arXiv subjects

Ioannis Gkeneralis

Publications and source records attributed to Ioannis Gkeneralis.

5 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 $\mathcal G(P)\congΓ(\operatorname{Ad}(P))$. 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↗

arXiv2026

Equivariant rigidity of complex and quaternionic moment--angle manifolds

We investigate equivariant rigidity properties of complex and quaternionic moment--angle manifolds. By reducing the classification problem to the equivariant rigidity of their quasitoric or quoric quotients and by using the associated principal bundle structures, we establish rigidity results within the category of locally linear actions. We prove that complex moment--angle manifolds are equivariantly rigid up to homeomorphism: any closed locally linear manifold equivariantly homotopy equivalent to a complex moment--angle manifold is equivariantly homeomorphic to it. In the quaternionic setting, under Hopkinson's globality condition on the characteristic data, we obtain the analogous equivariant homeomorphism rigidity for quaternionic moment--angle manifolds. These results show that, in the equivariant category considered here, the equivariant homotopy type of a moment--angle manifold determines its equivariant homeomorphism type.

math.AT↗

arXiv2026

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↗

arXiv2026

Remarks on Topological Rigidity of Real Moment-Angle Manifolds

We study topological rigidity of real moment-angle manifolds associated to flag simplicial complexes. Using the cubical geometry arising from the Davis construction, we identify the universal cover with the Davis complex and deduce that it admits a CAT(0) metric. As a consequence, its fundamental group satisfies the Farrell--Jones conjecture. Applying surgery theory, we deduce that real moment-angle manifolds of dimension at least five associated to flag complexes satisfy the Borel Conjecture. We also explain why this rigidity phenomenon is specific to the real case and fails for complex and quaternionic moment-angle complexes.

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↗