1
$\begingroup$

Suppose $\pi$ is the projection, $E$ total space, $B$ the base space, and $F$ the fiber.

A section of a fiber bundle is a continuous map $f\colon B \to E$ such that $\pi(f(x))=x$ for all $x \in B$.

Suppose that $M$ and $N$ are base spaces, and $\pi_E : E \to M$ and $\pi_F : F \to N$ are fiber bundles over $M$ and $N$, respectively. A bundle map consists of a pair of continuous functions

$\varphi\colon E\to F,\quad f\colon M\to N$

such that $\pi_F\circ \varphi = f\circ\pi_E$.

I wonder how the following is consistent with the definition of section above?

A bundle map from the base space itself (with the identity mapping as projection) to $E$ is called a section of $E$.

Specifically, how is the other fiber bundle like for the bundle map?

Added: I wonder if people think of a section more often as a "inverse" of projection, or as a bundle map? What is the purpose of viewing a section as a bundle map, which seems to me so indirect?

All quotes are from Wikipedia.

Thanks and regards!

  • 1
    When $\pi:E \to M$ is introduced, I don't think it is necessary to name the fibre here, but if it must be named, it would be better to give it a symbol that doesn't conflict with the $F$ that is the total space of the target bundle introduced subsequently.2011-08-16

0 Answers 0