \to V \subset \mathbb{R}^n$$\varphi \colon U \to V \subset \mathbb{R}^n$ around $x$$x$. Let $\gamma$$\gamma$ be a based loop in $M$$M$. The standard orientation $[e_1, \dots, e_n]$$[e_1, \dots, e_n]$ of $\mathbb{R}^n$$\mathbb{R}^n$ corresponds to an orientation $o$$o$ of $T_xM$$T_xM$ under the differential of $\varphi^{-1}$$\varphi^{-1}$ (cf. Orientation of manifolds). Fibre transport along $\gamma$$\gamma$ yields another orientation $o^\prime$$o^\prime$ of $T_xM$$T_xM$. Now set

$\displaystyle w^\prime([\gamma]) := \left\{ \begin{array}{rl} 1~~ & \text{if o^\prime=o,} \\ -1~~ & \text{if o^\prime=-o.} \end{array} \right.$

Using the comparison between homological and tangential orientation (see Orientation of manifolds), one can show the following:

Proposition 2.2. The map $w^\prime$$w^\prime$ is a well-defined group homomorphism and coincides with the orientation character $w$$w$.

In addition to this geometric characterisation, the orientation character also admits a description in terms of characteristic classes: Under the "evaluation" homomorphism of the Universal Coefficient Theorem $H^1(M;\Zz/2) \to \hom(H_1(M;\Zz), \Zz/2)$$H^1(M;\Zz/2) \to \hom(H_1(M;\Zz), \Zz/2)$, the first Stiefel-Whitney class $w_1(M)$$w_1(M)$ corresponds to a homomorphism $H_1(M;\Zz/2) \to \Zz/2$$H_1(M;\Zz/2) \to \Zz/2$ which we also call $w_1(M)$$w_1(M)$.

Proposition 2.3. Let $\rho \colon \pi_1(M) \to H_1(M;\mathbb{Z})$$\rho \colon \pi_1(M) \to H_1(M;\mathbb{Z})$ be the Hurewicz homomorphism. Then the orientation character of $M$$M$ coincides with the composition $w_1(M) \circ \rho$$w_1(M) \circ \rho$ via the canonical isomorphism $C_2 \cong \Zz/2$$C_2 \cong \Zz/2$.

Proof. All 1-manifolds are orientable, so suppose that $n = \textup{dim}(M) > 1$$n = \textup{dim}(M) > 1$. Let $\tau\colon M \to BO(n)$$\tau\colon M \to BO(n)$ classify the tangent bundle of $M$$M$ and consider the induced map $\tau_* \colon \pi_1(M) \to \pi_1(BO(n))$$\tau_* \colon \pi_1(M) \to \pi_1(BO(n))$. Now $\pi_1(BO(n)) \cong \pi_0(O(n)) \cong C_2$$\pi_1(BO(n)) \cong \pi_0(O(n)) \cong C_2$ is generated by the classifying map of the Möbius bundle; denote this class by $[\eta]$$[\eta]$. Hence, for a based loop $\gamma$$\gamma$ in $M$$M$ with homotopy class $[\gamma] \in \pi_1(M)$$[\gamma] \in \pi_1(M)$, we see the pullback $\gamma^*TM$$\gamma^*TM$ is non-trivial if and only if $\tau_*([\gamma]) = [\eta]$$\tau_*([\gamma]) = [\eta]$. By Proposition 2.1, we see that $w([\gamma]) = 1$$w([\gamma]) = 1$ if and only if $\tau_*([\gamma]) = [\eta]$$\tau_*([\gamma]) = [\eta]$. Applying the Universal Coefficient Theorem for $BO(n)$$BO(n)$, we see that the universal first Stiefel-Whitney class $w_1$$w_1$, which generates $H^1(BO(n); \Zz/2)$$H^1(BO(n); \Zz/2)$, has the property that $w_1(\rho([\eta])) = 1$$w_1(\rho([\eta])) = 1$. Hence, we see that $w([\gamma]) = 1$$w([\gamma]) = 1$ if and only if $\tau_*([\gamma]) = [\eta]$$\tau_*([\gamma]) = [\eta]$ if and only if $w_1(\rho(\tau_*[\gamma])) = 1$$w_1(\rho(\tau_*[\gamma])) = 1$ if and only if $w_1(M)(\rho([\gamma])) = 1$$w_1(M)(\rho([\gamma])) = 1$.

