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  "body": "\n## Definition\n\nLet $(M, g)$ be an [[oriented|913WEQERBE]] [[Riemannian manifold|7CBT4SM83Y]] of dimension $n \\ge 2$.\nLet $P_{SO}(M)$ be its bundle of oriented orthonormal frames, a principal $SO(n)$-bundle (see [[frame bundle|F4HWTRNXAP]]).\nLet $\\rho \\colon \\operatorname{Spin}(n) \\to SO(n)$ be the double covering homomorphism from the spin group.\nA *spin structure* on $(M, g)$ is a pair $(P, \\xi)$ such that:\n\n1. $P$ is a [[principal|8E37E74XVP]] $\\operatorname{Spin}(n)$-bundle over $M$.\n2. $\\xi \\colon P \\to P_{SO}(M)$ is a smooth map that commutes with the projections to $M$.\n3. $\\xi(p \\cdot a) = \\xi(p) \\cdot \\rho(a)$ for all $p \\in P$ and $a \\in \\operatorname{Spin}(n)$.\n\nTwo spin structures $(P, \\xi)$ and $(P', \\xi')$ are equivalent if there is a $\\operatorname{Spin}(n)$-equivariant bundle isomorphism $\\phi \\colon P \\to P'$ with $\\xi' \\circ \\phi = \\xi$.\n\n## Remarks\n\nClauses 1 to 3 imply that $\\xi$ is a two-sheeted covering map.\nAn oriented manifold admits a spin structure if and only if its second [[Stiefel–Whitney class|JQCK7A210W]] $w_2(TM)$ vanishes.\nIn that case the equivalence classes of spin structures are in bijection with $H^1(M; \\Z/2)$, after one class is fixed.\nExistence and the set of classes do not depend on the choice of the metric $g$.\n\nExample: the sphere $S^n$ with $n \\ge 2$ has $w_2 = 0$, because its tangent bundle becomes trivial after adding a trivial line bundle.\nSince $H^1(S^n; \\Z/2) = 0$, it has exactly one spin structure up to equivalence.\nFor $S^n$ it is $\\operatorname{Spin}(n+1) \\to SO(n+1) = P_{SO}(S^n)$.\n\nNon-example: the complex projective plane $\\C P^2$ is oriented but $w_2(T\\C P^2) \\ne 0$, so no pair $(P, \\xi)$ satisfies the clauses.\nA manifold with a spin structure is called a *spin manifold*.\n",
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