We develop a theory of Nöbeling manifolds similar to the theory of Hilbert space manifolds. We show that it reflects the theory of Menger manifolds developed by M. Bestvina [8] and is its counterpart in the realm of complete spaces. In particular we prove the Nöbeling manifold characterization conjecture.We define the n-dimensional universal Nöbeling space ν n to be the subset of R 2n+1 consisting of all points with at most n rational coordinates. To enable comparison with the infinite dimensional case we let ν ∞ denote the Hilbert space. We define an ndimensional Nöbeling manifold to be a Polish space locally homeomorphic to ν n . The following theorem for n = ∞ is the characterization theorem of H. Toruńczyk [36]. We establish it for n < ∞, where it was known as the Nöbeling manifold characterization conjecture.Characterization theorem An n-dimensional Polish ANE(n)-space is a Nöbeling manifold if and only if it is strongly universal in dimension n.The following theorem was proved by D. W. Henderson and R. Schori [23] for n = ∞. We establish it in the finite dimensional case.Topological rigidity theorem. Two n-dimensional Nöbeling manifolds are homeomorphic if and only if they are n-homotopy equivalent.We also establish the open embedding theorem, the Z-set unknotting theorem, the local Z-set unknotting theorem and the sum theorem for Nöbeling manifolds.

We show that Dranishnikov's asymptotic property C is preserved by direct products and the free product of discrete metric spaces. In particular, if G and H are groups with asymptotic property C, then both G×H and G * H have asymptotic property C. We also prove that a group G has asymptotic property C if 1 → K → G → H → 1 is exact, if asdim K < ∞, and if H has asymptotic property C. The groups are assumed to have left-invariant proper metrics and need not be finitely generated. These results settle questions of Dydak and Virk [15], of Bell and Moran [5], and an open problem in topology in [23].2010 Mathematics Subject Classification. 54F45 (primary), 20F69 (secondary).

Abstract. We show that a regular cover of a general topological space provides structure similar to a triangulation. In this general setting we define analogues of simplicial maps and prove their existence and uniqueness up to homotopy. As an application we give simple proofs of sharpened versions of nerve theorems of K. Borsuk and A. Weil, which state that the nerve of a regular cover is homotopy equivalent to the underlying space.Next we prove a nerve theorem for a class of spaces with uniformly bounded extension dimension. In particular we prove that the canonical map from a separable metric n-dimensional space into the nerve of its weakly regular open cover induces isomorphisms on homotopy groups of dimensions less than n.

Abstract. The coarse category was established by Roe [20] to distill the salient features of the large-scale approach to metric spaces and groups that was started by Gromov [13]. In this paper, we use the language of coarse spaces to define coarse versions of asymptotic property C [6] and decomposition complexity [16]. We prove that coarse property C implies coarse property A; we also show that these coarse versions share many of the features of their metric analogs such as preservation by products or unions.