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  "body": "\n## Summary\n\nA simple homotopy equivalence between finite CW complexes is a homotopy equivalence of a special kind.\nUp to homotopy, it is a finite sequence of elementary expansions and collapses, which add or remove a pair of cells in a trivial way.\nNot every homotopy equivalence is simple; the obstruction is the Whitehead torsion, which lies in the Whitehead group of the fundamental group.\nSimple homotopy type is finer than homotopy type and is the right notion for the s-cobordism theorem and for distinguishing lens spaces.\n\n## Explanations\n\n### L1 human\n\nImagine a shape made of cells, like a model built of balls and disks.\nAn elementary expansion glues on a new solid ball along half of its boundary.\nThis adds two cells but changes nothing up to deformation, since the ball can be squashed back flat.\nAn elementary collapse is the reverse move.\nTwo complexes are simple homotopy equivalent when one can pass from one to the other by finitely many such moves.\nA solid square and a single point are simple homotopy equivalent: collapse the square onto an edge, then the edge onto a point.\n",
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