Fiber Bundle

A fiber bundle (also called simply a Bundle) with Fiber $F$ is a Map $f:E\to B$ where $E$ is called the Total Space of the fiber bundle and $B$ the Base Space of the fiber bundle. The main condition for the Map to be a fiber bundle is that every point in the Base Space $b\in B$ has a Neighborhood $U$ such that $f^{-1}(U)$ is Homeomorphic to $U\times F$ in a special way. Namely, if

$$h: f^{-1}(U) \to U \times F$$

is the Homeomorphism, then

$${\rm proj}_{U} \circ h = f_{\vert f^{-1}(U)\vert},$$

where the Map ${\rm proj}_{U}$ means projection onto the $U$ component. The homeomorphisms $h$ which ``commute with projection'' are called local Trivializations for the fiber bundle $f$. In other words, $E$ looks like the product $B\times F$ (at least locally), except that the fibers $f^{-1}(x)$ for $x \in B$ may be a bit ``twisted.''

Examples of fiber bundles include any product $B\times F\to B$ (which is a bundle over $B$ with Fiber $F$), the Möbius Strip (which is a fiber bundle over the Circle with Fiber given by the unit interval [0,1]; i.e, the Base Space is the Circle), and $\Bbb{S}^3$ (which is a bundle over $\Bbb{S}^2$ with fiber $\Bbb{S}^1$). A special class of fiber bundle is the Vector Bundle, in which the Fiber is a Vector Space.

See also Bundle, Fiber Space, Fibration