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  "body": "\n## Summary\n\nThe primary obstruction is the first obstruction that appears when one tries to extend a map into an $(n-1)$-connected space over a CW complex.\nIt is a single cohomology class in degree $n + 1$ with coefficients in $\\pi_n$ of the target, and it does not depend on any choice.\nMany characteristic classes, such as the Euler class of an oriented sphere bundle, are primary obstructions.\n\n## Explanations\n\n### L1 human\n\nIf the target space $Y$ has no holes in dimensions below $n$, a map can always be extended over cells of dimension at most $n$.\nThe first real difficulty comes with the cells of dimension $n + 1$.\nThe primary obstruction is the cohomology class that measures this difficulty.\nTry to build a nonvanishing vector field on a closed oriented surface.\nIt can always be built over the vertices and edges, and the primary obstruction lives on the top cells.\nThere it equals the Euler characteristic of the surface times a generator.\n",
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