Fiber of a map - Imported Knowlpedia Content

Let $f : X \to Y$ be a map and let $y \in Y$ .

The fiber of $f$ over $y$ (also called the preimage fiber) is the subset

f^{-1}(y)=\{x\in X \mid f(x)=y\}\subseteq X.

When $f : M \to N$ is a smooth map between smooth manifolds, the fiber $f^{-1}(y)$ is a distinguished subset of $M$ whose geometry depends strongly on whether $y$ is a regular value. In particular, if $y$ is a regular value then $f^{-1}(y)$ is a smooth submanifold of $M$ ; if $f$ is a smooth submersion then every $y\in N$ is regular and all fibers are smooth submanifolds of the same dimension.

Examples

Projection fibers. For the projection $\pi_M : M\times F \to M$ , the fiber over $m\in M$ is $\pi_M^{-1}(m)=\{m\}\times F.$

Level sets of a function. For $f:\mathbb{R}^2\to \mathbb{R}$ given by $f(x,y)=x^2+y^2$ , the fibers are the level sets:

if $r>0$ , then $f^{-1}(r)=\{(x,y):x^2+y^2=r\}$ , a circle of radius $\sqrt r$ ;
if $r=0$ , then $f^{-1}(0)=\{(0,0)\}$ , a single point.

A fiber with multiple components. Let $h:S^1\to \mathbb{R}$ be the height function $h(\cos t,\sin t)=\sin t$ . Then the fiber over $0$ is $h^{-1}(0)=\{(1,0),(-1,0)\},$ which has two connected components.