## Generalized universal covering spaces and the shape group

### Volume 197 / 2007

#### Abstract

If a paracompact Hausdorff space $X$ admits a (classical)
universal covering space, then the natural homomorphism
$\varphi:\pi_1(X)\rightarrow \check{\pi}_1(X)$ from the
fundamental group to its first shape homotopy group is an
isomorphism.
We present a partial converse to this result:
a path-connected topological space $X$ admits a *generalized*
universal covering space if $\varphi:\pi_1(X)\rightarrow
\check{\pi}_1(X)$ is injective.

This generalized notion of universal covering $p:\widetilde{X}\rightarrow X$ enjoys most of the usual properties, with the possible exception of evenly covered neighborhoods: the space $\widetilde{X}$ is path-connected, locally path-connected and simply-connected and the continuous surjection $p:\widetilde{X}\rightarrow X$ is universally characterized by the usual general lifting properties. (If $X$ is first countable, then $p:\widetilde{X}\rightarrow X$ is already characterized by the unique lifting of paths and their homotopies.) In particular, the group of covering transformations $G=\mathop{\rm Aut}(\widetilde{X}\stackrel{p}{\rightarrow}X)$ is isomorphic to $\pi_1(X)$ and it acts freely and transitively on every fiber. If $X$ is locally path-connected, then the quotient $\widetilde{X}/G$ is homeomorphic to $X$. If $X$ is Hausdorff or metrizable, then so is $\widetilde{X}$, and in the latter case $G$ can be made to act by isometry. If $X$ is path-connected, locally path-connected and semilocally simply-connected, then $p:\widetilde{X}\rightarrow X$ agrees with the classical universal covering.

A necessary condition for the standard construction to yield a generalized universal covering is that $X$ be homotopically Hausdorff, which is also sufficient if $\pi_1(X)$ is countable. Spaces $X$ for which $\varphi:\pi_1(X)\rightarrow \check{\pi}_1(X)$ is known to be injective include all subsets of closed surfaces, all 1-dimensional separable metric spaces (which we prove to be covered by topological $\mathbb{R}$-trees), as well as so-called trees of manifolds which arise, for example, as boundaries of certain Coxeter groups.

We also obtain generalized regular coverings, relative to some special normal subgroups of $\pi_1(X)$, and provide the appropriate relative version of being homotopically Hausdorff, along with its corresponding properties.