Oriented cover

From Manifold Atlas
Jump to: navigation, search

This page has not been refereed. The information given here might be incomplete or provisional.


[edit] 1 Introduction

This page is based on [Ranicki2002, Definition 4.56]. Let 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). Let \pi denote the group of covering translations. Since \widetilde{X} is a regular cover \pi_1(\widetilde{X}) is a normal subgroup of \pi_1(X) and \pi \cong \pi_1(X)/\pi_1(\widetilde{X}) (See [Hatcher2002, Proposition 1.39]). Let 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_*,
which corresponds to the intuition that the cover \widetilde{X} is orientable if all loops in \widetilde{X} project to orientable loops in X.

Lemma 1.1. The space \widetilde{X} is orientable if and only if the orientation character w(X) factors through \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 such that w(X) = w\circ q, then w(\widetilde{X}) = w\circ q\circ p_* = 0. Conversely if w(\widetilde{X}) = 0 then the map

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

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


In light of this we make the following definition.

[edit] 2 Definition

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

[edit] 3 Lifts correspond to orientations

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

[edit] 4 Examples

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

[edit] 5 Orientations of the orientation double cover

Let M^m be a connected manifold and let (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]-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) 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 is m-dimensional we have that 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)).

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

[edit] 6 References

Personal tools