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Panagiotis Batakidis

Publications and source records attributed to Panagiotis Batakidis.

13 recordsLinked to original sources
arXiv2026

Obstructions to lifting quaternionic torus actions

We study the problem of lifting global and local quaternionic torus actions to principal quaternionic torus bundles. Let \(Q^k=(\operatorname{Sp}(1))^k\cong (S^3)^n\), let \(G\) be a compact Lie group acting on a connected, locally finite CW complex \(X\), and let \(Q^k\longrightarrow P\longrightarrow X\) be a principal \(Q^k\)-bundle. We first formulate a quaternionic analogue of the obstruction-theoretic framework of Hattori--Yoshida. The existence of a lifted \(G\)-action implies that the isomorphism class of \(P\) lies in the image of the restriction map induced by the Borel construction \(X_G=EG\times_GX\). In particular, the second Chern class admits an equivariant extension. Once a continuous pseudo-lift has been chosen, its failure to define a genuine action is measured by a factor set with values in the generally nonabelian gauge group \(\mathcal G(P)\cong\Gamma(\operatorname{Ad}(P))\). We obtain a necessary and sufficient lifting criterion in terms of the trivializability of this factor set, and show that, once a single lift exists, the set of all lifts modulo gauge conjugacy is classified by a pointed nonabelian \(H^1\)-set. We then apply this global theory to local quaternionic torus actions. Pulling back to the universal covering of the orbit space, untwists a local \(Q^n\)-action and produces a globally defined action on the pulled-back manifold. A preliminary lift of this global action need not be compatible with the deck transformations. We define a gauge-valued nonabelian descent defect, establish its crossed-cocycle identities and transformation law, and prove that the original local action lifts if and only if the global lifting obstruction vanishes and the descent defect is trivializable. In the abelian case, these constructions reduce to the classical obstruction theory for lifts of local torus actions.

math.DG↗
arXiv2025

The topology of local quaternionic toric actions

In this paper we examine the topology of manifolds equipped with a local quaternionic toric action modeled on the regular representation of the quaternionic torus $Q^n=(S^3)^n$. Building on our previous work, where the toric, differential and tetraplectic foundations were established, we show that the global topology of such manifolds is determined by the orbit space and its characteristic data. We construct Leray--Serre and Atiyah--Hirzebruch spectral sequences for the orbit projection, yielding explicit descriptions of the cohomology and $K$-theory of manifolds equipped with local quaternionic toric actions. In dimension four, we develop a quaternionic analogue of the Meyer signature formula and we briefly outline an $L$-theoretic interpretation of the resulting signature invariants. These results extend the methods of the classical (complex) toric topology to the quaternionic setting.

math.GT↗
arXiv2025

Tetraplectic structures compatible with local quaternionic toric actions

This paper introduces a quaternionic analogue of toric geometry by developing the theory of local $Q^n := Sp(1)^n$-actions on 4n-dimensional manifolds, modeled on the regular representation. We identify obstructions that measure the failure of local properties to globalize and define two invariants: a combinatorial invariant called the characteristic pair and a cohomological invariant called the Euler class, which together classify local quaternionic torus actions up to homeomorphism. We also study tetraplectic structures in quaternionic toric geometry by introducing locally generalized Lagrangian-type toric fibrations and show that such fibrations are locally modeled on $\mathbb{R}^n\times Q^n$ using a quaternionic version of the Arnold-Liouville theorem. In the last part, we show that orbit spaces of these actions acquire the structure of quaternionic integral affine manifolds with corners and Lagrangian overlaps, and we classify such spaces by establishing a quaternionic Delzant-type theorem.

math.GT↗
arXiv2024

Atiyah classes of Lie algebroid homotopy modules

For a Lie algebroid pair $A\hookrightarrow L$ we study cocycles constructed from the extension to $L$ of the higher connection forms of a representation up to homotopy $E$ of the Lie algebroid $A$. We show that there exists a cohomology class with values in the endomorphism bundle of $E$ that is independent of the extension above and vanishes whenever a homotopy $A$-compatible extension exists. Whenever the representation up to homotopy $E$ is the resolution of a Lie algebroid representation $K$ of $A$, it is shown that there exists a quasi-isomorphism sending the new Atiyah class to the classical one, associated to extensions to $L$ of the Lie algebroid representation $K$.

