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  "body": "\n## Summary\n\nNielsen fixed-point theory studies the least number of fixed points among all maps homotopic to a given self-map $f$ of a space.\nIt sorts the fixed points of $f$ into classes, using paths and their images, and counts the classes that cannot be removed by a homotopy.\nThis count is the Nielsen number $N(f)$.\nIt is a homotopy invariant and a lower bound for the number of fixed points of every map homotopic to $f$.\nIt refines the [[Lefschetz number|ZSYCA4EGEZ]], which can only show that at least one fixed point exists.\n\n## Explanations\n\n### L1 human\n\nThe Lefschetz number can show that a map must have a fixed point, but not how many.\nNielsen theory sorts the fixed points into groups.\nTwo fixed points are in the same group if the map carries some path joining them to a path that can be deformed back to it.\nDuring this deformation the two ends stay fixed.\nGroups with a nonzero total weight can never be removed, so their number is a guaranteed minimum.\nThe map $z \\mapsto z^3$ on the unit circle has two fixed points, $1$ and $-1$, in two essential classes.\nSo every map homotopic to it has at least two fixed points.\n",
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