Vanishing cycle

Remarks

When $B$ is a ball in the coordinates $w$, the class $\delta$ generates the reduced homology $\tilde H_n(f^{-1}(t) \cap B; \Z) \cong \Z$, and $\Sigma_t$ shrinks to the point $p$ as $t \to 0$. Let $f^{-1}(c)$ be a fibre further away, joined to $t$ by a path $\gamma$ of regular values. Transporting $\delta$ along $\gamma$ gives a vanishing cycle in $H_n(f^{-1}(c); \Z)$, which depends on the homotopy class of $\gamma$.

Example: $n = 0$ and $f(w) = w^2$. The fibre $f^{-1}(t)$ is the two points $\pm\sqrt{t}$, and $\delta = [\sqrt{t}] - [-\sqrt{t}]$ in $H_0$.

Example: $n = 1$ and $f(w) = w_0^2 + w_1^2$. The local fibre is an annulus, and $\delta$ is the class of its core circle.

Non-example: the critical point $0$ of $f(w) = w^3$ is degenerate, since the Hessian is $0$. Its local fibre is three points, with reduced $H_0$ of rank $2$, so it is not described by one vanishing cycle. After a small perturbation $w^3 - sw$ with $s \ne 0$, it splits into two nondegenerate critical points, each with its own vanishing cycle.