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M More generally, for EXE \to X any orthogonal group-principal bundle classified by a morphism E:XBOE : X \to \mathbf{B} O, the corresponding orientation double cover is the 2\mathbb{Z}_2-bundle classified by. } I think I got it right so far, but could't figure out the rest Of course, from my construction, if $M$ is orientable, then $M_{+}$ and $M_{-}$ are open subsets which cover $\tilde{M}$, so $\tilde{M}$ is not connected. U &=\underbrace{\varphi_\alpha\circ\varphi_\beta^{-1}}_{\text{smooth}}(x_1,,x_n). Indeed, suppose that an orientation of M is fixed. H 2 ) ; But Mbius strips, real projective planes, and Klein bottles are non-orientable. Such charts form an oriented atlas for M. This also makes $\pi$ a local diffeomorphism, since $\pi|_{U_\alpha^\pm}=\varphi_\alpha^{-1}\circ\varphi_\alpha^\pm$ and $\varphi_\alpha,\varphi_\alpha^\pm$ are diffeomorphisms. You may want to take a look at John Lee's Introduction to Smooth Manifolds. ; together with the function $\pi:\widetilde{M}\to M$ with $\pi((p,o_p))=p$. M { X-uUOx/%A 2z?%V15x7DiNl;r3BLrN73})73[>}u)ix3es|'W )3VNI1Qw0=}7Q pD3f7KfMsV~[|[NI3*Uh0^4ySb1?6 . Let U be an open subset of M chosen such that Any surface has a triangulation: a decomposition into triangles such that each edge on a triangle is glued to at most one other edge. In the latter case (which means there is an orientation-reversing path), the subgroup corresponds to a connected double covering; this cover is orientable by construction. ( Define the following subsets of $\widetilde{M}$: [6] In the context of general relativity, a spacetime manifold is space orientable if, whenever two right-handed observers head off in rocket ships starting at the same spacetime point, and then meet again at another point, they remain right-handed with respect to one another. Definition: if $M$ is a smooth manifold, define the orientable double cover of $M$ by: $$\widetilde{M}:=\{(p, o_p)\mid p\in M, o_p\in\{\text{orientations on }T_pM\}\}$$. direct sum of vector bundles, tensor product, external tensor product, fiber sequence/long exact sequence in cohomology. L 2 That is, the transition functions of the manifold induce transition functions on the tangent bundle which are fiberwise linear transformations. %PDF-1.5 By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. Step 2 (Differentiable Structure of $\widetilde{M}$): Define $\varphi_\alpha^+:U^+_\alpha\to \varphi_\alpha(U_\alpha)\subset\mathbb{R}^n$ by $\varphi^+_\alpha=\varphi_\alpha\circ\pi|_{U_\alpha^+}$ and, similarly, $\varphi_\alpha^-:U^-_\alpha\to\varphi_\alpha(U_\alpha)\subset\mathbb{R}^n$ by $\varphi^-_\alpha=\varphi_\alpha\circ\pi|_{U_\alpha^-}$. As $\widetilde{M}$ is orientable, in particular $U$ is orientable, so $M$ inherits an orientation from $U$ via $\pi|_U$. {\displaystyle w_{1}(M)\in H^{1}(M;\mathbf {Z} /2)} . Indeed, notice that for any $\alpha\in\Lambda$ we have $\pi^{-1}(U_\alpha)=U_\alpha^+\cup U_\alpha^-$ and $\pi(U_\alpha^\pm)=U_\alpha$. Last revised on February 15, 2019 at 07:08:01. (When n = 0, an orientation of M is a function M {1}.). ) More precisely, if S is orientable then H1(S) is a free abelian group, and if not then H1(S) = F + Z/2Z where F is free abelian, and the Z/2Z factor is generated by the middle curve in a Mbius band embedded in S. Let M be a connected topological n-manifold. When M is smooth, at each point p of M, the restriction of the tangent bundle of M to M is isomorphic to TpM R, where the factor of R is described by the inward pointing normal vector. + n Use MathJax to format equations. The topology on O is defined so that. A reflection of Rn through the origin acts by negation on When the Jacobian determinant is positive, the transition function is said to be orientation preserving. {^Fze3^XB-DOv3mY" D#Pk|)8GA:M&H; |1"5t&Dz8q-%crN\]tRHvj \varphi_\alpha^\pm\circ(\varphi_\beta^\pm)^{-1}(x_1,,x_n)&=\varphi_\alpha^\pm\left(\underbrace{\varphi_\beta^{-1}(x_1,,x_n)}_{=:p},\pm\left[\left.\frac{\partial }{\partial\varphi_\beta^1}\right|_p,,\left.