Non-abelian Fundamental Group Research Articles

The Structure of Vector Bundles on Non-primary Hopf Manifolds

Let X be a Hopf manifold with non-Abelian fundamental group and E be a holomorphic vector bundle over X, with trivial pull-back to ℂn − {0}. The authors show that there exists a line bundle L over X such that E ⊗ L has a nowhere vanishing section. It is proved that in case dim(X) ≥ 3, π*(E) is trivial if and only if E is filtrable by vector bundles. With the structure theorem, the authors get the cohomology dimension of holomorphic bundle E over X with trivial pull-back and the vanishing of Chern class of E.

Aharonov\u2013Bohm superselection sectors

We show that the Aharonov–Bohm effect finds a natural description in the setting of QFT on curved spacetimes in terms of superselection sectors of local observables. The extension of the analysis of superselection sectors from Minkowski spacetime to an arbitrary globally hyperbolic spacetime unveils the presence of a new quantum number labelling charged superselection sectors. In the present paper, we show that this “topological” quantum number amounts to the presence of a background flat potential which rules the behaviour of charges when transported along paths as in the Aharonov–Bohm effect. To confirm these abstract results, we quantize the Dirac field in the presence of a background flat potential and show that the Aharonov–Bohm phase gives an irreducible representation of the fundamental group of the spacetime labelling the charged sectors of the Dirac field. We also show that non-Abelian generalizations of this effect are possible only on spacetimes with a non-Abelian fundamental group.

The topology of dislocations in smectic liquid crystals

The order parameter of the smectic liquid crystal phase is the same as that of a superfluid or superconductor, namely a complex scalar field. We show that the essential difference in boundary conditions between these systems leads to a markedly different topological structure of the defects. Screw and edge defects can be distinguished topologically. This implies an invariant on an edge dislocation loop so that smectic defects can be topologically linked not unlike defects in ordered systems with non-Abelian fundamental groups.

Le lemme de Schwarz et la borne supérieure du rayon d’injectivité des surfaces

We study the injectivity radius of complete Riemannian surfaces (S, g) with bounded curvature \({|K(g)|\leq 1}\). We show that if S is orientable with nonabelian fundamental group, then there is a point \({p\in S}\) with injectivity radius R\({_p(g)\geq}\) arcsinh\({(2/\sqrt{3})}\). This lower bound is sharp independently of the topology of S. This result was conjectured by Bavard who has already proved the genus zero cases (Bavard 1984). We establish a similar inequality for surfaces with boundary. The proofs rely on a version due to Yau (J Differ Geom 8:369–381, 1973) of the Schwarz lemma, and on the work of Bavard (1984). This article is the sequel of Gendulphe (2014) where we studied applications of the Schwarz lemma to hyperbolic surfaces.

Pointwise partial hyperbolicity in three-dimensional nilmanifolds

We show the existence of a family of manifolds on which all (pointwise or absolutely) partially hyperbolic systems are dynamically coherent. This family is the set of 3-manifolds with nilpotent, non-abelian fundamental group. We further classify the partially hyperbolic systems on these manifolds up to leaf conjugacy. We also classify those systems on the 3-torus that do not have an attracting or repelling periodic 2-torus. These classification results allow us to prove some dynamical consequences, including existence and uniqueness results for measures of maximal entropy and quasi-attractors.

Cubic differentials and finite volume convex projective surfaces

We prove that there exists a natural bijection between the set of finite volume oriented convex projective surfaces with non-abelian fundamental group, and the set of finite volume hyperbolic Riemann surfaces endowed with a holomorphic cubic differential with poles of order at most 2 at the cusps.

Flat pseudo-Riemannian homogeneous spaces with non-abelian holonomy group

We construct homogeneous flat pseudo-Riemannian manifolds with non-abelian fundamental group. In the compact case, all homogeneous flat pseudo-Riemannian manifolds are complete and have abelian linear holonomy group. To the contrary, we show that there do exist non-compact and non-complete examples, where the linear holonomy is non-abelian, starting in dimensions ≥ 8 \geq 8 , which is the lowest possible dimension. We also construct a complete flat pseudo-Riemannian homogeneous manifold of dimension 14 with non-abelian linear holonomy. Furthermore, we derive a criterion for the properness of the action of an affine transformation group with transitive centralizer.

