Stellar Neighborhoods in Polyhedral Manifolds

Introduction. It is the purpose of this paper to give some weaker analogues for polyhedral manifolds of well-known properties of combinatorial manifolds. This is now possible primarily because of the recent work of Mazur [3 ; 4] and M. Brown [l ]. A crucial issue in this area is the extent to which these analogues may be improved for arbitrary triangulated manifolds. We shall prove a theorem which will be applied later on, after making some preliminary definitions. The join of two spaces X and F is represented by X o Y. A map / of X o F— F on itself is called ray preserving if for each ray p o q — q, p in X, q in F, fip o q — q) Qp o q — q. A subset A of a cone X o p is called starlike if pEK and each segment pox, xEX, meets A in a connected set; p is the center of A. Consider En to be the open cone over Sn_1 from the origin p, in the usual way. We coordinatize E" by (x, t), xES*1*1, t a real number with 0g¿<<»; (S"-1, 0)=p. Let Dn be the unit »-ball P o (S-1, 1).