Test map characterizations of local properties of fundamental groups

Abstract

Local properties of the fundamental group of a path-connected topological space can pose obstructions to the applicability of covering space theory. A generalized covering map is a generalization of the classical notion of covering map defined in terms of unique lifting properties. The existence of generalized covering maps depends entirely on the verification of the unique path lifting property for a standard covering construction. Given any path-connected metric space $X$, and a subgroup $H\le {\pi }_1(X,x_0)$, we characterize the unique path lifting property relative to $H$ in terms of a new closure operator on the ${\pi }_1$-subgroup lattice that is induced by maps from a fixed "test" domain into $X$. Using this test map framework, we develop a unified approach to comparing the existence of generalized coverings with a number of related properties.

44 pages, v.2 contains improved exposition and minor revisions

References in corpus (2)

• On small homotopies of loops
• Generalized covering space theories

Cited by in corpus (8)

• Transfinite product reduction in fundamental groupoids
• Scattered products in fundamental groupoids
• On the failure of the first Čech homotopy group to register geometrically relevant fundamental group elements
• Fibrations, unique path lifting, and continuous monodromy
• A simply connected universal fibration with unique path lifting over a Peano continuum with non-simply connected universal covering space
• Dense products in fundamental groupoids
• Sequential $n$-connectedness and infinite factorization in higher homotopy groups
• Nonlocal loss of first homotopy in polyhedral approximations of Peano continua