$\square$$\square$

## 3 Examples

1. Since $\mathbb RP^n$$\mathbb RP^n$ is orientable if and only if $n$$n$ is odd, the orientation character $w\colon\pi_1(\Rr P^n)\to C_2$$w\colon\pi_1(\Rr P^n)\to C_2$ is trivial if and only if $n$$n$ is odd. If $n$$n$ is even, $w\colon\pi_1(\Rr P^n) \to C_2$$w\colon\pi_1(\Rr P^n) \to C_2$ is an isomorphism.
2. The open Möbius strip has fundamental group $\Zz$$\Zz$ and is non-orientable. Therefore, the orientation character is given by the surjection $\Zz\twoheadrightarrow\Zz/2 \cong C_2$$\Zz\twoheadrightarrow\Zz/2 \cong C_2$.
3. If $i\colon N\to M$$i\colon N\to M$ is an embedding of a manifold of the same dimension (possibly with boundary), then the orientation character of $N$$N$ is given as the composition $\pi_1(N)\xrightarrow{i_*} \pi_1(M)\xrightarrow{w}C_2$$\pi_1(N)\xrightarrow{i_*} \pi_1(M)\xrightarrow{w}C_2$. This follows from the fact that the atlas of $M$$M$ with local orientations induces by restriction an atlas of $N$$N$ with local orientations. Therefore, the orientation covering of $N$$N$ is the orientation covering of $M$$M$ restricted to $N$$N$.
4. Let $n \geq 2$$n \geq 2$. The preimage of an embedded disk $D^n\subseteq M^n$$D^n\subseteq M^n$ in the orientation covering $\widehat M$$\widehat M$ is a disjoint union of two disks. The orientation covering of a connected sum $M\#N$$M\#N$ along $D^n$$D^n$ is the "double connected sum" of the orientation coverings along the preimages of $D^n$$D^n$ in the orientation coverings. How to pair the disks is determined by the local orientations. From this it follows that the orientation character of $M\#N$$M\#N$ is given by the pushout
$\displaystyle \xymatrix{\pi_1(S^{n-1})\ar[r]\ar[d]&\pi_1(M\backslash \text{int}(D^n))\ar[d]\ar[ddr]&\\ \pi_1(N\backslash \text{int}(D^n))\ar[r]\ar[drr]&\pi_1(M\#N)\ar@{-->}[dr]&\\&&C_2}$

## 4 The orientation character via Poincaré duality

Even though the original definition is very geometric, the orientation character is already completely determined by the homotopy type of a given closed manifold. Both the Hurewicz homomorphism and the first Stiefel-Whitney class are homotopy invariants (for the second point, see the Manifold Atlas page Wu class), so we already know this for differentiable manifolds by Proposition 2.3. In general this is most easily seen in the more abstract setting of Poincaré complexes, see [Lück2001, Section 3.1].

Lemma 4.1. If $(X, [X], w)$$(X, [X], w)$ is a connected Poincaré complex, then $w$$w$ is the only homomorphism $y\colon\pi_1(X) \to C_2$$y\colon\pi_1(X) \to C_2$ such that $X$$X$ is a Poincaré complex with orientation character $y$$y$.

Proof. Suppose $(X,[X],y)$$(X,[X],y)$ is a connected Poincaré complex. We first introduce some notation:

1. $\mathbb{Z}^y$$\mathbb{Z}^y$ denotes $\mathbb{Z}$$\mathbb{Z}$ as a right $\mathbb{Z}[\pi_1(X)]$$\mathbb{Z}[\pi_1(X)]$-module, where the $\pi_1(X)$$\pi_1(X)$-action is given by $z \cdot g = y(g) \cdot z$$z \cdot g = y(g) \cdot z$.
2. $\mathbb{Z}[\pi_1(X)]^y := \mathbb{Z}^y \otimes_{\mathbb{Z}[\pi_1(X)]}\mathbb{Z}[\pi_1(X)]$$\mathbb{Z}[\pi_1(X)]^y := \mathbb{Z}^y \otimes_{\mathbb{Z}[\pi_1(X)]}\mathbb{Z}[\pi_1(X)]$, where the tensor product is equipped with the diagonal action.
3. For a left $\mathbb{Z}[\pi_1(X)]$$\mathbb{Z}[\pi_1(X)]$-chain complex $C_*$$C_*$, we define $C_*^y$$C_*^y$ analogous to (2.).

Applying $H_*$$H_*$ to the $\mathbb{Z}[\pi_1(X)]$$\mathbb{Z}[\pi_1(X)]$-chain equivalence $? \cap [X] \colon C^{n-*}(\widetilde{X}) \to C_*(\widetilde{X})^y$$? \cap [X] \colon C^{n-*}(\widetilde{X}) \to C_*(\widetilde{X})^y$, we obtain an isomorphism of $\mathbb{Z}[\pi_1(X)]$$\mathbb{Z}[\pi_1(X)]$-modules

$\displaystyle H_n(C^{n-*}(\widetilde{X})) \xrightarrow{\cong} H_0(C_*(\widetilde{X})^y) \cong \mathbb{Z}^y.$

Note that the left hand side is independent of the orientation character $y$$y$. Therefore, $\Zz^y$$\Zz^y$ is isomorphic to $\Zz^w$$\Zz^w$ as a $\Zz[\pi_1(X)]$$\Zz[\pi_1(X)]$-module and $y$$y$ and $w$$w$ have to agree.

$\square$$\square$

Corollary 4.2. If $(X, [X], w)$$(X, [X], w)$ is a connected Poincaré complex, then $w$$w$ depends only on the homotopy type of $X$$X$.

Proof. Let $(X,[X],w)$$(X,[X],w)$ and $(X',[X'],w')$$(X',[X'],w')$ be connected Poincaré complexes and $f:X\rightarrow X'$$f:X\rightarrow X'$ be a homotopy equivalence. Then $(X,[X],w'\circ f_*\colon\pi_1(X)\to C_2)$$(X,[X],w'\circ f_*\colon\pi_1(X)\to C_2)$ is a Poincaré complex and $w$$w$ and $w'\circ f_*$$w'\circ f_*$ have to agree.

$\square$$\square$

Remark 4.3. The above statements are also true for connected Poincaré pairs.

Proposition 2.3 generalises to the following:

Proposition 4.4 [Byun1999, Lemma 4.3]. Using the Universal Coefficients Theorem and the Hurewicz map, the orientation character of a connected Poincaré complex coincides with the first Stiefel-Whitney class of its Spivak normal fibration.