Stable map

Remarks

Example: the height function $f \colon S^1 \to \R$, $f(x, y) = y$, on the unit circle is stable. Its critical points $(0, \pm 1)$ are nondegenerate with distinct values $\pm 1$. Any $g$ close enough to $f$ in the $C^2$ sense also has exactly two nondegenerate critical points, a maximum and a minimum. Their values are distinct, and such a $g$ is equivalent to $f$. Non-example: a constant map $c \colon S^1 \to \R$ is not stable. The maps $c + \varepsilon f$ come arbitrarily close to $c$ and have nonzero differential at some points, while every map equivalent to $c$ is constant. For a compact manifold $M$, a function $f \colon M \to \R$ is stable if and only if all its critical points are nondegenerate and its critical values are pairwise distinct. The equivalence used here is often called $\mathcal{A}$-equivalence, or left-right equivalence.