The simplicial homology of a simplicial complex is not a function of the complex, but of the complex together with a simplicial decomposition. The Hauptvermutung ("main conjecture") stated that simplicial homology could be proven to be an invariant of the complex by showing that any two decompositions have a common refinement.

When Milnor disproved this conjecture in 1962, the invariance of simplicial homology had long been proved using singular homology, and simplicial homology itself was regarded obsolete by many.