Deck Transformations Research Articles

Stability Conditions on Degenerated Elliptic Curves

Abstract We study stability conditions on reducible Kodaira curves obtained from degenerations of elliptic curves. We describe connected components of the spaces of stability conditions and compute the groups of deck transformations of those connected components.

$\mathbb{R}$-covered foliations and transverse pseudo-Anosov flows in atoroidal pieces

We study the transverse geometric behavior of 2-dimensional foliations in 3-manifolds. We show that an \mathbb{R} -covered, transversely orientable foliation with Gromov hyperbolic leaves in a closed 3-manifold admits a regulating, transverse pseudo-Anosov flow (in the appropriate sense) in each atoroidal piece of the manifold. The flow is a blow up of a one prong pseudo-Anosov flow. In addition we show that there is a regulating flow for the whole foliation. We also determine how deck transformations act on the universal circle of the foliation.

Seiberg–Witten Floer spectra and contact structures

In this article, the author defines an invariant of rational homology 3-spheres equipped with a contact structure as an element of a cohomotopy set of the Seiberg–Witten Floer spectrum as defined in Manolescu (Geometry Topol 7(2):889–932, 2003). Furthermore, in light of the equivalence established in Lidman and Manolescu (Astérisque 399:25, 2018) between the Borel equivariant homology of said spectrum and the Seiberg–Witten Floer homology of Kronheimer and Mrowka (Monopoles and three-manifolds, vol. 10, Cambridge University Press, Cambridge, 2007), the author shall show that this homotopy theoretic invariant recovers the already well known contact element in the Seiberg–Witten Floer cohomology (vid. e.g. Kronheimer et al. in Ann Math 20:457–546, 2007) in a natural fashion. Next, the behaviour of the cohomotopy invariant is considered in the presence of a finite covering. This setting naturally asks for the use of Borel cohomology equivariant with respect to the group of deck transformations. Hence, a new equivariant contact invariant is defined and its properties studied. The invariant is then computed in one concrete example, wherein the author demonstrates that it opens the possibility of considering scenarios hitherto inaccessible.

Bicritical Rational Maps With a Common Iterate

Abstract Let $f$ be a degree $d$ bicritical rational map with critical point set $\mathcal{C}_f$ and critical value set $\mathcal{V}_f$. Using the group $\textrm{Deck}(f^k)$ of deck transformations of $f^k$, we show that if $g$ is a bicritical rational map that shares an iterate with $f$, then $\mathcal{C}_f = \mathcal{C}_g$ and $\mathcal{V}_f = \mathcal{V}_g$. Using this, we show that if two bicritical rational maps of even degree $d$ share an iterate, then they share a second iterate, and both maps belong to the symmetry locus of degree $d$ bicritical rational maps.

Quadrilateral mesh generation III : Optimizing singularity configuration based on Abel–Jacobi theory

This work proposes a rigorous and practical algorithm for quad-mesh generation based the Abel–Jacobi theory of algebraic curves. We prove sufficient and necessary conditions for a flat metric with cone singularities to be compatible with a quad-mesh, in terms of the deck-transformation, then develop an algorithm based on the theorem. The algorithm has two stages: first, a meromorphic quartic differential is generated to induce a T-mesh; second, the edge lengths of the T-mesh are adjusted by solving a linear system to satisfy the deck transformation condition, which produces a quad-mesh.In the first stage, the algorithm pipeline can be summarized as follows: calculate the homology group; compute the holomorphic differential group; construct the period matrix of the surface and Jacobi variety; calculate the Abel–Jacobi map for a given divisor; optimize the divisor to satisfy the Abel–Jacobi condition by integer programming; compute a flat Riemannian metric with cone singularities at the divisor by Ricci flow; isometrically immerse the surface punctured at the divisor onto the complex plane and pull back the canonical holomorphic differential to the surface to obtain the meromorphic quartic differential; construct a motorcycle graph to generate a T-Mesh.In the second stage, the deck transformation constraints are formulated as a linear equation system of the edge lengths of the T-mesh. The solution provides a flat metric with integral deck transformations, which leads to the final quad-mesh.The proposed method is rigorous and practical. The T-mesh and quad-mesh results can be applied for constructing Splines directly. The efficiency and efficacy of the proposed algorithm are demonstrated by experimental results on surfaces with complicated topologies and geometries.

