Oriented cover

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1 Introduction

This page is based on [Ranicki2002, Definition 4.56]. Let $p:\widetilde{X} \to X$$\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}{^}p:\widetilde{X} \to X$ be a regular covering of a connected space with orientation character $w(X)\in H^1(X;\Zz_2) = \Hom(\pi_1(X),\Zz_2)$$w(X)\in H^1(X;\Zz_2) = \Hom(\pi_1(X),\Zz_2)$. Let $\pi$$\pi$ denote the group of covering translations. Since $\widetilde{X}$$\widetilde{X}$ is a regular cover $\pi_1(\widetilde{X})$$\pi_1(\widetilde{X})$ is a normal subgroup of $\pi_1(X)$$\pi_1(X)$ and $\pi \cong \pi_1(X)/\pi_1(\widetilde{X})$$\pi \cong \pi_1(X)/\pi_1(\widetilde{X})$ (See [Hatcher2002, Proposition 1.39]). Let $q: \pi_1(X) \to \pi$$q: \pi_1(X) \to \pi$ denote the quotient map. The orientation character of the cover factors as

$\displaystyle w(\widetilde{X}) = w(X)\circ p_*,$ $$\displaystyle w(\widetilde{X}) = w(X)\circ p_*,$$

which corresponds to the intuition that the cover $\widetilde{X}$$\widetilde{X}$ is orientable if all loops in $\widetilde{X}$$\widetilde{X}$ project to orientable loops in $X$$X$.

Lemma 1.1. The space $\widetilde{X}$$\widetilde{X}$ is orientable if and only if the orientation character $w(X)$$w(X)$ factors through $\pi$$\pi$.

Proof. Consider the diagram

$\displaystyle \xymatrix{ \pi_1(\widetilde{X}) \ar[dr]^-{w(\widetilde{X})} \ar[d] ^-{p_*}& \\ \pi_1(X) \ar[d]^-{q}\ar[r]^{w(X)} & \Zz_2 \\ \pi \ar@{-->}[ur]^-{w} & }$

If there exists a $w$$w$ such that $w(X) = w\circ q$$w(X) = w\circ q$, then $w(\widetilde{X}) = w\circ q\circ p_* = 0$$w(\widetilde{X}) = w\circ q\circ p_* = 0$. Conversely if $w(\widetilde{X}) = 0$$w(\widetilde{X}) = 0$ then the map

$\displaystyle \begin{array}{rcl} w:\pi & \to & \Zz_2 \\ {[\alpha]} & \mapsto & w(X)(\alpha)\end{array}$ $$\displaystyle \begin{array}{rcl} w:\pi & \to & \Zz_2 \\ {[\alpha]} & \mapsto & w(X)(\alpha)\end{array}$$

for any representative $\alpha\in \pi_1(X)$$\alpha\in \pi_1(X)$ of $[\alpha]\in \pi$$[\alpha]\in \pi$ is well defined and factors $w(X)$$w(X)$.

$\square$$\square$

In light of this we make the following definition.

2 Definition

An orientable cover $(\widetilde{X},\pi,w)$$(\widetilde{X},\pi,w)$ of a (connected) space $X$$X$ with an orientation character $w(X)\in H^1(X;\Zz_2) = \Hom(\pi_1(X),\Zz_2)$$w(X)\in H^1(X;\Zz_2) = \Hom(\pi_1(X),\Zz_2)$ is a regular covering of $X$$X$ with group of covering translations $\pi$$\pi$, together with an orientation character $w:\pi \to \Zz_2$$w:\pi \to \Zz_2$ such that

$\displaystyle \xymatrix{ w(X): \pi_1(X) \ar[r] & \pi \ar[r]^-{w} & \Zz_2. }$

An oriented cover is an orientable cover together with a choice of lift $\widetilde{b}$$\widetilde{b}$ of a basepoint $b\in X$$b\in X$ to $\widetilde{X}$$\widetilde{X}$.

3 Lifts correspond to orientations

Let $(\widetilde{M},\pi,w)$$(\widetilde{M},\pi,w)$ be an oriented cover of a connected manifold $M^m$$M^m$. A choice of lift $\widetilde{b}\in\widetilde{M}$$\widetilde{b}\in\widetilde{M}$ corresponds to a choice of $w$$w$-twisted fundamental class $[\widetilde{M}]\in H_m(M;\Z^w)$$[\widetilde{M}]\in H_m(M;\Z^w)$. Given a lift $\widetilde{b}$$\widetilde{b}$ a fundamental class $[\widetilde{M}]\in H_m(M;\Z^w)$$[\widetilde{M}]\in H_m(M;\Z^w)$ is uniquely determined by setting its restriction to $H_n(\widetilde{M},\widetilde{M}-\{\widetilde{b}\}; \Z)$$H_n(\widetilde{M},\widetilde{M}-\{\widetilde{b}\}; \Z)$ to be $1\in \pi$$1\in \pi$ and extending equivariantly. Conversely, given a fundamental class $[\widetilde{M}]\in H_m(M;\Z^w)$$[\widetilde{M}]\in H_m(M;\Z^w)$ define $\widetilde{b}$$\widetilde{b}$ to be the lift such that the restriction of $[\widetilde{M}]\in H_m(M;\Z^w)$$[\widetilde{M}]\in H_m(M;\Z^w)$ to $H_n(\widetilde{M},\widetilde{M}-\{\widetilde{b}\};\Z)$$H_n(\widetilde{M},\widetilde{M}-\{\widetilde{b}\};\Z)$ is $1\in \pi$$1\in \pi$.

