while it was not initially known, a group related to "k" had already been introduced in another context. henri poincaré had attempted to define the betti numbers of a manifold in terms of a triangulation. his methods, however, had a serious gap: poincaré could not prove that two triangulations of a manifold always yielded the same betti numbers. it was clearly true that betti numbers were unchanged by subdividing the triangulation, and therefore it was clear that any two triangulations that shared a common subdivision had the same betti numbers. what was not known was that any two triangulations admitted a common subdivision. this hypothesis became a conjecture known as the "hauptvermutung" (roughly "main conjecture"). the fact that triangulations were stable under subdivision led j.h.c. whitehead to introduce the notion of simple homotopy type. a simple homotopy equivalence is defined in terms of adding simplices or cells to a simplicial complex or cell complex in such a way that each additional simplex or cell deformation retracts into a subdivision of the old space. part of the motivation for this definition is that a subdivision of a triangulation is simple homotopy equivalent to the original triangulation, and therefore two triangulations that share a common subdivision must be simple homotopy equivalent. whitehead proved that simple homotopy equivalence is a finer invariant than homotopy equivalence by introducing an invariant called the "torsion". the torsion of a homotopy equivalence takes values in a group now called the "whitehead group" and denoted "wh"(π), where π is the fundamental group of the two complexes. whitehead found examples of non-trivial torsion and thereby proved that some homotopy equivalences were not simple. the whitehead group was later discovered to be a quotient of "k"(zπ), where zπ is the integral group ring of π. later john milnor used reidemeister torsion, an invariant related to whitehead torsion, to disprove the hauptvermutung.