It is an open question as to whether a generalized n-manifold, n > 5, that satisfies the disjoint disks property is a topological manifold. In this paper it is shown that any such space X satisfies general position properties for maps of polyhedra into X. The program to establish a topological characterization of euclidean space has been stalled, at least temporarily, by the discovery that Quinn's obstruction to resolving a generalized manifold may not be categorically zero as first asserted in [6]. (See [7].) Thus Cannon's conjecture [3] that a generalized n-manifold X, n > 5, is a topological manifold if and only if X satisfies the disjoint disks property remains open. Quinn's results [6, 7], however, do produce an obstruction a(X) that vanishes ii and only if X is a topological manifold, although it is not known at this time whether the obstruction can be nonzero. In this paper we show (Theorem 3) that any generalized n-manifold X, n > 5, having the disjoint disks property also satisfies general position properties for maps of arbitrary polyhedra into X. Thus, in a dimension theory sense X behaves like a topological manifold. Theorems of this type have also been obtained by John Walsh [9]. DEFINITIONS. A generalized n-manifold, n-gm, is a euclidean neighborhood retract X that is also a homology n-manifold; that is, H* (X, X x) H* (Rn, Rn 0) for all x E X. (All homology is understood to have integer coefficients.) The space X is said to satisfy the disjoint k-disks property, DDkP (or, simply, the disjoint disks property, DDP, when k = 2) if for every pair of maps fl, f2: Dk > X of the unit k-disk Dk of Rk into X and E > 0, there are maps /, f2: Dk X such that d(f',fi) 0, then f g means f is E-homotopic to g. A subset A of X is said to be 1-LCC in X (1-locally coconnected in X) if for every a E A and neighborhood U of a in X there is a neighborhood V of a in X such that the inclusion induced homomorphism 7r1 (V A) -7r1 (U A) is zero. A map (or embedding) f: A -X is said to be 1-LCC provided f(A) is 1-LCC in X. Main results. In [3] Cannon shows that if X is an n-gm, n > 5, having the DDP, then an arbitrary map of a 2-disk (hence, a 2-dimensional polyhedron) into Received by the editors September 24, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 57P05, 57N75; Secondary 57Q65. The author's research was partially funded by ONR grant #N00014-84-K-0761. (@)1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page