Let $f : X\rightarrow D$ be a family of smooth projective curves over the complex unit disk $D$, degenerating to a nodal curve $X_0$ above $0\in D$.

Let $\eta\in D - \{0\}$ be a general point, and let $\gamma\subset X_\eta$ be the vanishing cycle, which for me is a disjoint union of simple closed curves. Let $T\supset\gamma$ denote a tubular neighborhood of $\gamma$ in $X_\eta$, so $T$ is a disjoint union of annuli.

I believe there should be an exact sequence (in betti cohomology)

$$0\rightarrow H^1(X_0)\rightarrow H^1(X_\eta)\rightarrow H^1(T)\stackrel{\delta}{\rightarrow }H^2(X_0)\rightarrow H^2(X_\eta)\rightarrow H^2(T) = 0$$

The map $H^*(X_0)\rightarrow H^*(X_\eta)$ comes from the inclusion $X_\eta\rightarrow X$ using the isomorphism $H^*(X) = H^*(X_0)$.

- Is there a geometric description of the connecting map $\delta$?
- Does anyone have a good reference for this exact sequence?
- Am I correct in saying that this sequence is equivariant for the monodromy action of $\pi_1(D-\{0\},\eta)$, where $H^1(X_\eta)$ is the only term where the action is nontrivial?

To set some context, while I'd eventually want a deeper understanding of vanishing cycles, I'd like to begin by understanding this simple example, and it's hard to find a reference that doesn't immediately dive into perverse sheaves and such.