Let $Y$ be the Spec of a complete local Noetherian ring. If $f : X\rightarrow Y$ is a proper morphism with geometrically connected and reduced fibers, and $y_1\in Y$ is a geometric point specializing to $y_0$ (and let $x_1,x_0$ be lifts to $X$), then essentially by "topological invariance of the etale site", we obtain two exact sequences of fundamental groups $$\pi_1(X_{y_1},x_1)\rightarrow\pi_1(X,x_1)\rightarrow\pi_1(Y,y_1)\rightarrow 1$$ and $$1\rightarrow \pi_1(X_{y_0},x_0)\rightarrow\pi_1(X,x_0)\rightarrow\pi_1(Y,y_0)\rightarrow 1$$ By invariance under basepoints, we obtain vertical isomorphisms between the middle and right terms of the sequences above, and then by exactness we obtain a surjective map $$sp : \pi_1(X_{y_1})\rightarrow \pi_1(X_{y_0})$$ My questions are:

  1. How can we understand the specialization theorem analytically/topologically?

For example, I would like to say something like: Let $Y\subset\mathbb{C}$ be a small disk, let $\delta: [0,1]\rightarrow Y$ be a path, and $x$ a point in the fiber $X_{\delta(0)}$, then for every loop $\gamma\in\pi_1(X_{\delta(0)},x)$, as $t$ ranges over $[0,1]$, we can move $\gamma$ through the fibers $X_{\delta(t)}$ in a way which is unique up to homotopy in the fibers, and hence we for any $t$ we get a homomorphism of fundamental groups $$\pi_1(X_{\delta(0)})\rightarrow\pi_1(X_{\delta(t)})$$ I think this is true if $f$ is smooth, but how would one argue something like this? (The homotopy lifting property doesn't seem to say much about this situation) What if $f$ is not smooth? In either case, where does the properness of $f$ come into play?

  1. Is there a specialization homomorphism when $X$ is the complement of a relative normal crossings divisor (or a "suitably nice class of divisors" which includes sections) inside an $f : Z\rightarrow Y$ which is proper with geom. connected and reduced fibers?

My motivation for this is to understand the relation between fundamental groups of smooth punctured curves and the fundamental groups of their (stable) degenerations.

Part of my confusion is that I'm not really sure what general topological/analytic tools there are for handling fundamental groups when $f$ is not smooth (e.g., if $Y$ is a disk and $f : X\rightarrow Y$ is smooth away from the center, and is a degenerate curve at the center. In this case $f$ isn't even a fibration in general...), or even worse when $f$ ie neither smooth nor proper.

Relevant references would also be appreciated.


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