 Thus one cannot even compute the fundamental group of a given high-dimensional manifold, much less a classification.This ineffectiveness is a fundamental reason why surgery theory does not classify manifolds up to homeomorphism.Instead, for any fixed manifold "M" it classifies pairs formula_10 with "N" a manifold and formula_11 a "homotopy equivalence", two such pairs, formula_10 and formula_13, being regarded as equivalent if there exist a homeomorphism formula_14 and a homotopy formula_15.