4 Examples

The two most important examples of oriented covers are the universal cover $(\widetilde{X},\pi_1(X),w(X))$$(\widetilde{X},\pi_1(X),w(X))$ and the orientation double cover $(X^w,\Zz_2,\id_{\Zz_2})$$(X^w,\Zz_2,\id_{\Zz_2})$. These correspond to the two extreme cases of factoring the orientation character via $\pi_1(X)$$\pi_1(X)$ and $\Zz_2$$\Zz_2$ respectively. Every oriented cover is a regular cover of $X^w$$X^w$ and has $\widetilde{X}$$\widetilde{X}$ as a regular cover - this is a consequence of the fact that if $H_1, H_2$$H_1, H_2$ are normal subgroups of $G$$G$ with $H_1$$H_1$ a subgroup of $H_2$$H_2$ then $H_1$$H_1$ is a normal subgroup of $H_2$$H_2$. Thus for $(\overline{X},\pi,w)$$(\overline{X},\pi,w)$ any oriented cover of $X$$X$ the following diagram commutes:

$\displaystyle \xymatrix{ && \pi_1(\widetilde{X})=\{1\} \ar[dl] \ar[dd] && \\ & \pi_1(\overline{X})\ar[dr]^-{p_*} \ar[dl]&&&\\ \pi_1(X^w)\ar[rr] && \pi_1(X) \ar[dd]^{\id} \ar[dr]^-{q} \ar[rr]^{w(X)} && \Z_2\\ &&&\pi \ar[ur]^-{w} & \\ && \pi_1(X)\ar[ur]&& }$

5 Orientations of the orientation double cover

Let $M^m$$M^m$ be a connected manifold and let $(M^w,\Zz_2,\id_{\Zz_2})$$(M^w,\Zz_2,\id_{\Zz_2})$ be the orientation double cover. Following [Ranicki2002, Proposition 4.48] there is a short exact sequence of $\Z[\Z_2]$$\Z[\Z_2]$-modules

$\displaystyle \xymatrix@R=1mm{0 \ar[r] &\Z^{-} \ar[r] & \Z[\Z_2] \ar[r] & \Z \ar[r] &0 \\ & 1 \ar@{|->}[r] & 1-T, && \\ &&a+bT \ar@{|->}[r] & a+b. &}$

Applying $-\otimes_{\Z[\Z_2]} S(M^w)$$-\otimes_{\Z[\Z_2]} S(M^w)$ we obtain another short exact sequence which induces the following exact sequence in homology:

$\displaystyle \xymatrix{\ldots \ar[r] & H_{n+1}(M) \ar[r] & H_n(M;\Z^w) \ar[r] & H_n(M^w) \ar[r]^-{p_*} & H_n(M) \ar[r] & \ldots}$

As $M$$M$ is $m$$m$-dimensional we have that $H_{m+1}(M)=0$$H_{m+1}(M)=0$ and so by the long exact sequence

$\displaystyle H_m(M;\Z^w)\cong \ker(p_*:H_m(M^w)\to H_m(M)).$ $$\displaystyle H_m(M;\Z^w)\cong \ker(p_*:H_m(M^w)\to H_m(M)).$$

In other words, a $w$$w$-twisted fundamental class of a connected manifold $M$$M$ can be thought of as a fundamental class of the orientation double cover that projects to zero in $H_m(M)$$H_m(M)$. In the case that the manifold $M$$M$ is already orientatable, the orientation double cover $(M^w,\Zz_2,\id_{\Zz_2})$$(M^w,\Zz_2,\id_{\Zz_2})$ consists of two disjoint copies of $M$$M$ and a $w$$w$-twisted orientation corresponds to an orientation of $M^w$$M^w$ where the two copies of $M$$M$ are given opposite orientations. In the case that $M$$M$ is non-orientable $H_m(M;\Z)=0$$H_m(M;\Z)=0$ so a $w$$w$-twisted orientation is precisely an orientation of the cover $M^w$$M^w$.

6 References

• [Hatcher2002] A. Hatcher, Algebraic topology, Cambridge University Press, 2002. MR1867354 (2002k:55001) Zbl 1044.55001
• [Ranicki2002] A. Ranicki, Algebraic and geometric surgery, The Clarendon Press Oxford University Press, Oxford, 2002. MR2061749 (2005e:57075) Zbl 1003.57001