### Polyhedral quotient spaces

Jože Vrabec
1991 Colloquium Mathematicum
Throughout this paper, P will be an arbitrary polyhedron (all our polyhedra are compact), E ⊂ P × P will be an equivalence relation on P , and q will denote the quotient projection of P onto the quotient space P/E. We are interested in the following question: when is P/E a polyhedron such that q is PL? More precisely, when does there exist a polyhedron Q and a PL map f : P → Q inducing a homeomorphism P/E → Q? We shall answer this question for the case that the equivalence classes of E (the
more » ... asses of E (the fibers of q) are finite sets. Notice that a pair (Q, f ) satisfying the condition of the preceding paragraph -if such a pair exists -possesses the following universal property: for every polyhedron Q and every PL map f : It follows that such a pair (Q, f ) is essentially unique and that it can be considered the quotient of P (with respect to E) in the PL category. Therefore, when P/E admits a PL structure such that q is PL we shall say that P/E is a PL quotient space of P . It can easily happen that P/E admits polyhedral structures, but none with q being PL; e.g. the quotient space obtained by shrinking a subpolyhedron of P to a point is always a topological polyhedron but rarely a PL quotient space because a linear map cannot shrink a face of a simplex to a point and be injective on the rest of the simplex. By similar consideration we can convince ourselves that there are hardly any interesting PL quotient spaces with degenerate quotient projection. Therefore we shall from now on restrict our attention to the case that q is nondegenerate, i.e. the equivalence classes of E are finite sets. For future reference we state the following well-known triangulation criterion for PL quotient spaces. Proposition 1. Suppose that E has finite equivalence classes. P/E is a PL quotient space if and only if there exist a triangulation K of P and a labeling of the vertices of K such that (a) the endpoints of any 1-simplex of K are assigned different labels and