Math 396. Orientations on bundles and manifolds1. Orientation atlases and orientation formsRecall that an oriented trivializing atlas for E → M is a trivializing atlas{(Ui, φi: E|Ui' Ui× Rni)}such that whenever Ui∩ Ujis non-empty (so ni= nj) the ordered bases of E(m) induced by φiand φjfor each m ∈ Ui∩ Ujlie in the same orientation class (i.e., the change of basis matrix forthe m-fib e r has positive determinant). In other words, a trivializing atlas is oriented if it gives awell-defined orientation on each fiber E(m) (a non-trivial condition only for those m in a doubleoverlap Ui∩ Ujfor i 6= j). Thus, when given an oriented trivializing atlas we get an orientationon every E(m). (Note that this would not make sense if we allowed E(m) = 0.) We say that twooriented trivializing atlases are equivalent if, for each m ∈ M , they put the same orientation onthe fiber E(m). As we saw in class, this really is an equivalence relation, and in each equivalenceclass there is a unique maximal element (in the sense of containing all trivializing frames from alloriented atlases in the equivalence class). These maximal elements are called orientation atlasesfor E → M. Thus, two orientation atlases are equal if and only if they define the same orientationon each fiber E(m) (as equivalence is the same as equality for the maximal elements).We define an orientation form on E → M to be a nowhere-vanishing global section ω of det E,which is to say a trivializing section of the line bundle det E. The reason for the name is thatfor each m ∈ M the nonzero fiber value ω(m) ∈ det(E(m)) specifies a connected component ofdet(E(m)) − {0} and so puts an orientation on the vector space E(m). Two orientation forms ωand ω0on E → M are non-vanishing Cpsections of the same line bundle det(E), and so ω = fω0for a unique non-vanishing Cpfunction f on M. We say ω and ω0are equivalent (denoted ω ∼+ω0)if f is everywhere positive. This says exactly that the orientations on E(m) specified by ω(m) andω0(m) coincide for every m ∈ M. It is clear that ∼+is an equivalence relation, and the orientationforms in a common ∼+-equivalence class put the same orientation on each fiber E(m).We wish to set up a natural bijection between ∼+-equivalence classes of orientation forms onE → M and orientation atlases on E → M. The comparison will be via the orientations inducedon the fibers E(m) from each piece of data.Theorem 1.1. Pick an o rientation atlas on E → M , and let µmbe the resulting orientation onE(m) for each m ∈ M . There is a unique ∼+-equivalence class of orientation forms that inducethe orientation µmon E(m) for all m ∈ M .Pick a ∼+-equivalence class of orientation forms on E → M , and let µ0mbe the resulting orien-tation on E(m) for each m ∈ M . There is a unique orientation atlas on E → M that induces theorientation µ0mon E(m) for all m ∈ M .These procedures define inverse bijections between the set of orientation atlases and the set of∼+-equivalence classes of orientation forms on E → MProof. Since equality of orientation atlases and ∼+-equivalence of orientation forms may be checkedby considering orientations on fibers of E → M, the final part of the theorem (concerning inversebijections) follows from the rest. Similarly, the uniqueness aspects in the first two claims in thetheorem are clear; the only real issue is that of existence.Pick an orientation form ω ∈ (det E)(M), and let µmbe the resulting orientation on E(m) foreach m ∈ M. (This orientation of E(m) only depends on ω up to ∼+-equivalence.) We see kto find an oriented trivializing atlas for E → M that also induces the orientation µmon E(m)for each m ∈ M. Consider connected open sets U ⊆ M over which E is trivial; such opens do12cover M. Let {Ui} be a collection of such opens that cover M. For each i, let {s1i, . . . , sni,i} inE(Ui) be a trivialization of E|Ui, so s1i∧ · · · ∧ sni,iis a non-vanishing section of (det E)(Ui). Thus,s1i∧ · · · ∧ sni,i= fiω|Uifor a non-vanishing Cpfunction fion Ui. But Uiis connected, so fihasconstant sign. Replacing s1iwith −s1iif necessary, we can arrange that fi> 0 on Ui, so we havebuilt a trivializing frame over Uithat induces the orientation µmon E(m) for each m ∈ Ui. Letφi: E|Ui' Ui× Rnibe the trivialization of E|Uithat we have just built.I claim that the data {(Ui, φi: E|Ui' Ui× Rni)} defined by the above trivializations is anoriented trivializing atlas that induces the orientation µmon E(m) for each m ∈ M (and so theassociated orientation atlas gives what we need). Since two ordered bases of a vector space havechange-of-basis matrix with positive determinant if and only if they give rise to the same orientation,the trivializing atlas is oriented if for each Uiand m ∈ Ui, the orientation put on E(m) by thetrivialization φiover Uidepends only on m and not on the particular i such that Uicontains m(so if m ∈ Ui∩ Ujthen the ith and jth trivializations put the same orientation on E(m)). We cando better, as is required: the ith trivialization puts the orientation µmon E(m) for all m ∈ Ui.This follows from how we built the trivialization of E|Uiabove. This completes the passage fromequivalence classes of orientation forms to orientation atlases.To go in the reverse direction, we pick an orientation atlas on E → M, say putting orientationµmon E(m) for each m ∈ M , and we seek to build a trivializing section ω ∈ (det E)(M) such thatω(m) ∈ det(E(m)) − {0} lies in the component distinguished by µmfor each m ∈ M. We firstconsider the local version of the problem, and then we will globalize using a Cppartition of unity(hence the need to assume M is second countable and Hausdorff, not merely a premanifold withcorners). Let {Ui} be the opens in the chosen orientation atlas, so they are an open covering ofM and we have trivializations E|Ui' Ui× Rnigiving rise to the orientation µmon E(m) for eachm ∈ Ui. Passing to the top exterior power, we get a trivialization (det E)|Ui' Ui× ∧ni(Rni) suchthat for each m ∈ Uithe induced isomorphism on m fibers is det(E(m)) ' ∧ni(Rni) with the µm-component on the left going over to the “standard” component on the right (for e1∧ · · · ∧ eni, with{ej} the standard basis of Rni). Thus, the constant section e1∧ · · · ∧ enigoes over to a trivializingsection ωiof (det E)|Uisuch that on each fiber over m ∈ Uithe vector ωi(m) ∈