Coverings and fundamental groups: a new approach

Abstract

Classical fundamental groups behave reasonably well for Poincare spaces (i.e.. semy-locally simply connected spaces). One has a construction of the universal covering for such spaces. For arbitrary spaces it is a different matter. We define monodromy groups $\pi (p,b_{0})$ for any map $p: E \rightarrow B$ with the path lifting property and any $b_{0} \in B$. $p$ is called a $\mathcal{P}$-covering, where $\mathcal{P}$ is
a class of Peano spaces (i.e., connected and locally path connected spaces), if it has existence and uniqueness of lifts of maps $f: X \rightarrow B$ for any $X \in \mathcal{P}$. 
For any $B$ there is the maximal $\mathcal{P}$-covering $p_{\mathcal{P}}:B_{\mathcal{P}}\rightarrow B$ and its monodromy group is called the $\mathcal{P}$-fundamental group of $(B,b_{0})$. In case of $\mathcal{P}$ consisting of all disk-hedgehogs we construct a universal covering theory of all spaces in analogy to the classical covering theory of Poincare spaces.