A Smooth Shioda-Tate Isomorphism for Elliptically Fibered 4-Manifolds

Maria Morariu

Home/Archive/arXiv

arXiv · 2611.05177

A Smooth Shioda-Tate Isomorphism for Elliptically Fibered 4-Manifolds

Maria Morariu

View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper studies a subgroup of the mapping class group of an elliptically fibered $4$-manifold $π: M\to B$, inspired by number theory. The smooth Mordell-Weil group $\mathrm{MW}(π)$ is the group of isotopy classes of smooth sections of $π$. We construct a smooth analog of the Shioda-Tate isomorphism, identifying $\mathrm{MW(π)}$ with the quotient of $H^2(M;\mathbb{Z})$ by the trivial lattice of $π$. This shows that $\mathrm{MW}(π)$ can be computed explicitly from $H^2(M;\mathbb{Z})$. As an application, we show that the natural inclusion of the holomorphic Mordell-Weil group into the smooth one is an isomorphism if and only if $M$ is a rational elliptic surface. As another application, we show that $\mathrm{MW}(π)$ satisfies a strong version of Nielsen realization: any torsion element of $\mathrm{MW}(π)$ is represented by a holomorphic translation of $M$ of the same order.

Explore related subjects

math.GT

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Maria Morariu. 2026-11-03. A Smooth Shioda-Tate Isomorphism for Elliptically Fibered 4-Manifolds. https://arxiv.org/abs/2611.05177

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

2-Linearizability of Geometric 3-Manifold Groups Over Commutative Rings

The fundamental groups of compact 3-manifolds are known to be residually finite. Feng Luo conjectured that a stronger statement is true, by only allowing finite groups of the form $\mathrm{PGL}(2,R)$, where $R$ is a finite commutative ring. In earlier work, this conjecture was disproven in full generality. The conjecture arose in the context of orientable connected compact 3-manifolds which are geometrizable. By constructing explicit faithful linear representations using rings with nilpotent elements, we demonstrate that the conjecture holds for six of the eight Thurston model geometries, namely all but $\mathbb{S}^3$ and $\widetilde{\mathrm{SL}_2}$. In the case of $\mathbb{S}^3$, the conjecture holds if we replace $\mathrm{PGL}(2,R)$ with $\mathrm{GL}(2,R)$. A spherical counterexample for the projective variant is the Poincaré homology sphere $Σ(2,3,5)$. In the case of $\widetilde{\mathrm{SL}_2}$, the conjecture fails to hold for both the projective and non-projective variants; a counterxample is provided by the Brieskorn sphere $Σ(2,3,7)$.

math.GT↗

arXiv2026

Erratum to "Linear representations of 3-manifold groups over rings"

The fundamental groups of compact $3$-manifolds are known to be residually finite. Feng Luo conjectured that a stronger statement is true, by allowing only finite groups of the form $\mathrm{PGL}_2(R),$ where $R$ is a finite commutative ring with identity. In a prior paper, "Linear representations of $3$-manifold groups over rings", we, together with two co-authors, gave an equivalent formulation of Luo's conjecture via faithful representations, investigated variants of the conjecture where the target group is of type $\mathrm{SL}_2, \mathrm{GL}_2, \mathrm{PSL}_2$, or $\mathrm{PGL}_2$ as in the original conjecture, and claimed to provide a $3$-manifold group which violated all four variants of the conjecture. We have discovered a flaw in the work, such that the existing proof is valid only for $\mathrm{SL}_2$ and $\mathrm{GL}_2$. Here, we provide a modified argument for the $\mathrm{PSL}_2$ and $\mathrm{PGL}_2$ cases.

math.GT↗

arXiv2026

Universal bounds for decompositions of PL manifolds

The geometric dimension of a nonempty compact PL manifold $N$ is the smallest nonnegative integer $d$ for which $N$ admits a handle decomposition with all handles of index at most $d$. A PL filling of a compact PL manifold is a decomposition into compact codimension $0$ PL submanifolds, called sectors, with pairwise disjoint interiors and intersections described by standard local models. We advance Singhof's program of simplifying the sectors of a filling and their intersections. Specifically, for integers $m, \ell\ge 1$ and nonnegative integers $a_1\le\cdots\le a_\ell$, we prove that every closed PL $m$-manifold admits a filling into $\ell$ sectors whose nonempty $r$-fold intersections have geometric dimension at most $a_r$ for each $1\le r\le \ell$ if and only if \[ a_{λ_1}+\cdots+a_{λ_q}\ge m-\ell+1 \] for every partition $λ_1+\cdots+λ_q=\ell$ into positive integers. Under the same inequalities, any prescribed $\ell$-sector filling of the boundary of a compact PL $m$-manifold extends to a filling with the corresponding relative handle index bounds. Necessity follows from the cup-product obstructions of Courte, Moussard, Ren, and Zhou. To prove sufficiency, we develop compatible handle trades that simplify intersections in increasing order of depth. For nondecreasing tuples of nonnegative integers, our theorem resolves the PL classification problem posed by Courte, Moussard, Ren, and Zhou. In particular, we obtain the bound $(2,2,4)$ for $3$-sector fillings in dimension $6$ and improve the $5$-sector bound $(1,2,3,3,4)$ of Rubinstein and Tillmann to $(1,2,2,3,4)$ in dimension $8$.

math.GT↗