Simple homotopy and the vanishing criterion are owned by AT

The letter "s" in "s-cobordism" refers to simple homotopy equivalence. AT-22 owns the definition of simple homotopy equivalence (Simple homotopy equivalence) and the theorem that a finite CW homotopy equivalence is simple if and only if its Whitehead torsion vanishes (Whitehead torsion is the complete obstruction to finite CW simple homotopy, Whitehead torsion of a finite CW homotopy equivalence); this page cites both and does not mint a second definition, no new expansion–collapse calculus is introduced here, and the parity and basis conventions are those of AT-22.

For a fixed presentation $H$ the condition ${\tau }_H(W,M_0)=0$ of Presentation-indexed Whitehead torsion of an h-cobordism is equivalent to the inclusion $M_0\hookrightarrow W$ being simple for the associated finite CW structures, by the AT-22 criterion. The presentation-relative s-cobordism theorem asked on this page cares whether at least one such presentation exists; it does not identify the torsion classes of presentations that are not related by the elementary handle modifications of this page, and it does not promote a vanishing class in one presentation to a simple homotopy equivalence of arbitrary CW models. The h-cobordism hypothesis used throughout is the one of h-Cobordism.