math.DG↗
arXiv2024

Courant-Dorfman algebras of differential operators and Dorfman connections of Courant algebroids

We construct an algebra and a complex of multidifferential operators on tensor products of a Courant algebroid E with values in the endomorphism bundle of a smooth vector bundle B, predual of E, extending the standard complex of the Courant-Dorfman algebra of E. Also, we study Dorfman connections of E on B, and show that the Cartan calculus, curvatures of induced connections and basic differential geometric identities of them make sense in this algebra.

math.DG↗
arXiv2021

Poisson structures of near-symplectic manifolds and their cohomology

We connect Poisson and near-symplectic geometry by showing that there is a singular Poisson structure on a near-symplectic 4-manifold. The Poisson structure $π$ is defined on the tubular neighbourhood of the singular locus $Z_ω$ of the 2-form $ω$, it is of maximal rank 4 and it vanishes on a degeneracy set containing $Z_ω$. We compute its smooth Poisson cohomology, which depends on the modular vector field and it is finite dimensional. We conclude with a discussion on the relation between the Poisson structure $π$ and the overtwisted contact structure associated to a near-symplectic 4-manifold.

math.SG↗
arXiv2020

Poisson Cohomology of Broken Lefschetz Fibrations

We compute the formal Poisson cohomology of a broken Lefschetz fibration by calculating it at fold and Lefschetz singularities. Near a fold singularity the computation reduces to that for a point singularity in 3 dimensions. For the Poisson cohomology around singular points we adapt techniques developed for the Sklyanin algebra. As a side result, we give compact formulas for the Poisson coboundary operator of an arbitrary Jacobian Poisson structure in 4 dimensions.

math.DG↗
arXiv2017

Atiyah classes and dg-Lie algebroids for matched pairs

For every Lie pair $(L,A)$ of algebroids we construct a dg-manifold structure on the $\mathbb{Z}$-graded manifold $\mathcal M=L[1]\oplus L/A$ such that the inclusion $ι: A[1] \to \mathcal M$ and the projection $p:\mathcal M\to L[1]$ are morphisms of dg-manifolds. The vertical tangent bundle $T^p\mathcal M$ then inherits a structure of dg-Lie algebroid over $\mathcal M$. When the Lie pair comes from a matched pair of Lie algebroids, we show that the inclusion $ι$ induces a quasi-isomorphism that sends the Atiyah class of this dg-Lie algebroid to the Atiyah class of the Lie pair. We also show how (Atiyah classes of) Lie pairs and dg-Lie algebroids give rise to (Atiyah classes of) dDG-algebras.

math.DG↗
arXiv2016

W- algebras and Duflo Isomorphism

We prove that when Kontsevich's deformation quantization is applied on weight homogeneous Poisson structures, the operators in the $\ast-$ product formula are weight homogeneous. We then consider the linear Poisson case $X=\mathfrak{g}^\ast$ for a semi simple Lie algebra $\mathfrak{g}$. As an application we provide an isomorphism between the Cattaneo-Felder-Torossian reduction algebra $H^0(\mathfrak{g},\mathfrak{m},χ)$ and the $W-$ algebra $(U(\mathfrak{g})/U(\mathfrak{g})\mathfrak{m}_χ)^\mathfrak{m}$. We also show that in the $W-$ algebra setting, $(S(\mathfrak{g})/S(\mathfrak{g})\mathfrak{m}_χ)^\mathfrak{m}$ is polynomial. Finally, we compute generators of $H^0(\mathfrak{g},\mathfrak{m},χ)$ as a deformation of $(S(\mathfrak{g})/S(\mathfrak{g})\mathfrak{m}_χ)^\mathfrak{m}$.

math.QA↗