Configuration-mapping spaces and homology stability

For a given bundle \(\xi :E \rightarrow M\) over a manifold, configuration-section spaces parametrise finite subsets \(z \subseteq M\) equipped with a section of \(\xi \) defined on \(M \smallsetminus z\), with prescribed “charge” in a neighbourhood of the points z. These spaces may be interpreted physically as spaces of fields that are permitted to be singular at finitely many points, with constrained behaviour near the singularities. As a special case, they include the Hurwitz spaces, which parametrise branched covering spaces of the 2-disc with specified deck transformation group. We prove that configuration-section spaces are homologically stable (with \(\mathbb {Z}\) coefficients) whenever the underlying manifold M is connected and has non-empty boundary and the charge is “small” in a certain sense. This has a partial intersection with the work on Hurwitz spaces of Ellenberg, Venkatesh and Westerland.

Monodromy of rational curves on toric surfaces

For an ample line bundle $\mathcal{L}$ on a complete toric surface $X$, we consider the subset $V_{\mathcal{L}} \subset \vert \mathcal{L} \vert$ of irreducible, nodal, rational curves contained in the smooth locus of $X$. We study the monodromy map from the fundamental group of $V_{\mathcal{L}}$ to the permutation group on the set of nodes of a reference curve $C \in V_{\mathcal{L}}$. We identify a certain obstruction map $\varPsi_{X}$ defined on the set of nodes of $C$ and show that the image of the monodromy is exactly the group of deck transformations of $\varPsi_{X}$, provided that $\mathcal{L}$ is sufficiently big (in a sense we precise below). Along the way, we provide a handy tool to compute the image of the monodromy for any pair $(X, \mathcal{L})$. Eventually, we present a family of pairs $(X, \mathcal{L})$ with small $\mathcal{L}$ and for which the image of the monodromy is strictly smaller than expected.

Locally connected Smale spaces, pinched spectrum, and infra-nilmanifolds

We show that if (X,f) is a locally connected Smale space (e.g., a basic set of an Axiom-A diffeomorphism) such that the local product structure on X can be lifted by a covering with virtually nilpotent group of deck transformations to a global direct product, then (X,f) is topologically conjugate to a hyperbolic infra-nilmanifold automorphism. We use this result to give a generalization to Smale spaces of a theorem of M. Brin and A. Manning on Anosov diffeomorphisms with pinched spectrum, and to show that every locally connected codimension one Smale space is topologically conjugate to a hyperbolic automorphism of a torus.

GRAPH IMMERSIONS, INVERSE MONOIDS AND DECK TRANSFORMATIONS

If $f:\tilde{\unicode[STIX]{x1D6E4}}\rightarrow \unicode[STIX]{x1D6E4}$ is a covering map between connected graphs, and $H$ is the subgroup of $\unicode[STIX]{x1D70B}_{1}(\unicode[STIX]{x1D6E4},v)$ used to construct the cover, then it is well known that the group of deck transformations of the cover is isomorphic to $N(H)/H$ , where $N(H)$ is the normalizer of $H$ in $\unicode[STIX]{x1D70B}_{1}(\unicode[STIX]{x1D6E4},v)$ . We show that an entirely analogous result holds for immersions between connected graphs, where the subgroup $H$ is replaced by the closed inverse submonoid of the inverse monoid $L(\unicode[STIX]{x1D6E4},v)$ used to construct the immersion. We observe a relationship between group actions on graphs and deck transformations of graph immersions. We also show that a graph immersion $f:\tilde{\unicode[STIX]{x1D6E4}}\rightarrow \unicode[STIX]{x1D6E4}$ may be extended to a cover $g:\tilde{\unicode[STIX]{x1D6E5}}\rightarrow \unicode[STIX]{x1D6E4}$ in such a way that all deck transformations of $f$ are restrictions of deck transformations of $g$ .

Real submanifolds of maximum complex tangent space at a CR singular point, II

We study a germ of real analytic n-dimensional submanifold of $C^n$ that has a complex tangent space of maximal dimension at a CR singularity. Under the condition that its complexification admits the maximum number of deck transformations, we first classify holomorphically its quadratic CR singularity. We then study its transformation to a normal form under the action of local (possibly formal) biholomorphisms at the singularity. We first conjugate formally its associated reversible map $\sigma$ to suitable normal forms and show that all these normal forms can be divergent. We then construct a unique formal normal form under a non degeneracy condition.