On rational homology disk smoothings of valency 4 surface singularities

Thanks to the recent work of Bhupal, Stipsicz, Szabo, and the author, one has a complete list of resolution graphs of weighted homogeneous complex surface singularities admitting a rational homology disk ("QHD") smoothing, i.e., one with Milnor number 0. They fall into several classes, the most interesting of which are the three classes whose resolution dual graph has central vertex with valency 4. We give a uniform "quotient construction" of the QHD smoothings for these classes; it is an explicit Q-Gorenstein smoothing, yielding a precise description of the Milnor fibre and its non-abelian fundamental group. This had already been done for two of these classes in a previous paper; what is new here is the construction of the third class, which is far more difficult. In addition, we explain the existence of two different QHD smoothings for the first class. We also prove a general formula for the dimension of a QHD smoothing component for a rational surface singularity. A corollary is that for the valency 4 cases, such a component has dimension 1 and is smooth. Another corollary is that "most" H-shaped resolution graphs cannot be the graph of a singularity with a QHD smoothing. This result, plus recent work of Bhupal-Stipsicz, is evidence for a general Conjecture: The only complex surface singularities with a QHD smoothing are the (known) weighted homogeneous examples.

Classification of free actions on complete intersections of four quadrics

In this paper we classify all free actions of finite groups on Calabi-Yau complete intersection of 4 quadrics in $\PP^7$, up to projective equivalence. We get some examples of smooth Calabi-Yau threefolds with large nonabelian fundamental groups. We also observe the relation between some of these examples and moduli of polarized abelian surfaces.

Plane sextics with a type $\bold{E}_8$ singular point

We construct explicit geometric models for and compute the fundamen- tal groups of all plane sextics with simple singularities only and with at least one type E8 singular point. In particular, we discover four new sextics with nonabelian fundamental groups; two of them are irreducible. The groups of the two irreducible sextics found are finite. The principal tool used is the reduction to trigonal curves and Grothendieck's dessins d'enfants. 1.1. Principal results. This paper is a continuation of my paper (6), where we started the study of the equisingular deformation families and the fundamental groups of plane sextics (i.e., curves B ⊂ P 2 of degree six) with a distinguished triple singular point, using the representation of such sextics via trigonal curves in Hirzebruch surfaces and Grothendieck's dessins d'enfants. (All varieties in the paper are over C and are considered in their Hausdorff topology.) Recall that, in spite of the fact that the deformation classification of sextics can be reduced to a relatively simple, although tedious, arithmetical problem, see (2), the geometry of the pairs (P 2 , B) remains a terra incognita, as the construction relies upon the global Torelli theorem for K3-surfaces and is quite implicit. On the contrary, the approach suggested in (6), although not resulting in a defining equation for B, gives one a fairly good understanding of the topology of (P 2 , B); in particular, it is sufficient for the computation of the fundamental groupπ1(P 2 r B). A few other applications of this approach and more motivation can be found in (6); for a brief overview of the latest achievements on the subject, see C. Eyral, M. Oka (7). In the present paper, we deal with the case when the distinguished triple point in question is of type E8. (The case of a type E7 singular point was considered in (6), and the case of E6 is the subject of a forthcoming paper.) As in (6), a simple trick with the skeletons reduces most sextics B ⊂ P 2 to certain trigonal curves ¯ Bin �2 (instead of the original surface �3); this simplifies dramatically the classification of the sextics and the computation of their fundamental groups. It is still unclear if there is a simple geometric relation between B and ¯ B ' . Throughout the paper, we consider a plane sextic B ⊂ P 2 satisfying the following conditions: (∗) B has simple singularities only, and B has a distinguished singular point P of type E8.

On the connection between fundamental groups and pencils with multiple fibers

We present two results about the relationship between fundamental groups of quasiprojective manifolds and linear systems on a projectivization. We prove the existence of a plane curve with non-abelian fundamental group of the complement which does not admit a mapping onto an orbifold with non-abelian fundamental group. We also find an affine manifold whose irreducible components of its characteristic varieties do not come from the pull-back of the characteristic varieties of an orbifold.

On Calabi-Yau threefolds with large nonabelian fundamental groups

In this short note we construct Calabi-Yau threefolds with nonabelian fundamental groups of order 64 as quotients of the small resolutions of certain complete intersections of quadrics in P 7 that were first considered by M. Gross and S. Popescu.

Solving four-dimensional surgery problems using controlled theory

In this paper, the controlled surgery sequence of Ranicki, Pedersen, and Quinn is applied to the solution of surgery problems in dimension four when the fundamental group is not known to be good. Our examples concern free non-Abelian fundamental groups, surface fundamental groups, and special knot groups. Using results from our earlier paper (joint with Spaggiari), we state a general result from which our examples follow.

Applications of controlled surgery in dimension 4: Examples

The validity of Freedman's disk theorem is known to depend only on the fundamental group. It was conjectured that it fails for nonabelian free fundamental groups. If this were true then surgery theory would work in dimension four. Recently, Krushkal and Lee proved a surprising result that surgery theory works for a large special class of 4-manifolds with free nonabelian fundamental groups. The goal of this paper is to show that this also holds for other fundamental groups which are not known to be good, and that it is best understood using controlled surgery theory of Pedersen-Quinn-Ranicki. We consider some examples of 4-manifolds which have the fundamental group either of a closed aspherical surface or of a 3-dimensional knot space. A more general theorem is stated in the appendix.

