Below is an excerpt from part V of Deligne's Étale cohomology - starting points.
Let $X$ be a complex analytic variety and $f:X\to D$ a morphism from $X$ to the disk. We denote by $[0,t]$ the closed line segment with endpoints $0$ and $t$ in $D$ and by $(0,t]$ the semi-open segment. If $f$ is smooth, then the inclusion $$j:f^{-1}(0,t]\hookrightarrow f^{-1}[0,t]$$ is a homotopy equivalence: we can push the special fiber $f^{-1}(0)$ into $f^{-1}[0,t]$. [I think this should be $f^{-1}(0,t]$.]
In practice, for $t$ small enough, $f^{-1}(0,t]$ will be a fiber bundle on $(0,t]$ so that the inclusion $$f^{-1}(t)\hookrightarrow f^{-1}(0,t]$$ will also be a homotopy equivalence. We then define the cospecialization morphism to be the homotopy class of maps $$\mathrm{cosp}:f^{-1}(0)\hookrightarrow f^{-1}[0,t]\overset{\simeq}{\leftarrow}f^{-1}(0,t]\overset{\simeq}{\hookleftarrow}f^{-1}(t).$$ This construction can be expressed in pictorial terms by saying that for a smooth morphism, the general fiber swallows the special fiber.
Let us assume no longer that $f$ is necessarily smooth (but assume that $f^{-1}(0,t]$ is a fiber bundle over $(0,t]$. We can still define a morphism $\mathrm{cosp}^\bullet$ on cohomology provided $j_\ast\mathbf Z=\mathbf Z$ and $\mathrm R^qj_\ast \mathbf Z=0$ for $q>0$. Under these assumptions, the Leray spectral sequence for $j$ shows that we have $$\mathrm H^\bullet(f^{-1}[0,t],\mathbf Z)\overset{\cong}{\to}\mathrm H^\bullet(f^{-1}(0,t],\mathbf Z)$$ and $\mathrm{cosp}^\bullet$ is the composite morphism $$\mathrm H^\bullet(f^{-1}(0),\mathbf Z)\leftarrow \mathrm H^\bullet(f^{-1}[0,t],\mathbf Z)\overset{\cong}{\to}\mathrm H^\bullet(f^{-1}(0,t],\mathbf Z)\overset{\cong}{\to}\mathrm H^\bullet(f^{-1}(t),\mathbf Z).$$
Questions.
- How to use smoothness of $f$ to get a deformation retraction of $j$ (the inclusion)? If I understand correctly, smoothness implies the underlying $C^\infty$ map is a submersion and therefore admits an Ehresmann connection. This allows to use parallel transport to obtain a horizontal flow but without properness it may not be defined on entire fibers, so but I don't see how to get a deformation retraction...
- What's a "canonical" example of a non-smooth $f:X\to D$ which is well behaved (say, a fiber bundle over the punctured disk) but with the inclusion $j$ not a homotopy equivalence?
- What is the geometric interpretation of the conditions in the definition of the cohomological cospecialization map? What is an example where they fail?
