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On nonorientable $4$--manifolds

R. İnanç Baykur, Porter Morgan

Abstract

We present several structural results on closed, nonorientable, smooth $4$--manifolds, extending analogous results and machinery for the orientable case. We prove the existence of simplified broken Lefschetz fibrations and simplified trisections on nonorientable $4$--manifolds, yielding descriptions of them via factorizations in mapping class groups of nonorientable surfaces. With these tools in hand, we classify low genera simplified broken Lefschetz fibrations on nonorientable $4$--manifolds. We also establish that every closed, smooth $4$--manifold is obtained by surgery along a link of tori in a connected sum of copies of $\mathbb{CP}^2$, $S^1 \times S^3$ and $S^1\widetilde{\times} S^3$. Our proofs make use of topological modifications of singularities, handlebody decompositions, and mapping classes of surfaces.

On nonorientable $4$--manifolds

Abstract

We present several structural results on closed, nonorientable, smooth --manifolds, extending analogous results and machinery for the orientable case. We prove the existence of simplified broken Lefschetz fibrations and simplified trisections on nonorientable --manifolds, yielding descriptions of them via factorizations in mapping class groups of nonorientable surfaces. With these tools in hand, we classify low genera simplified broken Lefschetz fibrations on nonorientable --manifolds. We also establish that every closed, smooth --manifold is obtained by surgery along a link of tori in a connected sum of copies of , and . Our proofs make use of topological modifications of singularities, handlebody decompositions, and mapping classes of surfaces.

Paper Structure

This paper contains 16 sections, 16 theorems, 11 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lefschetz, fold and cusp singularities
  4. Simplified broken Lefschetz fibrations
  5. Simplified trisections
  6. Nonorientable Kirby diagrams
  7. Topology of nonorientable SBLFs
  8. Mapping class group factorizations for SBLFs
  9. Simplified BLFs and trisections on nonorientable $4$--manifolds
  10. Existence of SBLFs
  11. Simplified trisections
  12. Classification of low genera fibrations
  13. Klein bottle bundles over the 2-sphere
  14. Nonorientable genus two SBLFs with only indefinite fold singularity.
  15. Nonorientable genus two SBLFs with Lefschetz singularities.
  16. ...and 1 more sections

Key Result

Theorem 1

Every closed, connected, nonorientable $4$--manifold $X$ admits a simplified broken Lefschetz fibration that can be obtained by homotoping a given generic map $X \to S^2$.

Figures (14)

  • Figure 1: Left: a Kb--bundle over $S^2$. Middle: the standard Kirby diagram for $\mathbb{RP}^4\#\mathbb{RP}^4$. Right: a diagram of $S^1\Tilde{\times} S^3\#S^1\Tilde{\times} S^3$, obtained from the middle diagram after a pair of nonorientable blow-downs.
  • Figure 2: The orientation double cover of $\mathbb{RP}^4\#\mathbb{RP}^4$. The Kirby diagram shown has two $0$--handles, two twisted $1$--handles, two untwisted $1$--handles, and four $2$--handles. The vertical dashed line distinguishes the two $0$--handles, and the $2$--handles are drawn in black, gray, red, and blue. Rotating one of the $0$--handles, all of the $1$--handles become untwisted.
  • Figure 3: Left: the 2-skeleton $R_n$. On the upper half of the Kirby diagram, the higher $n-1$ strands run parallel, and the lowest one goes behind all of them. In the lower half of the diagram, all $n$ strands run parallel. Right: the rational homology ball $B_n$. The $2$--handle passes over the $1$--handle $n$ times. In both Kirby diagrams shown, the $2$--handles have blackboard framing.
  • Figure 4: The orientation double cover of $R_n$.
  • Figure 5: A BLF on $(\#_{g-k} S^1\Tilde{\times} S^3) \# (\#_k \mathbb{RP}^4)$. Figure \ref{['fig:notall_htpy']} shows the base diagram of this BLF when $k=0$.
  • ...and 9 more figures

Theorems & Definitions (41)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 5
  • Example 6
  • Proposition 7
  • proof
  • Example 8
  • Example 9
  • ...and 31 more