N. Brodskiy

Covering maps for locally path-connected spaces

We define Peano covering maps and prove basic properties analogous to classical covers. Their domain is always locally path-connected but the range may be an arbitrary topological space. One of characterizations of Peano covering maps is via the uniqueness of homotopy lifting property for all locally path-connected spaces. Regular Peano covering maps over path-connected spaces are shown to be identical with generalized regular covering maps introduced by Fischer and Zastrow [15]. If X is path-connected, then every Peano covering map is equivalent to the projection e X/H → X, where H is a subgroup of the fundamental group of X and e X equipped with the topology used in [2], [15] and introduced in [23, p.82]. The projection e X/H → X is a Peano covering map if and only if it has the unique path lifting property. We define a new topology on e X for which one has a characterization of e X/H → X having the unique path lifting property if H is a normal subgroup of π1(X). Namely, H must be closed in π1(X). Such groups include π(U , x0) (U being an open cover of X) and the kernel of the natural homomorphism π1(X, x0) → π̌1(X, x0).

A Hurewicz theorem for the Assouad–Nagata dimension

Given a function

Assouad-Nagata Dimension of Wreath Products of Groups

f\colon X\to Y

Coarse dimensions and partitions of unity

of metric spaces, its {\it asymptotic dimension}

Dimension zero at all scales

\asdim(f)

Rips complexes and covers in the uniform category

is the supremum of

Group actions and covering maps in the uniform category

\asdim(A)

Assouad-Nagata dimension via Lipschitz extensions

such that

Compression of uniform embeddings into Hilbert space

A\subset X

Coarse structures and group actions

and