Cerf theory

Definition

Let $M$ be a compact smooth manifold of dimension $n$ without boundary. Cerf theory is the study of smooth one-parameter families $f_t \colon M \to \R$, $t \in [0, 1]$, of smooth functions, with $(x, t) \mapsto f_t(x)$ smooth. Its basic notion is a generic path, also called a Cerf path. A family $(f_t)$ is a generic path if it satisfies the following conditions.

1. For all but finitely many $t$, $f_t$ is a Morse function with distinct critical values.
2. At each remaining time $t_0$, one of two cases occurs.
3. In the first case, $f_{t_0}$ is Morse, exactly two critical points share a critical value, and these two values cross transversally at $t_0$.
4. In the second case, $f_{t_0}$ has distinct critical values and exactly one degenerate critical point $p_0$.
5. In the second case, near $p_0$ there are local coordinates $(x_1, \dots, x_n)$ and a local time parameter $s$ with $s = 0$ at $t_0$ with the following form. For some $0 \le \lambda \le n - 1$ and a smooth function $c$, one has $f_t = c(s) + x_1^3 + s x_1 - x_2^2 - \cdots - x_{\lambda+1}^2 + x_{\lambda+2}^2 + \cdots + x_n^2$.