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Geography of irreducible 4-manifolds with order two fundamental group

Mihail Arabadji, Porter Morgan

TL;DR

The paper addresses the geography problem for irreducible smooth 4-manifolds with order-two fundamental group and odd intersection form by constructing irreducible copies of R_{m,n} for large regions in the (b_2^+, b_2^−) plane. It develops and combines a toolkit of techniques—torus/Luttinger surgeries, symplectic fiber sums, rational blow-downs, and, crucially, a Z_2-construction and fiber-reversing doubles of Lefschetz fibrations—to realize many lattice points in the geography plane, including points with even and odd b_2^+. The main result shows that, except for seven lattice points, every R_{m,n} with a suitable region constraints admits an irreducible smooth structure, thereby significantly extending the known irreducible π_1 = \mathbb{Z}_2 geography. These constructions also provide a path toward a broader program for irreducible exotic 4-manifolds and highlight methods that preserve irreducibility under intricate gluings and quotients, with potential extensions to other cyclic fundamental groups.

Abstract

Let $R$ be a closed, oriented topological 4-manifold whose Euler characteristic and signature are denoted by $e$ and $σ$. We show that if $R$ has order two $π_1$, odd intersection form, and $2e + 3σ\geq 0$, then for all but seven $(e, σ)$ coordinates, $R$ admits an irreducible smooth structure. We accomplish this by performing a variety of operations on irreducible simply-connected 4-manifolds to build 4-manifolds with order two $π_1$. These techniques include torus surgeries, symplectic fiber sums, rational blow-downs, and numerous constructions of Lefschetz fibrations, including a new approach to equivariant fiber summing.

Geography of irreducible 4-manifolds with order two fundamental group

TL;DR

The paper addresses the geography problem for irreducible smooth 4-manifolds with order-two fundamental group and odd intersection form by constructing irreducible copies of R_{m,n} for large regions in the (b_2^+, b_2^−) plane. It develops and combines a toolkit of techniques—torus/Luttinger surgeries, symplectic fiber sums, rational blow-downs, and, crucially, a Z_2-construction and fiber-reversing doubles of Lefschetz fibrations—to realize many lattice points in the geography plane, including points with even and odd b_2^+. The main result shows that, except for seven lattice points, every R_{m,n} with a suitable region constraints admits an irreducible smooth structure, thereby significantly extending the known irreducible π_1 = \mathbb{Z}_2 geography. These constructions also provide a path toward a broader program for irreducible exotic 4-manifolds and highlight methods that preserve irreducibility under intricate gluings and quotients, with potential extensions to other cyclic fundamental groups.

Abstract

Let be a closed, oriented topological 4-manifold whose Euler characteristic and signature are denoted by and . We show that if has order two , odd intersection form, and , then for all but seven coordinates, admits an irreducible smooth structure. We accomplish this by performing a variety of operations on irreducible simply-connected 4-manifolds to build 4-manifolds with order two . These techniques include torus surgeries, symplectic fiber sums, rational blow-downs, and numerous constructions of Lefschetz fibrations, including a new approach to equivariant fiber summing.

Paper Structure

This paper contains 37 sections, 26 theorems, 57 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Topological classification of $4$-manifolds with $\mathbf{\pi_1\cong\mathbb{Z}_2}$.
  4. Lefschetz fibrations and surgeries
  5. The $\mathbf{\mathbb{Z}_2}$-construction
  6. Preliminary constructions and computations
  7. Fiber-reversing double of Lefschetz fibrations
  8. Product of surfaces minus many tori.
  9. Irreducible copies of $R_{a,b}$ when $a$ is even
  10. $\mathbf{(b_2^+,b_2^-) = (2\chi, 10\chi-c)}$ for $\mathbf{0< c\leq 8\chi-2}$ (except finitely many points)
  11. $c$ is even
  12. $c$ is odd
  13. Irreducible $R_{2+2k,5+2k}$, where $k\geq 1$, i.e. $\sigma = -3$
  14. Small $\mathbf{c \in \{0,1,2,3,4\}}$
  15. $R_{2k, 10k+4}$ for $k\geq 1$ (i.e. $c=0$)
  16. ...and 22 more sections

Key Result

Theorem 1

Let $R_{m,n}$ be the closed, oriented, smoothable $4$--manifold with $b_2^+=m$, $b_2^-=n$, odd intersection form, and order two fundamental group. Let $\mathcal{R}$ be the region Unless $m=n\leq 7$, $R_{m,n}$ admits an irreducible smooth structure.

Figures (7)

  • Figure 1: Based paths for the doubled fibration
  • Figure 2: Based path for the total monodromy
  • Figure 3: The curves on $F\times G\cong \Sigma_k\times (T^2-D^2)$.
  • Figure 4: The curves on $F\times G\cong \Sigma_k \times \Sigma_2$.
  • Figure 5: Curves on $\Sigma_g$.
  • ...and 2 more figures

Theorems & Definitions (48)

  • Theorem 1
  • Proposition 1
  • proof
  • Proposition 2
  • Proposition 3
  • Definition 2.1
  • Theorem 2: Usher usher2007minimality
  • Theorem 3: Hamilton and Kotschick
  • Definition 2.2
  • Proposition 4: Tian, Proposition 3.1
  • ...and 38 more