Understanding the Concept of Monodromy; case of Lefschetz Fibrations. My question is on the concept of monodromy around  critical points in a Lefschetz fibration $p: M^4 \rightarrow S^2$  (and monodromy in general), where $M^4$ is a 4-manifold and $S^2$ is the 2-sphere.
It is  known that the monodromy about any  critical point in a Lefschetz fibration (there are finitely-many critical points) is described by a Dehn twist $D$. Is the following an accurate description of the concept of monodromy by D about a critical point x in this type of fibration:
Let x be a critical  point in the base $S^2$. Let  $x':= p^{-1}(x)$ be the lift of x by the fibration map p ,x' lives in the 4-manifold M. Then, if we draw a small loop $B(x,r)$ (small-enough so that $B(x,r)$ does not intersect any other critical point in the base; critical points for a Lefschetz fibration are isolated ) in the base, centered at x, and we wind around the loop once, say from $a=0$ to $b=2\pi$, then $a':= p^{-1}(a)$ is related to $b':=p^{-1}(b)$ by the Dehn twist $D$, i.e., is it the case that $a'=D(b')$?
Thanks for any help, references, etc
 A: This is how I understand monodromy: consider the small ball $B(x,r)$ within a  trivialized neighborhood that you described. Then this ball, since it is  assumed to be inside of a trivialized neighborhood, is an $S^1-$ bundle $S^1 \times F$ , where $F$ is the fiber. Now, if the monodromy is trivial, you do get a trivialized $S^1 \times F$. Basically, as you wind around the boundary $S^1$ of the ball $B(x,r)$ , each point lifts to a fiber $F$ , but, as you complete a loop around $S^1$ , you end up gluing fibers $F$ to each other when $0 \rightarrow 2\pi$ . This gluing is done by an element of the mapping class group MCG(F) of the fiber. The choice of map, up to isotopy, is the monodromy.
In other words, the monodromy dictates that you glue the lift $S^1 \times F$ along a Dehn twist D , so that you end up with the lifted bundle $S^1 \times F /$~ , where '~' is the relationship $(x,0)$~$(D(x),1)$ , where $D(x)$ is the Dehn twist associated to the monodromy. So you end up gluing your fiber along the monodromy map $D$.
