Applications of general position properties of dendrites to Hilbert space topology

Let A be a dendrite whose endpoints are dense and let A be the complement in A of a dense o-compact collection of endpoints of A . We investigate the general position properties that products of A and A possess and apply these to Hilbert space topology. In particular, it is shown that Aⁿ × [-1, 1] is a compact (n+1)-dimensional AR that satisfies the disjoint n-cells property, Aⁿ⁺¹ is a compact (n+1)-dimensional AR that satisfies the stronger general position property that maps of n-dimensional compacta into Aⁿ⁺¹ are approximable by Z-maps, and A is a nowhere locally compact topologically complete (n+1)-dimensional AR that satisfies the discrete n-cells property. As for applications, we use A to build a hierarchy of examples of fake boundary sets in the Hilbert cube that satisfy higher and higher orders of local connectivity and whose complements, though not homeomorphic to the pseudo-interior s of the Hilbert cube, share many of the topological properties of s . Also, it is shown that A stabilizes those complete separable ANR's that are l2-manifolds off of compact subsets with infinite codimension and this theorem applies to stabilize the fake that are constructed here in using two different techniques, one using A . Finally, some partial results on characterizing l-2 manifolds based on their homological structure are included.