Irreducible 4-manifolds with order two fundamental group and even intersection form

TL;DR

This work constructs irreducible smooth 4-manifolds with order-two fundamental group and even intersection forms, i.e., spin double covers, by a $\mathbb{Z}_2$-construction applied to spin simply-connected building blocks built from Lefschetz fibrations and symplectic geometry. The authors develop spin-tracking machinery to ensure the double covers remain spin and to classify manifolds up to homeomorphism by $e$, $\sigma$, and spin type, yielding two main geographies: $w_2$-type (ii) (spin) and $w_2$-type (iii) (non-spin with spin double cover). For $w_2$-type (ii), irreducible examples realize all but 17 coordinates in the $(e,\sigma)$-plane with $b_2^+\!$ even, and all but 24 coordinates with $b_2^+$ odd, under appropriate parity constraints; for $w_2$-type (iii), irreducible examples realize all but 24 coordinates in the same plane, with further parity and $\,\sigma\bmod 16$ constraints guiding the construction. The results expand the irreducible geography of 4-manifolds with finite cyclic fundamental groups and demonstrate how spin-structure preservation interacts with equivariant fiber sums and torus surgeries to control topology and homeomorphism types. These constructions provide a framework to systematically realize large swaths of the geography while ensuring irreducibility via spin-universal covers, with potential implications for understanding exotic smooth structures in dimension four.

Abstract

We construct smooth manifolds with order two $π_1$ and even intersection forms which are irreducible, meaning they do not decompose into non-trivial connected sums. Their intersection forms being even implies that their universal covers admit spin structures. Such manifolds are determined up to homeomorphism by their Euler characteristic $e$, signature $σ$, and whether they themselves are also spin. In the case that the manifold is spin, we construct irreducible manifolds for all but $17$ realizable coordinates in the region of the $(e,σ)$-plane with $c_1^2 = 2e+3σ\geq 0$ up to orientation. In the case that the manifold is non-spin, we construct irreducible manifolds for all but $24$ realizable coordinates in the region of the $(e,σ)$-plane with $σ/8<-8$ and $c_1^2/4>9$, again up to orientation. We construct these manifolds by taking equivariant fiber sums of Lefschetz fibrations and other symplectic manifolds which are simply-connected and spin. Along the way, we develop machinery to track when the spin structure is preserved during these operations.

Figures (8)

• Figure 1: The lines on the left restricting $Q_{\tilde{X}}$ are $y\geq x+1$, $y\geq \frac{3}{2}x$, and $y\geq 2x-1$. The lines on the right restricting $Q_{X}$ are $b\geq a$, $b\geq \frac{3a-1}{2}$, and $b\geq 2a-1$. Recall that $x=2a$ and $y=2b+1$.
• Figure 2:
• Figure 3:
• Figure 4: Even $b_2^+$ and $w_2$-type $(ii)$
• Figure 5: Odd $b_2^+$ and $w_2$-type $(ii)$

Theorems & Definitions (38)

• Theorem 1
• Theorem 2
• Definition 2.1
• Theorem 3: HambletonKreck
• Proposition 1
• Definition 2.2
• Theorem 4
• Theorem 5
• Theorem 6: Hamilton and Kotschick
• Lemma 1