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  "body": "\n## Definition\n\nLet $M$ be a [[topological manifold|QQQ9B1DK4P]] of dimension $n \\ge 1$, and use integral [[relative homology|4BVWH0BN28]].\nFor each $x \\in M$, the [[excision theorem|SGHMSBHHA0]] gives $H_n(M, M \\setminus \\set{x}) \\cong H_n(\\R^n, \\R^n \\setminus \\set{0}) \\cong \\Z$.\nA *local orientation* of $M$ at $x$ is a choice of generator $\\mu_x$ of $H_n(M, M \\setminus \\set{x})$.\nAn *orientation* of $M$ is a function $x \\mapsto \\mu_x$ assigning a local orientation to each point, with the following local consistency.\nEach $x \\in M$ has a chart neighbourhood $U \\cong \\R^n$ containing an open ball $B$ of finite radius about $x$, and a class $\\mu_B \\in H_n(M, M \\setminus B)$.\nFor every $y \\in B$, the image of $\\mu_B$ in $H_n(M, M \\setminus \\set{y})$ is $\\mu_y$.\n$M$ is *orientable* if it has an orientation.\n\n## Remarks\n\nA connected orientable manifold has exactly two orientations.\nFor a [[smooth manifold|F45VQ47GPV]], an orientation in this sense is the same as a choice of [[orientation|0QJJAQ6ZYZ]] of each tangent space $T_x M$ that is locally given by a chart.\nIt is also the same as an atlas whose transition maps have positive Jacobian determinant.\nReplacing $\\Z$ by a commutative ring $R$ gives $R$-orientations; every manifold has a unique $\\Z/2$-orientation.\n\nExample: the [[sphere|T62369RFH6]] $S^n$ is orientable.\nFix a generator $\\mu \\in H_n(S^n) \\cong \\Z$; its image in each $H_n(S^n, S^n \\setminus \\set{y})$ is a generator, and one takes this image as $\\mu_y$.\nThe condition holds with $\\mu_B$ the image of $\\mu$ in $H_n(S^n, S^n \\setminus B)$ for every chart ball $B$.\n\nNon-example: the open [[Möbius strip|04AJE0M271]] is not orientable.\nFollowing a local orientation once around the central circle returns the opposite generator, so no continuous choice exists.\n",
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