Higher Braided Mixed Sums, Intersection Lattices, and Symplectic Four-Manifolds
Abstract
We study mixed symplectic sums along connected unbranched cyclic multisections of degrees p,q≥ 2 in ruled surfaces. With p(g−1)=q(h−1)=n, the resulting four-manifold has signature zero and Euler characteristic e=4n(1−1/p−1/q). We classify the types having the rational cohomology ring of #k(S2× S2). They occur only for odd k and, with t=(k+1)/2, a=g−1, and b=h−1, are exactly the solutions of n=a+b+t, a| b+t, b| a+t. For each odd k there are finitely many types; we give an explicit parametrization and tables for k≤ 15. Every type admits an equatorially product-framed symplectic realization and, possibly with different framings, a spin realization whose free integral intersection form is kH. We prove minimality, boundary incompressibility, an integral Mayer--Vietoris presentation, and a regular cyclic cover of degree lcm(p,q). The boundary subgroup π1(ΣG)×ℤ embeds in the fundamental group; this gives a nontrivial Bass--Serre splitting and rules out word-hyperbolicity, cocompact real- or complex-hyperbolic lattices, and torsion-free cocompact irreducible bidisk lattices. There are five types for k=3 and seven for k=5; for k=1 we recover the degree pairs (2,3), (2,4), and (3,3).
Related
- Braided Multisections and Symplectic Four-Manifolds with the Rational Cohomology of S2× S2
- Equivariant Quotients of Derived Symplectic Spaces and Legendrian Intersection Theorem
- Braided Surfaces and Small Exotic 4-Manifolds
- Diophantine Approximation by Lagrangian Frames in Symplectic Lattices
- The intersection of two real flag manifolds in a complex flag manifold, and the canonical form of a compact symmetric triad
BibTeX
@misc{akhmedov2026higher,
title = {{Higher Braided Mixed Sums, Intersection Lattices, and Symplectic Four-Manifolds}},
author = {Anar Akhmedov and Azer Akhmedov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03756},
eprint = {2610.03756},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03756},
note = {CC0},
}
RIS
TY - UNPB
TI - Higher Braided Mixed Sums, Intersection Lattices, and Symplectic Four-Manifolds
AU - Anar Akhmedov
AU - Azer Akhmedov
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.03756
UR - https://arxiv.org/abs/2610.03756
AB - We study mixed symplectic sums along connected unbranched cyclic multisections of degrees p,q≥ 2 in ruled surfaces. With p(g−1)=q(h−1)=n, the resulting four-manifold has signature zero and Euler characteristic e=4n(1−1/p−1/q). We classify the types having the rational cohomology ring of #_k(S^2× S^2). They occur only for odd k and, with t=(k+1)/2, a=g−1, and b=h−1, are exactly the solutions of n=a+b+t, a| b+t, b| a+t. For each odd k there are finitely many types; we give an explicit parametrization and tables for k≤ 15. Every type admits an equatorially product-framed symplectic realization and, possibly with different framings, a spin realization whose free integral intersection form is kH. We prove minimality, boundary incompressibility, an integral Mayer--Vietoris presentation, and a regular cyclic cover of degree lcm(p,q). The boundary subgroup π_1(Σ_G)×ℤ embeds in the fundamental group; this gives a nontrivial Bass--Serre splitting and rules out word-hyperbolicity, cocompact real- or complex-hyperbolic lattices, and torsion-free cocompact irreducible bidisk lattices. There are five types for k=3 and seven for k=5; for k=1 we recover the degree pairs (2,3), (2,4), and (3,3).
ER -