nLab Seifert fibration

Definition

A Seifert 3-manifold is a 3-manifold which is the total space of a circle-fiber bundle (circle bundle) over a 2-dimensional orbifold. Since the circle-fibration structure may not be unique, one also speaks, more explicitly, of Seifert-fibered manifolds.

If already the total space is a 3-orbifold, one correspondingly speaks of a Seifert orbifold etc. (e.g. Mecchia & Seppi 2020 Def. 2.6).

Fully generally one speaks of Seifert fibrations.

(Beware that there is also the un-related notion of Seifert surfaces, also considered in higher dimensions: These are coboundaries of knots.)

Accordingly, a general Seifert fibration is locally an orbifold quotient fibration of the form

$$\big( D^2 \times S^1 \big) \sslash \mathbb{Z}_n \xrightarrow{ pr_{D^2} \sslash \mathbb{Z}_n } D^2 \sslash \mathbb{Z}_n \,,$$

where the cyclic group $\mathbb{Z}_n \coloneqq \mathbb{Z}/n$ acts diagonally via rotations by

1. the angle $2\pi\tfrac{1}{a}$ on the 2-disk $D^2 = \big\{ z \in \mathbb{C} \,\big\vert\, {\vert z\vert} \lt 1 \big\}$,

2. an angle $-2\pi\tfrac{b}{a}$ on the circle $S^1 \simeq \big\{ \xi \in \mathbb{C} \,\big\vert\, {\vert \xi \vert} = 1 \big\}$,

$$(z, \xi) \,\sim\, \big( z \, q ,\, \xi \, q^{-b} \big) \,, \;\;\;\; q \coloneqq \exp\big(2\pi \mathrm{i} \tfrac{1}{a}\big) \,.$$

with $a,b \in \mathbb{Z}$, $a \geq 2$ (not necessarily coprime), their fraction $b/a$ also called the slope modulo 1 of the fibration at this point:

This orbifold quotient is a non-singular (Seifert-fibered) smooth manifold iff $a$ and $b$ are coprime integers, otherwise it is an orbifold whose orbi-singular (fixed) locus has isotropy group $\mathbb{Z}_{gcd}$ determined by the greatest common divisor $gcd \coloneqq gcd(a,b)$.

(Bonahon & Siebenmann 2010 p 36 (18 of 67))

Related concepts

• lens space

Beware that there is also the un-related concept of

• Seifert surfaces

References

Seifert 3-manifolds

The original article:

• Herbert Seifert: Topology of 3-dimensional fibered spaces, in: A textbook of topology, Pure Appl. Math. 89, Academic Press (1980) 139–152 [pdf]

Lecture notes:

• Mark Jankins, Walter Neumann: Lectures on Seifert Manifolds, Brandeis University (1981) [pdf, pdf]

Monographs:

• Kyung Bai Lee, Frank Raymond: Seifert Fiberings, Mathematical Surveys and Monographs (2010) 166 [doi:10.1090/surv/166, toc:pdf]

Further survery:

• Kyung Bai Lee, Frank Raymond: Seifert Manifolds [arXiv:math/0108188]

Discussion via diffeological spaces:

• Patrick Iglesias-Zemmour, Jean-Paul Mohsen: Seifert Orbifolds (2015) [pdf, pdf]

See also:

• Wikipedia, Seifert fiber space

• Encyclopedia of Mathematics: Seifert fibration

Seifert 3-orbifolds

Generalization to the case that already the total space is an orbifold (not just its $S^1$-quotient):

• Francis Bonahon, Laurent C. Siebenmann: The classification of Seifert fibred 3-orbifolds, in: Low Dimensional Topology, Cambridge University Press (2010) 19-85 [doi:10.1017/CBO9780511662744.002]

See also:

• Mattia Mecchia, Andrea Seppi: On the diffeomorphism type of Seifert fibered spherical 3-orbifolds, Rend. Istit. Mat. Univ. Trieste 52 (2020) 1-39 [arXiv:2005.12023, doi:10.13137/2464-8728/30920]

In field theory

On the 3d-3d correspondence for Seifert manifolds:

• Sergei Gukov, Du Pei, pp 7 in: Equivariant Verlinde formula from fivebranes and vortices, Commun. Math. Phys. 355 (2017) 1–50 [arXiv:1501.01310, doi:10.1007/s00220-017-2931-9]

• Du Pei: 3d-3d correspondence for Seifert manifolds, PhD thesis (2016) [spire:1469350, pdf]

Claim of geometric engineering of topological order on M5-branes in the guise of fusion categories:

• Gil Young Cho, Dongmin Gang, Hee-Cheol Kim: M-theoretic Genesis of Topological Phases, J. High Energ. Phys. 2020 115 (2020) [arXiv:2007.01532, doi:10.1007/JHEP11(2020)115]

  (by wrapping M5-branes on, among others, Seifert manifolds)

• Shawn X. Cui, Yang Qiu, Zhenghan Wang, From Three Dimensional Manifolds to Modular Tensor Categories, Commun. Math. Phys. 397 (2023) 1191–1235 [doi:10.1007/s00220-022-04517-4, arXiv:2101.01674]

• Federico Bonetti, Sakura Schäfer-Nameki, Jingxiang Wu: $MTC[M_3,G]$: 3d Topological Order Labeled by Seifert Manifolds [arXiv:2403.03973]

See also:

• Nathan Paul Moore: String Theory, Chern-Simons Theory and the Fractional Quantum Hall Effect, PhD thesis (2014) [spire:1783390]