
FUNDAMENTAL GROUPS OF TOPOLOGICAL STACKS
... The purpose of this paper is to prove a basic formula for the fundamental group of the coarse moduli stack of a topological stack (Theorem 8.3). This result has consequences in classical algebraic topology which seem, surprisingly, to be new. (Some special cases have appeared previously in [Ar1, Ar2 ...
... The purpose of this paper is to prove a basic formula for the fundamental group of the coarse moduli stack of a topological stack (Theorem 8.3). This result has consequences in classical algebraic topology which seem, surprisingly, to be new. (Some special cases have appeared previously in [Ar1, Ar2 ...
Midterm solutions.
... True and False (ugh! mea culpa) I intended to say that the subsets under consideration are non-empty in which case it is true (one of the discarded middle thirds will eventually lie between two distinct points). But the empty set is by denition connected..... ...
... True and False (ugh! mea culpa) I intended to say that the subsets under consideration are non-empty in which case it is true (one of the discarded middle thirds will eventually lie between two distinct points). But the empty set is by denition connected..... ...
Covering space
In mathematics, more specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space, C, to a topological space, X, such that each point in X has an open neighbourhood evenly covered by p (as shown in the image); the precise definition is given below. In this case, C is called a covering space and X the base space of the covering projection. The definition implies that every covering map is a local homeomorphism.Covering spaces play an important role in homotopy theory, harmonic analysis, Riemannian geometry and differential topology. In Riemannian geometry for example, ramification is a generalization of the notion of covering maps. Covering spaces are also deeply intertwined with the study of homotopy groups and, in particular, the fundamental group. An important application comes from the result that, if X is a ""sufficiently good"" topological space, there is a bijection between the collection of all isomorphism classes of connected coverings of X and the conjugacy classes of subgroups of the fundamental group of X.