The fundamental group of preimage of covering map

Question

Define i: B $\to$ Y is an inclusion, p: X $\to$ Y is a covering map. Define $D=p^{-1}(B)$. We assume here B and Y are locally path-connected and semi-locally simply connected. Then if B,Y, X are path-connected in what case D is path-connected (dependent on the fundamental groups)? Also what's the fundamental group of D at some point?

Answer 1

The question with regard to pathconnected is answered in this stackexchange question, which refers to the book Topology and Groupoids (T&G). A paper was published on this as Groupoids and the Mayer-Vietoris sequence Journal of Pure and Applied Algebra 30 (1983) 109-129.

The first step is to translate the problem, into one on groupoids, since a covering map of spaces $p: X \to Y$ induces a covering morphism of $\pi_1(p): \pi_1(X) \to \pi_1(Y)$ of fundamental groupoids, which is a special case of a fibration of groupoids dealt with in the paper. Covering morphisms of groupoids carry all the algebraic information usually obtained from covering maps.

In that algebraic model it is easier to analyse the situation, and to get the type of exact sequence required.

The conclusion with regard to fundamental groups is that they can be seen as pullbacks of other fundamental groups.