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  "body": "\n## Summary\n\nObstruction theory studies when a map defined on part of a [[CW complex|5W95QFHXPR]] extends to the whole complex, or when a fibration has a section.\nIt works by extending one skeleton at a time.\nAt each step the problem of going one cell dimension higher is measured by a cohomology class, the obstruction.\nThe method turns geometric extension problems into computations in [[singular cohomology|6N7BC70CX0]] with coefficients in [[homotopy groups|P976WYEFMB]].\nCharacteristic classes such as the Euler class and the Stiefel–Whitney classes arise as obstructions of this kind.\n\n## Explanations\n\n### L1 human\n\nTo extend a map over a CW complex, one builds it cell by cell, in order of dimension.\nA map on the boundary sphere of a new cell extends over the cell exactly when it is null-homotopic.\nRecording, for each cell, the homotopy class of the map on its boundary gives a cochain, the obstruction cocycle.\nIf its cohomology class vanishes, one can change the map on lower cells so that every cell of the next dimension can be filled.\nFor example, the identity of $S^n$ does not extend over the disc $D^{n+1}$, and the obstruction is the generator of $H^{n+1}(D^{n+1}, S^n; \\Z) \\cong \\Z$.\n",
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