More Zariski pairs and finite fundamental groups of curve complements

We give a recipe for finding new examples of plane curves with a finite non-abelian fundamental group of the complement and new examples of Zariski pairs of plane curves.

Weak approximation and non-abelian fundamental groups

We introduce a new obstruction to weak approximation which is related to étale non-abelian coverings of a proper and smooth algebraic variety X defined over a number field k. This enables us to give some counterexamples to weak approximation which are not accounted for by the Brauer–Manin obstruction, for example bielliptic surfaces.

A curve of degree five with non-abelian fundamental group

We compute the fundamental group of an irreducible rational quintic curve applying Cremona transformations and covering methods. We find a non-abelian group which is also the fundamental group of a Q -homology sphere of dimension 3.

Nonabelian vortices

A detailed study is presented of classical field configurations describing nonabelian vortices in two spatial dimensions, when a global SO(3) symmetry is spontaneously broken to a discrete group K isomorphic to the group of integers mod 4. The vortices in this model are characterized by the nonabelian fundamental group π 1( SO(3)/ K) ≽ Q 8 , which is isomorphic to the group of quaternions. We present an ansatz describing isolated vortices in this system and prove that it is stable to perturbations. Kinematic constraints are derived which imply that at a finite temperature, only two species of vortices are stable to decay, due to “dissociation”. The latter process is the nonabelian analogue of the well-known instability of charge | q| > 1 abelian vortices to dissociation into those with charge | q| = 1. The energy of configurations containing at maximum two vortex-antivortex pairs, is then computed. When the pairs are all of the same type, we find the usual Coulombic interaction energy as in the abelian case. When they are different, one finds novel interactions which are a departure from Coulomb-like behaviour. These results allow one to compute the grand canonical partition function (GCPF) for thermal pair creation of nonabelian vortices, in the approximation where the fugacities for vortices of each type are small. It is found that the vortex fugacities depend on a real continuous parameter a which characterize the degeneracy of the vacuum in the model considered. Depending on the relative sizes of these fugacities, the vortex gas will be dominated by one of either of the two types mentioned above. In these regimes, we expect the standard Kosterlitz-Thouless phase transitions to occur, as in systems of abelian vortices in two dimensions. Between these two regimes, the gas contains pairs of both types, so nonabelian effects will be important. The results presented should provide a starting point from which to investigate screening properties of the nonabelian vortex gas.

Idempotent Multiplications on Surfaces and Aspherical Spaces

Every continuous idempotent multiplication on a space induces an idempotent comultiplication on its cohomology algebra over a commutative ring and a homomorphic idempotent multiplication on each homotopy group. We classify all idempotent comultiplications on any graded anticommutative algebra A∗ over a principal ideal domain K up to degree 2 provided the degree 1 component A1 is torsion free and the degree 2 component A2 is of rank 1. All algebraic possibilities can be topologically realized. We also describe all homomorphic idempotent multiplications on arbitrary groups. This allows a complete classification up to homotopy of all idempotent multiplications on aspherical CW-complexes. For surfaces we obtain an explicit list. Notably, the Klein bottle allows infinitely many nonhomotopic idempotent multiplications, but all other surfaces with nonabelian fundamental group have only the projections as idempotent multiplications (up to homotopy). Introduction. Idempotent multiplications on sets and topological spaces have been considered by many authors, for instance as an axiomatic approach to the averaging operation (sample: [2, 3, 10]). If X denotes a connected topological space, then the existence of H-space structures places severe restrictions on the structure of X. (See, for instance, [6] or [16].) This is due to the presence of homotopy identities on both sides. If, however, one considers idempotent multiplications μ : X ×X → X, that is, multiplications which satisfy μ(x, x) = x for all x ∈ X, then no restriction follows from the presence of such multiplications, since every space X allows the two idempotent multiplications p1, p2 : X × X → X, p(x, y) = x and q(x, y) = y for all x, y ∈ X. These are the so-called trivial multiplications. On the other hand, the existence of nontrivial idempotent multiplication again forces restrictions on the space. We wish to illustrate this by discussing idempotent multiplications on suitable classes of spaces. The Received by the editors on September 1, 1988, and in revised form on October 24, 1988. Copyright c ©1991 Rocky Mountain Mathematics Consortium

Failure of averaging on multiply connected domains

We show that for every open Riemann surface X with non-abelian fundamental group there is a multiple-valued function f on X such that the fiberwise convex hull of the graph of f fails to contain the graph of a single-valued holomorphic function on X.