TITLE: Is the direct image of a constant sheaf a constant sheaf?
QUESTION [7 upvotes]: Is the direct image of a constant sheaf a constant sheaf? I'm not an expert on sheaf theory and can't find this anywhere

REPLY [7 votes]: When the map is well behaved, say a locally trivial fibration over a base $B$ with fiber $F$, then the direct image of a constant sheaf will be locally constant (all of the fibers of this sheaf will be equal to the cohomology of the constant sheaf on $F$), but won't be constant in general.
The obstruction is the monodromy, the defect of compatibility between all local trivializations
along a loop in the base space.