Effectiveness - non effectiveness in semialgebraic and PL geometry
type: Item
instance of: scholarly article
title: Effectiveness - non effectiveness in semialgebraic and PL geometry
zbMATH Open document ID: 0729.14040
DOI: 10.1007/BF01233424
published in: Inventiones Mathematicae
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143831
review text: 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.
Mathematics Subject Classification ID: 14P10
Mathematics Subject Classification ID: 14Q20
Mathematics Subject Classification ID: 57Q25
Mathematics Subject Classification ID: 03D25
zbMATH DE Number: 4204560
zbMATH Keywords: effectiveness of the semi-algebraic ``Hauptvermutung''
zbMATH Keywords: PL-ball
zbMATH Keywords: nondecidability
author: Francesca Acquistapace
author: R. Benedetti
author: Fabrizio Broglia
MaRDI profile type: Publication
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OpenAlex ID: W1999099039
description: scientific article; zbMATH DE number 4204560