Geometriae Dedicata 56: 257-262, 1995. 257 © 1995 KluwerAcademic Publishers. Printed in the Netherlands. A Note About Bistellar Operations on PL-Manifolds with Boundary* MARIA RITA CASALI Dipartimento di Matematica Pura ed Applicata, Universit?tdi Modena, Via Campi 213B, 1-41100 Modena, Italy (Received: 5 May 1994) Abstract. In 1990, U. Pachner proved that simplicial triangulations of the same PL-manifold (with boundary) are always connected by a finite sequence of transformations belonging to two different groups: shelling operations (and their inverses), which work mostly with the boundary triangulations, and bistellar operations, which affect only the interior of the triangulations. The purpose of this note is to prove that, in case of simplicial triangulations coinciding on the boundary, bistellar operations are sufficient to solve the homeomorphism problem. Mathematics Subject Classifications (1991): 57Q 15, 57N15, 52B70. 1. Introduction In 1990, Pachner [P3] proved that simplicial triangulations of the same PL-manifold (with boundary) are always connected by a finite sequence of transformations, belonging to two different groups: shelling operations (and their inverses), which work mostly with the boundary triangulations, and bistellar operations, which affect only the interior of the triangulations. Further, in 1991, Pachner himself [P4] succeeded in showing that the first group of transformations is sufficient to realize the homeomorphism of PL-manifolds with boundary. This note is devoted to proving that, in case of simplicial triangulations coin- ciding on the boundary, bistellar operations have the same universal property: MAIN THEOREM. Let M and M ~ be two PL-manifolds, and let K (resp. K t) be a simplicial triangulation ofM (resp. M ~) with OK = OK I. Then, M and M ~ are PL-homeomorphic if and only if K and K ~ are bistellar equivalent. Note that our Main Theorem extends to the boundary case a previous result of Pachner [P1], stating that bistellar equivalence is a simplicial realization of PL homeomorphism among closed PL-manifolds. * Work performed under the auspicies of the GNSAGA of the CNR (National Research Council of Italy) and financially supported by MURST of Italy (project 'Geometria Reale e Complessa').