TITLE: Fundamental group of a topological pullback
QUESTION [26 upvotes]: This should be such an elementary problem in algebraic topology that I'm almost too embarrassed to ask, but here goes.
Let $f: X\to Z$ be a surjective fibration, and let $g: Y\to Z$ be any map. Assume all spaces are path-connected, and base points $x,y,z$ chosen so that $f(x)=g(y)=z$. Form the pullback in the topological category,
$$
\begin{array}{ccc}
E & \to & X \newline
\downarrow & & \downarrow \newline
Y & \to & Z.
\end{array}
$$ 
Note that $E$ need not be path-connected.

Is it possible to express $\pi_1(E,e)$ for a given choice of base point $e\in E$, in terms of $f_\sharp: \pi_1(X,x)\to \pi_1(Z,z)$ and $g_\sharp: \pi_1(Y,y)\to \pi_1(Z,z)$?

REPLY [10 votes]: Expressed in terms of the homotopy pullback $N(f,g)$ of a pair of based maps $f\colon X\longrightarrow A$ and $g\colon Y\longrightarrow A$, the long exact sequence is Corollary 2.2.3 of May and Ponto ``More concise algebraic topology''.  The result of which it is a 
corollary, Proposition 2.2.2, describes the pointed set of based maps $[Z,N(f,g)]$
for any based space $Z$. The dual result for homotopy pushouts is Proposition 2.1.2.  The corollary is used heavily in the study of fracture theorems for localization and completion later on in the book.