Let $X,Y$ be path-connected finite CW complexes with base points $x\_0,y\_0$, let $f\colon X\to Y$ be a surjective continuous map, such that for every $y\in Y$, the fiber $f^{-1}(y)$ is path connected. In this case, is the induced map $$f\_\*\colon\pi\_1(X,x\_0)\to\pi\_1(Y,y\_0)$$ on topological fundamental groups necessarily surjective? 


(If this is not true in general, will this be true in the case when $Y$ is a complex algebraic manifold, $X\subset Y\times\mathbb{P}^n$ a quasi-projective variety, and $f=\mathrm{pr}\_1$?)


[I think one sufficient condition is that $f$ satisfies the "arc-lifting property": Sufficiently short arc $(-\epsilon,\epsilon)$ centered at any $y\in Y$ can be lifted to an arc in $X$. For then we can cover a path in $X$ by finitely many arcs in $X$, and join the arcs by path in the fibers. But I am not sure if this is always doable?]



Morphism with connected fibers induce surjection on fundamental groups?
This answer is a complement to Andy's one. If $X$ and $Y$ are complex algebraic varieties then you have the following fact (see more generally Kollár "Shafarevich maps and automorphic forms" Proposition 2.10.2):


If $X$, $Y$ are irreducible algebraic varieties with $Y$ normal and $f:X\to Y$ is a dominant morphism such that the geometric generic fiber is connected then $f\_\*:\pi\_1(X)\to \pi\_1(Y)$ is surjective.


In case $Y$ is not normal then the above fact is not true. Take $Y=$ nodal cubic, $X=$ normalization of $Y$ minus one of the two preimages of the node. This situation is realized topologically as follows: it's the map from the sphere minus North Pole to the sphere with North Pole and South Pole identified. (Deleting one of the two points is not necessary to provide a counterexample to the above fact but it provides a counterexample with connected fibers, which is the case you are interested in).


