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Seifert fibrations of lens spaces over non-orientable bases

Hansjörg Geiges, Christian Lange

TL;DR

This note corrects a gap in the earlier classification of Seifert fibrations of lens spaces over non-orientable bases by showing the base must be $RP^2(n)$ with at most one singular fibre and by identifying the missing cases. The main result shows that the only lens spaces with such fibrations are $L(4n,2n±1)$ (for $n≥1$), each with a unique fibration over $RP^2(n)$; for $n≥2$ the Seifert invariant is forced to $β=±1$ and the diffeomorphism type is explicit via Hopf and SO(4) constructions. The analysis combines orbifold fundamental group considerations, Euler-characteristic arguments, and explicit Seifert constructions to complete the classification, correcting gaps from the previous work and clarifying the role of non-orientable bases. These results refine the landscape of Seifert fibrations on lens spaces and connect to classical results on Klein bottles in lens spaces and their diffeomorphism types.

Abstract

We classify the Seifert fibrations of lens spaces where the base orbifold is non-orientable. This is an addendum to our earlier paper `Seifert fibrations of lens spaces'. We correct Lemma 4.1 of that paper and fill the gap in the classification that resulted from the erroneous lemma.

Seifert fibrations of lens spaces over non-orientable bases

TL;DR

This note corrects a gap in the earlier classification of Seifert fibrations of lens spaces over non-orientable bases by showing the base must be with at most one singular fibre and by identifying the missing cases. The main result shows that the only lens spaces with such fibrations are (for ), each with a unique fibration over ; for the Seifert invariant is forced to and the diffeomorphism type is explicit via Hopf and SO(4) constructions. The analysis combines orbifold fundamental group considerations, Euler-characteristic arguments, and explicit Seifert constructions to complete the classification, correcting gaps from the previous work and clarifying the role of non-orientable bases. These results refine the landscape of Seifert fibrations on lens spaces and connect to classical results on Klein bottles in lens spaces and their diffeomorphism types.

Abstract

We classify the Seifert fibrations of lens spaces where the base orbifold is non-orientable. This is an addendum to our earlier paper `Seifert fibrations of lens spaces'. We correct Lemma 4.1 of that paper and fill the gap in the classification that resulted from the erroneous lemma.

Paper Structure

This paper contains 3 sections, 2 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. The base of the fibration
  3. Lens spaces fibring over $\mathbb{R}\mathrm{P}^2(n)$

Key Result

Lemma 2.1

If a lens space admits a Seifert fibration over a non-orientable base, the base is $\mathbb{R}\mathrm{P}^2$ (as a topological surface), and there is at most one singular fibre. In other words, the base orbifold is $\mathbb{R}\mathrm{P}^2(n)$ for some $n\in\mathbb{N}=\{1,2,\ldots\}$.

Theorems & Definitions (6)

  • Lemma 2.1
  • proof
  • Claim
  • Proposition 3.1
  • proof
  • Remark 3.2