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Smooth structures on non-orientable $4$-manifolds via twisting operations

Valentina Bais, Rafael Torres

TL;DR

The paper investigates smooth structures on non-orientable 4-manifolds via twisting operations and Gluck twists, linking these constructions to Cappell–Shaneson mapping tori and CP$^2$-stabilization. Central contributions include constructing a manifold $Y$ from $\mathbb{R}P^2\times S^2$ that is homeomorphic but not diffeomorphic to the nontrivial $S^2$-bundle over $\mathbb{R}P^2$, and showing a 5D cobordism with a single 2-handle to a mapping torus related to exotic $\mathbb{R}P^4$; twisting inside $Y$ yields new examples like $\mathbb{R}P^4\#_{S^1}\mathbb{R}P^4$ with the same homeomorphism type but distinct smooth structures. The authors introduce an η-invariant approach (Stolz) in the Pin$^+$-setting to distinguish smooth structures, including families with Euler characteristic $1$, and demonstrate that CP$^2$-stabilization often renders these inequivalent structures diffeomorphic. They further explore knotted 2-spheres in non-orientable 4-manifolds, revealing nontrivial distinctions between complements that contrast with orientable cases. Overall, the work extends the landscape of exotic smooth structures in non-orientable 4-manifolds and highlights the effectiveness of Gluck twists, RP$^2$ twisting, and η-invariants in this setting.

Abstract

Four observations compose the main results of this note. The first records the existence of a smoothly embedded 2-sphere $S$ inside $\mathbb{R} P^2\times S^2$ such that performing a Gluck twist on $S$ produces a manifold $Y$ that is homeomorphic but not diffeomorphic to the total space of the non-trivial 2-sphere bundle over the real projective plane $S(2γ\oplus \mathbb{R})$. The second observation is that there is a 5-dimensional cobordism with a single 2-handle between the 4-manifold $Y$ and a mapping torus that was used by Cappell-Shaneson to construct an exotic $\mathbb{R} P^4$. This construction of $Y$ is similar to the one of the Cappell-Shaneson homotopy 4-spheres. The third observation is that twisting an embedded real projective plane inside $Y$ produces a manifold that is homeomorphic but not diffeomorphic to the circle sum of two copies of $\mathbb{R}P^4$. Knotting phenomena of 2-spheres in non-orientable 4-manifolds that stands in glaring contrast with known phenomena in the orientable domain is pointed out in the fourth observation.

Smooth structures on non-orientable $4$-manifolds via twisting operations

TL;DR

The paper investigates smooth structures on non-orientable 4-manifolds via twisting operations and Gluck twists, linking these constructions to Cappell–Shaneson mapping tori and CP-stabilization. Central contributions include constructing a manifold from that is homeomorphic but not diffeomorphic to the nontrivial -bundle over , and showing a 5D cobordism with a single 2-handle to a mapping torus related to exotic ; twisting inside yields new examples like with the same homeomorphism type but distinct smooth structures. The authors introduce an η-invariant approach (Stolz) in the Pin-setting to distinguish smooth structures, including families with Euler characteristic , and demonstrate that CP-stabilization often renders these inequivalent structures diffeomorphic. They further explore knotted 2-spheres in non-orientable 4-manifolds, revealing nontrivial distinctions between complements that contrast with orientable cases. Overall, the work extends the landscape of exotic smooth structures in non-orientable 4-manifolds and highlights the effectiveness of Gluck twists, RP twisting, and η-invariants in this setting.

Abstract

Four observations compose the main results of this note. The first records the existence of a smoothly embedded 2-sphere inside such that performing a Gluck twist on produces a manifold that is homeomorphic but not diffeomorphic to the total space of the non-trivial 2-sphere bundle over the real projective plane . The second observation is that there is a 5-dimensional cobordism with a single 2-handle between the 4-manifold and a mapping torus that was used by Cappell-Shaneson to construct an exotic . This construction of is similar to the one of the Cappell-Shaneson homotopy 4-spheres. The third observation is that twisting an embedded real projective plane inside produces a manifold that is homeomorphic but not diffeomorphic to the circle sum of two copies of . Knotting phenomena of 2-spheres in non-orientable 4-manifolds that stands in glaring contrast with known phenomena in the orientable domain is pointed out in the fourth observation.
Paper Structure (20 sections, 20 theorems, 51 equations, 4 figures)

This paper contains 20 sections, 20 theorems, 51 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Background results, tools and constructions
  4. Handlebody diagrams of non-orientable 4-manifolds
  5. Gluck twists
  6. 2-sphere bundles over the real projective plane
  7. Circle sums
  8. Twisting an embedded $\mathbb{R} P^2$
  9. $\mathop{\mathrm{Pin}}\nolimits^{\pm}$-structures on a mapping torus and framings on its cross-section
  10. An invariant to distinguish smooth structures
  11. An inequivalent smooth structure and an embedded 2-sphere
  12. Inequivalent smooth structures on non-orientable 4-manifolds.
  13. A result of Akbulut
  14. Connected sums with $\mathbb{CP}^2$
  15. Proofs of main results
  16. ...and 5 more sections

Key Result

Theorem 1

There is a smoothly embedded 2-sphere $S\subset \mathbb{R} P^2\times S^2$ with trivial normal bundle $\nu( S) = D^2\times S^2$ such that the complements are not homotopy equivalent, and doing a Gluck twist on $S$ yields a 4-manifold $Y$ that is homeomorphic but not diffeomorphic to the non-orientable total space $S(2\gamma \oplus \mathbb{R})$ of the non-trivial 2-sphere bundle over the real projec

Figures (4)

  • Figure 1: Handlebodies of (\ref{['Decomposition Y']}) and (\ref{['Decomposition Z']}) are drawn in Figure 4. The value $p = - 1$ corresponds to $Y$, and $p = 1$ to $Z$. Figure 4 corrects [Torres].
  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (35)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 25 more