On the Spectrum of Differential Operators Under Riemannian Coverings

For a Riemannian covering p :M_{2} rightarrow M_{1}, we compare the spectrum of an essentially self-adjoint differential operator D_{1} on a bundle E_{1} rightarrow M_{1} with the spectrum of its lift D_{2} on p^{*}E_{1} rightarrow M_{2}. We prove that if the covering is infinite sheeted and amenable, then the spectrum of D_{1} is contained in the essential spectrum of any self-adjoint extension of D_{2}. We show that if the deck transformations group of the covering is infinite and D_{2} is essentially self-adjoint (or symmetric and bounded from below), then D_{2} (or the Friedrichs extension of D_{2}) does not have eigenvalues of finite multiplicity and in particular, its spectrum is essential. Moreover, we prove that if M_{1} is closed, then p is amenable if and only if it preserves the bottom of the spectrum of some/any Schrödinger operator, extending a result due to Brooks.

Von Neumann dimension, Hodge index theorem and geometric applications

A reformulation of the Hodge index theorem within the framework of Atiyah’s $$L^2$$ -index theory is provided. More precisely, given a compact Kahler manifold (M, h) of even complex dimension 2m, we prove that $$\begin{aligned} \sigma (M)=\!\!\sum _{p,q=0}^{2m}\!(-1)^ph_{(2),\Gamma }^{p,q}(M) \end{aligned}$$ where $$\sigma (M)$$ is the signature of M and $$h_{(2),\Gamma }^{p,q}(M)$$ are the $$L^2$$ -Hodge numbers of M with respect to a Galois covering having $$\Gamma $$ as group of deck transformations. Likewise we also prove an $$L^2$$ -version of the Frolicher index theorem, see (3). Afterwards we give some applications of these two theorems and finally we conclude this paper by collecting other properties of the $$L^2$$ -Hodge numbers.

Computing stable epimorphisms onto finite groups

Let U be a finitely presented group with a normal subgroup F, and let H be a finite group. We present an effective method for constructing all U-stable epimorphisms from F onto H; that is, epimorphisms whose kernels are normal in U. Our strategy is an adaptation of the algorithm for determining all epimorphisms from a finitely presented group onto a finite group. As an application, we construct all admissible regular graph covering projections with a given deck transformation group.

On G-rigid surfaces

Rigid algebraic varieties form an important class of complex varieties that exhibit interesting geometric phenomena. In this paper we propose a natural extension of rigidity to complex projective varieties with a finite group action (G-varieties) and focus on the first nontrivial case, namely, on G-rigid surfaces that can be represented as desingularizations of Galois coverings of the projective plane with Galois group G. We obtain local and global G-rigidity criteria for these G-surfaces and present several series of such surfaces that are rigid with respect to the action of the deck transformation group.

The monodromy theorem for compact Kähler manifolds and smooth quasi-projective varieties

Given any connected topological space X, assume that there exists an epimorphism \(\phi {:}\; \pi _1(X) \rightarrow {\mathbb {Z}}\). The deck transformation group \({\mathbb {Z}}\) acts on the associated infinite cyclic cover \(X^\phi \) of X, hence on the homology group \(H_i(X^\phi , {\mathbb {C}})\). This action induces a linear automorphism on the torsion part of the homology group as a module over the Laurent ring \({\mathbb {C}}[t,t^{-1}]\), which is a finite dimensional \({\mathbb {C}}\)-vector space. We study the sizes of the Jordan blocks of this linear automorphism. When X is a compact Kahler manifold, we show that all the Jordan blocks are of size one. When X is a smooth complex quasi-projective variety, we give an upper bound on the sizes of the Jordan blocks, which is an analogue of the Monodromy Theorem for the local Milnor fibration.

WHAT IS… A Laminated Deck Transformation?

WHAT IS… A Laminated Deck Transformation?

