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Fiber-preserving and orientation-reversing involutions of Seifert fibered 3-manifolds

Benjamin Peet

TL;DR

This work classifies fiber-preserving, orientation-reversing involutions on orientable Seifert fibered 3-manifolds. The authors construct a special class $\Psi$ of such involutions that act trivially on the base by gluing a product involution across copies of $V(2,2;-1)$, and prove that $\Psi$ forms a single conjugacy class under fiber-preserving diffeomorphisms. The main result shows every fiber-preserving, orientation-reversing involution can be written as $\psi\circ g$ with $g$ fiber-preserving and orientation-preserving and $\psi\in\Psi$, reducing the problem to the orientation-preserving case, with an extension to manifolds whose base orbifold is non-orientable via the orientable base-space double cover. The paper also provides a concrete construction, analysis of fixed-point and Dehn-filling behavior, and an explicit example illustrating the six conjugacy-inequivalent involutions arising in a representative manifold. This yields a complete framework for understanding these involutions in terms of extended product actions and clarifies the interplay with base-space orientation and non-orientable bases.

Abstract

We consider fiber-preserving, orientation-reversing involutions on orientable Seifert fibered 3-manifolds and the conditions on a manifold for admissibility of such involutions. We construct a class $Ψ$ of fiber-preserving, orientation-reversing involutions that act trivially on the base. Each element of $Ψ$ is obtained by extending a product involution across Seifert pieces of type $V(2,2;-1)$ - a solid torus with three fibers filled according to Seifert invariants $(2,1)$, $(2,1)$, and $(1,-1)$. We show that $Ψ$ forms a single conjugacy class under fiber-preserving diffeomorphisms. Our main result establishes that any fiber-preserving, orientation-reversing involution factors as $ψ\circ g$, where $g$ is fiber-preserving and orientation-preserving and $ψ\inΨ$, thus reducing the problem to the previously known orientation-preserving case. Through the orientable base-space double covering, we further extend the classification to manifolds with non-orientable base orbifold.

Fiber-preserving and orientation-reversing involutions of Seifert fibered 3-manifolds

TL;DR

This work classifies fiber-preserving, orientation-reversing involutions on orientable Seifert fibered 3-manifolds. The authors construct a special class of such involutions that act trivially on the base by gluing a product involution across copies of , and prove that forms a single conjugacy class under fiber-preserving diffeomorphisms. The main result shows every fiber-preserving, orientation-reversing involution can be written as with fiber-preserving and orientation-preserving and , reducing the problem to the orientation-preserving case, with an extension to manifolds whose base orbifold is non-orientable via the orientable base-space double cover. The paper also provides a concrete construction, analysis of fixed-point and Dehn-filling behavior, and an explicit example illustrating the six conjugacy-inequivalent involutions arising in a representative manifold. This yields a complete framework for understanding these involutions in terms of extended product actions and clarifies the interplay with base-space orientation and non-orientable bases.

Abstract

We consider fiber-preserving, orientation-reversing involutions on orientable Seifert fibered 3-manifolds and the conditions on a manifold for admissibility of such involutions. We construct a class of fiber-preserving, orientation-reversing involutions that act trivially on the base. Each element of is obtained by extending a product involution across Seifert pieces of type - a solid torus with three fibers filled according to Seifert invariants , , and . We show that forms a single conjugacy class under fiber-preserving diffeomorphisms. Our main result establishes that any fiber-preserving, orientation-reversing involution factors as , where is fiber-preserving and orientation-preserving and , thus reducing the problem to the previously known orientation-preserving case. Through the orientable base-space double covering, we further extend the classification to manifolds with non-orientable base orbifold.
Paper Structure (24 sections, 16 theorems, 52 equations, 4 figures)

This paper contains 24 sections, 16 theorems, 52 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminary definitions
  3. Admissible Seifert manifolds
  4. Previous results
  5. Fixed point free involutions
  6. Summary of admissible Seifert manifolds
  7. Involutions on genus $g$ surfaces
  8. Orientation-preserving
  9. Orientation-reversing
  10. Theorem
  11. Preliminary results
  12. Construction of the fiber- and orientation-reversing involutions $\Psi$
  13. Construction of $V(2,2;-1)$
  14. Construction of $M=(g,o_1|(2,1),\ldots,(2,1),(1,-\frac{n}{2}))$
  15. Construction of fiber- and orientation-reversing involution
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $M$ be a closed, compact, and orientable Seifert $3$-manifold that fibers over an orientable base space. Let $\varphi:G\rightarrow Diff_{+}^{fp}(M)$ be a finite group action on $M$ such that the obstruction class can be expressed as for a collection of fibers $\{\alpha_{1},\ldots,\alpha_{m}\}$ and integers $\{b_{1},\ldots,b_{m}\}$. Then $\varphi$ is an extended product action.

Figures (4)

  • Figure 1: $3I\times3I$ less three discs
  • Figure 2: Dehn twist annuli
  • Figure 3: Loop in base space
  • Figure 4: Base space

Theorems & Definitions (26)

  • Theorem 1.1
  • Lemma 3.1
  • Corollary 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Theorem 4.1
  • Example 4.1
  • Proposition 5.1
  • proof
  • Proposition 5.2
  • ...and 16 more