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$\text{Pin}^{\pm}$-structures on non-oriented 4-manifolds via Lefschetz fibrations

Valentina Bais

Abstract

We study necessary and sufficient conditions for a 4-dimensional Lefschetz fibration over the 2-disk to admit a $\text{Pin}^{\pm}$-structure, extending the work of A. Stipsicz in the orientable setting. As a corollary, we get existence results of $\text{Pin}^{+}$ and $\text{Pin}^-$-structures on closed non-orientable 4-manifolds and on Lefschetz fibrations over the 2-sphere. In particular, we show via three explicit examples how to read-off $\text{Pin}^{\pm}$-structures from the Kirby diagram of a 4-manifold. We also provide a proof of the well-known fact that any closed 3-manifold $M$ admits a $\text{Pin}^-$-structure and we find a criterion to check whether or not it admits a $\text{Pin}^+$-structure in terms of a handlebody decomposition. We conclude the paper with a characterization of $\text{Pin}^+$-structures on vector bundles.

$\text{Pin}^{\pm}$-structures on non-oriented 4-manifolds via Lefschetz fibrations

Abstract

We study necessary and sufficient conditions for a 4-dimensional Lefschetz fibration over the 2-disk to admit a -structure, extending the work of A. Stipsicz in the orientable setting. As a corollary, we get existence results of and -structures on closed non-orientable 4-manifolds and on Lefschetz fibrations over the 2-sphere. In particular, we show via three explicit examples how to read-off -structures from the Kirby diagram of a 4-manifold. We also provide a proof of the well-known fact that any closed 3-manifold admits a -structure and we find a criterion to check whether or not it admits a -structure in terms of a handlebody decomposition. We conclude the paper with a characterization of -structures on vector bundles.
Paper Structure (9 sections, 11 theorems, 65 equations, 10 figures)

This paper contains 9 sections, 11 theorems, 65 equations, 10 figures.

Table of Contents

  1. Introduction
  2. $\text{Pin}^{\pm}$-structures and embedded surfaces
  3. Non-orientable Lefschetz fibrations
  4. $\text{Pin}^-$-structures on 4-manifolds via Lefschetz fibrations
  5. $\text{Pin}^+$-structures on Lefschetz fibrations over the 2-disk
  6. Some examples
  7. $\text{Pin}^{\pm}$-structures on Lefschetz fibrations over $S^2$
  8. $\text{Pin}^+$ and $\text{Pin}^-$-structures on 3-manifolds
  9. Interpretation of $\text{Pin}^+$-structures à la Milnor

Key Result

Theorem 1

Let $X$ be a smooth 4-manifold and $f: X \to D^2$ a Lefschetz fibration with regular fiber $\Sigma$. There is no $\text{Pin}^-$-structure on $X$ if and only if there are $k+1$ vanishing cycles $c_0, c_1, \dots, c_k$ such that $[c_0]=\sum_{i=1}^k [c_i] \in H_1(\Sigma; \mathbb{Z}_2)$ and $k+\sum_{1 \l

Figures (10)

  • Figure 1: A Kirby diagram of $\mathbb{R} \mathbb{P}^4$.
  • Figure 2: The Lefschetz fibration associated to the 2-handlebody of $\mathbb{R} \mathbb{P}^4$.
  • Figure 3: A Kirby diagram of $S^2 \mathbin{\widetilde{{\times}}} \mathbb{R} \mathbb{P}^2$.
  • Figure 4:
  • Figure 5: The 2-handle- body of $S^2 \mathbin{\widetilde{{\times}}} \mathbb{R} \mathbb{P}^2$ as a Lefschetz fibration over $D^2$.
  • ...and 5 more figures

Theorems & Definitions (19)

  • Theorem 1
  • Theorem 2
  • Theorem 3: Miller-Ozbagci [MillerOzbagci]
  • Example 4: $\mathbb{R} \mathbb{P}^4$
  • Definition 5
  • Remark 6
  • Theorem 7: Kirby-Taylor, [KirbyTaylor]
  • Lemma 8
  • Lemma 9
  • Remark 10
  • ...and 9 more