---
tag: 9V1GF5WMKA
kind: concept
name: Seifert fiber space
msc: [57K30, 57M50]
specializes: [R20TB2KEQQ]
sources:
  - url: https://en.wikipedia.org/wiki/Seifert_fiber_space
same_as:
  wikidata: Q1935541
  wikipedia: Seifert fiber space
---

## Summary

A Seifert fiber space is a [[3-manifold|R20TB2KEQQ]] decomposed into disjoint circles, called fibers, that fit together locally like a standard fibered solid torus.
Almost all fibers are regular, and around each of the finitely many exceptional fibers the nearby fibers wind several times.
The space of fibers is a $2$-dimensional [[orbifold|RPAVY0W0XS]], and the manifold is classified by this base and a few integer invariants.
Seifert fiber spaces account for six of the eight Thurston geometries and are one of the two kinds of pieces in the JSJ decomposition.

## Explanations

### L1 human

Picture a $3$-dimensional space filled by circles, like a bundle of loops of string that never touch.
Near most circles, the neighbors run parallel to it, as in a product of a disk and a circle.
Near a few special circles, the neighbors wind around several times before closing up, like the strands of a twisted rope.
The product $S^2 \times S^1$ and the $3$-sphere, filled by the circles of the Hopf fibration, are examples.