# Section

From Encyclopedia of Mathematics

2010 Mathematics Subject Classification: *Primary:* 14-XX [MSN][ZBL]

A *section* or
*section surface* of a
surjective (continuous) map or of a
fibre space $p:X\to Y$* is*
a (continuous) mapping $s:Y\to X$ such that $p\circ s={\rm id}_Y$.

If $(X,p,Y)$ is a Serre fibration, then

$$\pi_n(X) = \pi_n(p^{-1}(pt))\oplus \pi_n(Y).$$ For a principal fibre bundle the existence of a section implies its triviality. A vector bundle always possesses the so-called zero section.

#### References

[Sp] | E.H. Spanier, "Algebraic topology", McGraw-Hill (1966) pp. 77 MR0210112 MR1325242 Zbl 0145.43303 |

**How to Cite This Entry:**

Section.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Section&oldid=30774

This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article