The authors study the problem of effectiveness of the semi-algebraic “Hauptvermutung” of Shiota-Yokoi. Namely, let K, L be two finite simplicial complexes in \({\mathbb{R}}^ n\) with at most k simplices. Let f: \(| K| \to | L|\) be a semi-algebraic homeomorphism whose graph is defined (as a semi-algebraic set) with polynomials of degree \(\leq q\). Then there is an algorithm which constructs subdivisions \(K'\) and \(L'\) of K and L, and a simplicial isomorphism from \(K'\) to \(L'.\)
The problem adressed by the authors is the possibility of the existence of an effective bound on the number of simplices of \(K'\) and \(L'\) in terms of (n,p,q,k). - The answer is yes if \(m=\dim | K| =\dim | L| \leq 3\), and no if \(m\geq 6\), even in the case where K is a PL-ball, and L the standard simplex \(\Delta_ m.\)
The proof uses a result of Novikov, which is a topological counter-part of the “nondecidability of the triviality for finitely presented groups”. The use of such results in semi-algebraic geometry is due to Nabutovsky.