Braided Multisections and Symplectic Four-Manifolds with the Rational Cohomology of S2× S2
Abstract
We construct symplectic four-manifolds by taking mixed fiber sums along explicit cyclic multisections in ruled surfaces. For a connected unbranched degree-p multisection in Σg× S2, we determine the first homology and fundamental group of the complement and prove that its boundary is incompressible. It follows that no direct gluing of two such complements can be simply connected; moreover, the first homology of every direct sum retains finite quotients determined by the covering degrees. We classify the mixed sums having Euler characteristic $4$ and signature $0$. Up to interchanging the two summands, exactly three possibilities occur, corresponding to the degree pairs (2,3), (2,4), and (3,3). For each of these cases, suitable adapted product-framed symplectic gluings have the rational cohomology ring of S2× S2. Varying the gluing by symplectic transvections produces infinitely many pairwise nondiffeomorphic examples, distinguished by the unbounded orders of their finite first homology groups. We also construct the twisted ruled analogue of the (2,4) case. Explicit finite-holonomy multisections give connected square-zero symplectic surfaces in the classes 2S3−F3 and 4S2−2F2 in the nontrivial S2-bundles over Σ3 and Σ2, respectively. More generally, for a square-zero degree-p multisection in the nontrivial bundle the complement has first homology ℤ2g⊕ℤ/(p/2). Suitable gluings in the twisted (2,4) case have b1=0, b2=2, and signature zero, and every such sum is non-spin. Hence they have the rational cohomology ring of ℂP2#ℂP̅ 2. We compare these constructions with the author's 2006 construction of minimal symplectic four-manifolds having the integral cohomology S2× S2, obtained via knot surgery and twisted fiber sums.
Related
- Higher Braided Mixed Sums, Intersection Lattices, and Symplectic Four-Manifolds
- Gurtas Lefschetz Fibrations, Rational Blowdowns, and Exotic Symplectic Four-Manifolds
- Braided Multisections and Aspherical Stein Fillings
- Braided Multisections and Akhmedov's Fake Quadrics
- Nonexistence of degree two rational multisections of conic bundles over the plane
BibTeX
@misc{akhmedov2026braided,
title = {{Braided Multisections and Symplectic Four-Manifolds with the Rational Cohomology of $S^2\times S^2$}},
author = {Anar Akhmedov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11727},
eprint = {2609.11727},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11727},
note = {CC0},
}
RIS
TY - UNPB
TI - Braided Multisections and Symplectic Four-Manifolds with the Rational Cohomology of S^2× S^2
AU - Anar Akhmedov
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11727
UR - https://arxiv.org/abs/2609.11727
AB - We construct symplectic four-manifolds by taking mixed fiber sums along explicit cyclic multisections in ruled surfaces. For a connected unbranched degree-p multisection in Σ_g× S^2, we determine the first homology and fundamental group of the complement and prove that its boundary is incompressible. It follows that no direct gluing of two such complements can be simply connected; moreover, the first homology of every direct sum retains finite quotients determined by the covering degrees. We classify the mixed sums having Euler characteristic $4$ and signature $0$. Up to interchanging the two summands, exactly three possibilities occur, corresponding to the degree pairs (2,3), (2,4), and (3,3). For each of these cases, suitable adapted product-framed symplectic gluings have the rational cohomology ring of S^2× S^2. Varying the gluing by symplectic transvections produces infinitely many pairwise nondiffeomorphic examples, distinguished by the unbounded orders of their finite first homology groups. We also construct the twisted ruled analogue of the (2,4) case. Explicit finite-holonomy multisections give connected square-zero symplectic surfaces in the classes 2S_3−F_3 and 4S_2−2F_2 in the nontrivial S^2-bundles over Σ_3 and Σ_2, respectively. More generally, for a square-zero degree-p multisection in the nontrivial bundle the complement has first homology ℤ^2g⊕ℤ/(p/2). Suitable gluings in the twisted (2,4) case have b_1=0, b_2=2, and signature zero, and every such sum is non-spin. Hence they have the rational cohomology ring of ℂP^2#ℂP̅^ 2. We compare these constructions with the author's 2006 construction of minimal symplectic four-manifolds having the integral cohomology S^2× S^2, obtained via knot surgery and twisted fiber sums.
ER -