\frac{\partial }{\partial\varphi_\beta^n}\right|_p\right]\right)\\ Connect and share knowledge within a single location that is structured and easy to search. @PaulFrost, you have a point. Conversely, M is orientable if and only if the structure group of the tangent bundle can be reduced in this way. M H The best answers are voted up and rise to the top, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, $\{(\widetilde{U}_{\alpha}, \widetilde{\phi}_{\alpha})\}$. For XX a manifold, not necessarily oriented or even orientable, write, for any choice of orthogonal structure. [1] A space is orientable if such a consistent definition exists. The existence of a volume form implies a reduction of the structure group of the tangent bundle or the frame bundle to GL+(n, R). H The codomain of this group has two generators, and maps to one of them. ( Z M M If M is a manifold with boundary, then an orientation of M is defined to be an orientation of its interior. 1 p Let U Rn+ be a chart at a boundary point of M which, when restricted to the interior of M, is in the chosen oriented atlas. K Various equivalent formulations of orientability can be given, depending on the desired application and level of generality. ( {\displaystyle H_{n}\left(M,M\setminus \{p\};\mathbf {Z} \right)} ; An exposition in a broader context is in the section higher spin structures at. n 1 (p, o_p)&\mapsto \phi(p) , For example, a torus embedded in. A space-orientation of a pseudo-Riemannian manifold is identified with a section of the associated bundle, where O(M) is the bundle of pseudo-orthogonal frames. ) H H For the orientation of a shape in a space, see, Orientability of differentiable manifolds, Homology and the orientability of general manifolds, Orientation of manifolds in generalized cohomology theories, https://en.wikipedia.org/w/index.php?title=Orientability&oldid=1082487729, Short description is different from Wikidata, Creative Commons Attribution-ShareAlike License 3.0, The Encyclopedia of Mathematics article on, This page was last edited on 13 April 2022, at 12:47. [2][3], The most intuitive definitions require that M be a differentiable manifold. Some of these definitions require that M has extra structure, like being differentiable. ( On a two-dimensional manifold, it corresponds to a choice of clockwise and counter-clockwise. Step 3 (Orientability of $\widetilde{M}$): Let's construct a pointwise orientation $O:(p,o_p)\mapsto O_{(p,o_p)}$ on $\widetilde{M}$. an etale space with local sections the 2-element set. M In particular, if the first cohomology group with Z/2 coefficients is zero, then the manifold is orientable. / By the excision theorem, A manifold M is orientable if and only if the first StiefelWhitney class There is a canonical map : O M that sends a local orientation at p to p. It is clear that every point of M has precisely two preimages under . ) p Another way to construct this cover is to divide the loops based at a basepoint into either orientation-preserving or orientation-reversing loops. Moreover, any other chart around p is related to the first chart by an orientation preserving transition function, and this implies that the two charts yield the same generator, whence the generator is unique. (see also Chern-Weil theory, parameterized homotopy theory), topological vector bundle, differentiable vector bundle, algebraic vector bundle, real vector bundle, complex vector bundle, direct sum of vector bundles, tensor product of vector bundles, inner product of vector bundles, dual vector bundle, universal principal bundle, universal principal -bundle, universal vector bundle, universal complex line bundle, groupal model for universal principal -bundles, prequantum circle bundle, prequantum circle n-bundle. ; M )#f{b7-L%Cl>ODgo5 zK}} =>[!'F{b}5Dp~^Eg 3LZz3S ;f`DXksO:'-Od ~ "[+ More precisely, let O be the set of all local orientations of M. To topologize O we will specify a subbase for its topology. Do weekend days count as part of a vacation? ( } Since $(p,o_p)$ is arbitrary, this means that $O$ is continuous. M Most surfaces encountered in the physical world are orientable. In step 3 it would perhaps be more transparent to say that for each $(p,o) \in \tilde M$ we get a linear isomorphism $d_{(p,o)}\pi : T_{(p,o)} \tilde M \to T_pM$, and an orientation of $\tilde M$ is given by taking on $T_{(p,o)} \tilde M$ the unique orientation $\omega_{(p,o)}$ such $d_{(p,o)}\pi(\omega_{(p,o)}) = o$. Z Both $\varphi_\alpha^+,\varphi_\alpha^-$ are homeomorphisms, because $\varphi_\alpha$ and $\pi|_{U_\alpha^\pm}$ are homeomorphisms. This raises the question of what exactly such