Homology, Homotopy and Applications

Volume 5 (2003)

Number 2

Volume of a Workshop at Stanford University

Dicovering spaces

Pages: 1 – 17

DOI: http://dx.doi.org/10.4310/HHA.2003.v5.n2.a1

Author

Lisbeth Fajstrup (Department of Mathematics, Aalborg Universitet, Aalborg, Denmark)

Abstract

For a local po-space $X$ and a base point $x_0 \in X$, we define the universal dicovering space $\Pi: \tilde{X}_{x_0} \to X$. The image of $\Pi$ is the future $\uparrow x_0$ of $x_0$ in $X$ and $\tilde{X}_{x_0}$ is a local po-space such that $|\stackrel{\rightarrow}{\pi}_1(\tilde{X},[x_0],x_1)|=1$ for the constant dipath $[x_0]\in\Pi^{-1}(x_0)$ and $x_1\in \tilde{X}_{x_0}$. Moreover, dipaths and dihomotopies of dipaths (with a fixed starting point) in $\uparrow x_0$ lift uniquely to $\tilde{X}_{x_0}$. The fibers $\Pi^{-1}(x)$ are discrete, but the cardinality is not constant. We define dicoverings $P:\hat{X}\to X_{x_0}$ and construct a map $\phi:\tilde{X}_{x_0}\to\hat{X}$ covering the identity map. Dipaths and dihomotopies in $\hat{X}$ lift to $\tilde{X}_{x_0}$, but we give an example where $\phi$ is not continuous.

Keywords

covering spaces, abstract homotopy theory, dihomotopy theory

2010 Mathematics Subject Classification

51H15, 54Exx, 57M10

Full Text (PDF format)

An erratum to this article is available as HHA 13(1) pp. 403-406.