Critical point theory

Remarks

The basic mechanism is a deformation along a gradient flow. If $[a, b]$ contains no critical value and a compactness condition holds, it shows that $M^a$ is a deformation retract of $M^b$. So the topology of the sublevel sets changes only at critical values.

Example: let $M = S^1 \subset \R^2$ and $f(x, y) = y$. The critical points are $(0, 1)$ and $(0, -1)$, with critical values $1$ and $-1$. The sublevel set $M^c$ is empty for $c < -1$, a point for $c = -1$, an arc for $-1 < c < 1$, and the whole circle for $c \ge 1$. So its topology changes exactly at the two critical values.

Non-example: for $f(x) = e^x$ on $\R$, every real number is a regular value and $f$ has no critical point. The sublevel set $M^c$ is empty for $c \le 0$ and is the half-line $(-\infty, \log c]$ for $c > 0$. So its topology changes at $c = 0$, although $0$ is not a critical value, and $f$ has no minimum. This is possible because the compactness condition, the Palais–Smale condition, fails. Along $x_n = -n$ the values $f(x_n)$ are bounded and $f'(x_n) \to 0$, but $(x_n)$ has no convergent subsequence.

The two classical frameworks are Morse theory, through Morse functions, and Lusternik–Schnirelmann theory.