transition functions are preserving. M [5] This characterization of orientability extends to orientability of general vector bundles over M, not just the tangent bundle. A computation with the long exact sequence in relative homology shows that the above homology group is isomorphic to ) Step 1 (Topology of $\widetilde{M}$): Take an atlas $\{(U_\alpha,\varphi_\alpha)\}_{\alpha\in\Lambda}$ such that $\{U_\alpha\}_{\alpha\in\Lambda}$ is a countable basis for $M$. M On a one-dimensional manifold, a local orientation around a point p corresponds to a choice of left and right near that point. Thanks for contributing an answer to Mathematics Stack Exchange! Moreover, for an arbitrary $p\in M$, any open set $U_\alpha$ containing $p$ is such that $\pi^{-1}(U_\alpha)=U_\alpha^+\cup U_\alpha^-$ (disjoint union) and $\pi|_{U_\alpha^\pm}:U_\alpha^\pm\to U_\alpha$ is a homeomorphism, which shows that $\pi$ is a double covering. An abstract surface (i.e., a two-dimensional manifold) is orientable if a consistent concept of clockwise rotation can be defined on the surface in a continuous manner. B If a spacetime is time-orientable then the two observers will always agree on the direction of time at both points of their meeting. n \varphi_\alpha^\pm\circ(\varphi_\beta^\pm)^{-1}(x_1,,x_n)&=\varphi_\alpha^\pm\left(\underbrace{\varphi_\beta^{-1}(x_1,,x_n)}_{=:p},\pm\left[\left.\frac{\partial }{\partial\varphi_\beta^1}\right|_p,,\left.\frac{\partial }{\partial\varphi_\beta^n}\right|_p\right]\right)\\ >> If this is done in such a way that, when glued together, neighboring edges are pointing in the opposite direction, then this determines an orientation of the surface. Otherwise the surface is non-orientable. n $_\blacksquare$. They cannot be preserving an orientation of the manifold because an orientation of the manifold is an atlas, and it makes no sense to say that a transition function preserves or does not preserve an atlas of which it is a member. But the opposite implication is still not clear to me. p ) ( G For an orientable surface, a consistent choice of "clockwise" (as opposed to counter-clockwise) is called an orientation, and the surface is called oriented. ) n {\displaystyle GL^{+}(n)} . $\{(U_\alpha,\varphi_\alpha)\}_{\alpha\in\Lambda}$, $$U_\alpha^+:=\left\{(p,o_p)\in\widetilde{M}\mid p\in U_\alpha,\, o_p=\left[\left.\frac{\partial }{\partial\varphi_\alpha^1}\right|_p,,\left.\frac{\partial }{\partial\varphi_\alpha^n}\right|_p\right]\right\},$$, $$U_\alpha^-:=\left\{(p,o_p)\in\widetilde{M}\mid p\in U_\alpha,\,o_p=-\left[\left.\frac{\partial }{\partial\varphi_\alpha^1}\right|_p,,\left.\frac{\partial }{\partial\varphi_\alpha^n}\right|_p\right]\right\}.$$, $\{U_\alpha^+,U_\alpha^-\}_{\alpha\in\Lambda}$, $\pi^{-1}(U_\alpha)=U_\alpha^+\cup U_\alpha^-$, $\pi|_{U_\alpha^\pm}:U_\alpha^\pm\to U_\alpha$, $\varphi_\alpha^+:U^+_\alpha\to \varphi_\alpha(U_\alpha)\subset\mathbb{R}^n$, $\varphi^+_\alpha=\varphi_\alpha\circ\pi|_{U_\alpha^+}$, $\varphi_\alpha^-:U^-_\alpha\to\varphi_\alpha(U_\alpha)\subset\mathbb{R}^n$, $\varphi^-_\alpha=\varphi_\alpha\circ\pi|_{U_\alpha^-}$, \begin{align*} U Occasionally, n = 0 must be made into a special case. ; Another way to define orientations on a differentiable manifold is through volume forms. H About the orientability, I guess it will have something to do with the orientability of the atlas $\{(\widetilde{U}_{\alpha}, \widetilde{\phi}_{\alpha})\}$, but since I can't figure out the definition of $\widetilde{\phi}$, I'm stuck. An oriented atlas is one for which all transition functions are orientation preserving, M is orientable if it admits an oriented atlas, and when n > 0, an orientation of M is a maximal oriented atlas. That is to say that a loop going around one way on the surface can never be continuously deformed (without overlapping itself) to a loop going around the opposite way. stream n Formally, the pseudo-orthogonal group O(p,q) has a pair of characters: the space orientation character + and the time orientation character , Their product =+ is the determinant, which gives the orientation character. A local orientation of M around a point p is a choice of generator of the group. If X1, , Xn is a basis of tangent vectors at a point p, then the basis is said to be right-handed if (X1, , Xn) > 0.

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