# Defining the cospecialization in topology

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.

1. 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...
2. 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?
3. What is the geometric interpretation of the conditions in the definition of the cohomological cospecialization map? What is an example where they fail?