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  "body": "\n## Summary\n\nThe Bockstein homomorphism is the connecting map in the long exact sequence of (co)homology that comes from a short exact sequence of coefficient groups.\nIt raises cohomological degree by one.\nFor the sequence $0 \\to \\Z/p \\to \\Z/p^2 \\to \\Z/p \\to 0$ it gives a natural operation on mod $p$ cohomology that detects $p$-torsion in integral cohomology.\n\n## Explanations\n\n### L1 human\n\nCohomology with coefficients in $\\Z/p$ loses some information about integral cohomology.\nThe Bockstein homomorphism recovers part of it by asking whether a mod $p$ class lifts to a mod $p^2$ class.\nWhen it does not lift, the Bockstein gives a nonzero class one degree higher.\nFor the real projective plane, the generator of $H^1(\\R P^2; \\Z/2)$ does not lift to $\\Z/4$ coefficients, and its Bockstein is the generator of $H^2(\\R P^2; \\Z/2)$.\n",
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