Fiber of a map

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

Definition

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

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

1. Projection fibers. For the projection $\pi_M : M\times F \to M$, the fiber over $m\in M$ is

   $$\pi_M^{-1}(m)=\{m\}\times F.$$

2. 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.

3. 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.