Homological finiteness of abelian covers

We present a method for deciding when a regular abelian cover of a finite CW-complex has finite Betti numbers. To start with, we describe a natural parameter space for all regular covers of a finite CW-complex X, with group of deck transformations a fixed abelian group A, which in the case of free abelian covers of rank r coincides with the Grassmanian of r-planes in H^1(X,\Q). Inside this parameter space, there is a subset \Omega_A^i(X) consisting of all the covers with finite Betti numbers up to degree i. Building on work of Dwyer and Fried, we show how to compute these sets in terms of the jump loci for homology with coefficients in rank 1 local systems on X. For certain spaces, such as smooth, quasi-projective varieties, the generalized Dwyer-Fried invariants that we introduce here can be computed in terms of intersections of algebraic subtori in the character group. For many spaces of interest, the homological finiteness of abelian covers can be tested through the corresponding free abelian covers. Yet in general, abelian covers exhibit different homological finiteness properties than their free abelian counterparts.

Stability Conditions for Slodowy Slices and Real Variations of Stability

We provide examples of an explicit submanifold in Bridgeland stabilities space of a local Calabi-Yau, and propose a new variant of definition of stabilities on a triangulated category, which we call a "real variation of stability conditions". We discuss its relation to Bridgeland's definition; the main theorem provides an illustration of such a relation. We also state a conjecture by the second author and Okounkov relating this structure to quantum cohomology of symplectic resolutions and establish its validity in some special cases. More precisely, let X be the standard resolution of a transversal slice to an adjoint nilpotent orbit of a simple Lie algebra over C. An action of the affine braid group on the derived category of coherent sheaves on X and a collection of t-structures on this category permuted by the action have been constructed in arXiv:1101.3702 and arXiv:1001.2562 respectively. In this note we show that the t-structures come from points in a certain connected submanifold in the space of Bridgeland stability conditions. The submanifold is a covering of a submanifold in the dual space to the Grothendieck group, and the affine braid group acts by deck transformations. In the special case when dim (X)=2 a similar (in fact, stronger) result was obtained in arXiv:math/0508257.

REMARKS ON SIMPLY k-CONNECTIVITY AND k-DEFORMATION RETRACT IN DIGITAL TOPOLOGY

Abstract. To study a deformation of a digital space from the view-point of digital homotopy theory, we have often used the notions ofa weak k-deformation retract [20] and a strong k-deformation re-tract [10, 12, 13]. Thus the papers [10, 12, 13, 16] firstly developedthe notion of a strong k-deformation retract which can play an im-portant role in studying a homotopic thinning of a digital space.Besides, the paper [3] deals with a k-deformation retract and itshomotopic property related to a digital fundamental group. Thus,as a survey article, comparing among a k-deformation retract in [3],a strong k-deformation retract in [10, 12, 13], a weak deformationk-retract in [20] and a digital k-homotopy equivalence [5, 24], we ob-serve some relationships among them from the viewpoint of digitalhomotopy theory. Furthermore, the present paper deals with someparts of the preprint [10] which were not published in a journal (seeProposition 3.1). Finally, the present paper corrects Boxer’s paper[3] as follows: even though the paper [3] referred to the notion of adigital homotopy equivalence (or a same k-homotopy type) whichis a special kind of a k-deformation retract, we need to point outthat the notion was already developed in [5] instead of [3] and fur-ther corrects the proof of Theorem 4.5 of Boxer’s paper [3] (see theproof of Theorem 4.1 in the present paper). While the paper [4]refers some properties of a deck transformation group (or an auto-morphism group) of digital covering space without any citation, thestudy was early done by Han in his paper (see the paper [14]).Received May 15, 2014. Accepted June 10, 2014.2010 Mathematics Subject Classification. 55Q70, 52Cxx, 55P15, 68R10, 68U05.Key words and phrases. simply k-connected, digital isomorphism, strong k-deformation, weak k-deformation retract, k-homotopy equivalence (same k-homotopytype), digital topology.

Symmetry, isotopy, and irregular covers

We say that a covering space of surfaces \(S\rightarrow X\) has the Birman–Hilden property if the subgroup of the mapping class group of \(X\) consisting of mapping classes that have representatives that lift to \(S\) embeds in the mapping class group of \(S\) modulo the group of deck transformations. We identify one necessary condition and one sufficient condition for when a covering space has this property. We give new explicit examples of irregular branched covering spaces that do not satisfy the necessary condition as well as explicit covers that satisfy the sufficient condition. Our criteria are conditions on simple closed curves, and our proofs use the combinatorial topology of curves on surfaces.