Inventiones mathematicae

, Volume 196, Issue 1, pp 163–232 | Cite as

O-minimal Hauptvermutung for polyhedra I

  • Masahiro ShiotaEmail author


Milnor discovered two compact polyhedra which are homeomorphic but not PL homeomorphic (a counterexample to the Hauptvermutung). He constructed the homeomorphism by a finite procedure repeated infinitely often. Informally, we call a procedure constructive if it consists of an explicit procedure that is repeated only finitely many times. In this sense, Milnor did not give a constructive procedure to define the homeomorphism between the two polyhedra. In the case where the homeomorphism is semialgebraic, the author and Yokoi proved that the polyhedra in R n are PL homeomorphic. In that article, the required PL homeomorphism was not constructively defined from the given homeomorphism. In the present paper we obtain the PL homeomorphism by a constructive procedure starting from the homeomorphism. We prove in fact that for any ordered field R equipped with any o-minimal structure, two definably homeomorphic compact polyhedra in R n are PL homeomorphic (the o-minimal Hauptvermutung theorem 1.1). Together with the fact that any compact definable set is definably homeomorphic to a compact polyhedron we can say that o-minimal topology is “tame”.

Mathematics Subject Classification (2000)

03C64 57Q05 57Q25 


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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Graduate School of MathematicsNagoya UniversityChikusa, NagoyaJapan

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