The topological fundamental group and generalized covering spaces

By Daniel K. Biss. 2002

74 citations recorded by the source corpus.

It is a basic result of topology that a space X has a simply connected cover if and only if X is semilocally simply connected; that is, if for each x ∈ X, there is an open neighborhoodU of x such that the inclusion of U in X induces the trivial map onπ1. Of course, a covering space is simply a fibration all of whose fibers are discrete, and in the case of a simply connected cover, there is a one-to-one correspondence between the discrete fiber andπ1(X). This suggests a syllogism: X is semilocally simply connected if and only ifπ1(X) is discrete. Unfortunately, since π1(X) is not a topological space, that sentence makes no sense. In this paper, we introduce a topology on π1(X), and prove the above statement. Furthermore, we are able to completely classify the connected covers of any spaceX in terms of the topology of π1(X). We also construct fibrations with fiber π1(X) (as well asπ1(X)/π for some subgroups π of π1(X)) and develop a theory of a special type of fibration—which we call rigid covering fibrations—which behaves in most ways like a covering space, but exists in a significantly more general setting. In addition to simply generalizing the classical theory of covering spaces, these results give a much clearer idea of what goes wrong when a space fails to be semilocally simply connected. These results all reduce to classical theorems when X is a CW complex; thus, it is desirable to define a new category of spaces to provide a natural home for our ideas. We do this by finding a suitable enlargement of the CW category, which is small enough that

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a4dcac6500b5759f8c29eff13c35b0628a302cea