Finite Fundamental Group Research Articles

Metrics of positive Ricci curvature on quotient spaces

One of the classical problems in differential geometry is the investigation of closed manifolds which admit Riemannian metrics with given lower bounds for the sectional or the Ricci curvature and the study of relations between the existence of such metrics and the topology and geometry of the underlying manifold. Despite many efforts during the past decades, this problem is still far from being understood. For example, so far the only obstructions to the existence of a metric with positive Ricci curvature come from the obstructions to the existence of metrics with positive scalar curvature ([Li], [Hi], [Ro], [SchY], [Ta]) and the Bonnet-Myers theorem which implies that the fundamental group of a closed manifold with positive Ricci curvature must be finite. Fruitful constructions of metrics with positive Ricci curvature on closed manifolds have so far been established by techniques that include deformation of metrics ([Au], [Eh], [We]), Kahler geometry ([Yau1], [Yau2]), bundles and warping ([Po], [Na], [BB], [GPT]), special kinds of surgery ([SY], [Wr]), metrical glueing ([GZ2]), and Sasakian geometry ([BGN]). Particularly large classes of examples of manifolds with positive Ricci curvature are given by all compact homogeneous spaces with finite fundamental group ([Na]) and all closed cohomogeneity one manifolds with finite fundamental group ([GZ2]). In this article we present several new classes of manifolds which admit metrics of positive or nonnegative Ricci curvature. The idea is to consider quotients of manifolds (M,g) of positive or nonnegative Ricci curvature by a free isometric action. While taking such a quotient non-decreases the sectional curvature, it may well happen in general that the Ricci curvature of the quotient is inferior to the one of M . Our first results, however, state that the quotient of M does admit metrics of positive Ricci curvature if M belongs to one of the aforementioned classes of homogeneous spaces or spaces with a cohomogeneity one action. More precisely, we prove:

Complex multiplication symmetry of black hole attractors

We show how Moore's observation, in the context of toroidal compactifications in type IIB string theory, concerning the complex multiplication structure of black hole attractor varieties, can be generalized to Calabi–Yau compactifications with finite fundamental groups. This generalization leads to an alternative general framework in terms of motives associated to a Calabi–Yau variety in which it is possible to address the arithmetic nature of the attractor varieties in a universal way via Deligne's period conjecture.

Hyperbolic manifolds are geodesically rigid

We show that if all geodesics of two non-proportional metrics on a closed manifold coincide (as unparameterized curves), then the manifold has a finite fundamental group or admits a local-product structure. This implies that, if the manifold admits a metric of negative sectional curvature, then two metrics on the manifold have the same geodesics if and only if they are proportional.

Polyhedra with finite fundamental group dominate finitely many different homotopy types

In 1968 K. Borsuk asked: Does every polyhedron dominate only finitely many different shapes? In this question the notion of shape can be replaced by the notion of homotopy type. We showed earlier that the answer to the Borsuk question is no. However, in a

Foliations by minimal surfaces and contact structures on certain closed 3‐manifolds

Let (M, g) be a closed, connected, oriented C∞ Riemannian 3‐manifold with tangentially oriented flow F. Suppose that F admits a basic transverse volume form μ and mean curvature one‐form κ which is horizontally closed. Let {X, Y} be any pair of basic vector fields, so μ(X, Y) = 1. Suppose further that the globally defined vector 𝒱[X, Y] tangent to the flow satisfies [Z.𝒱[X, Y]] = fZ𝒱[X, Y] for any basic vector field Z and for some function fZ depending on Z. Then, 𝒱[X, Y] is either always zero and H, the distribution orthogonal to the flow in T(M), is integrable with minimal leaves, or 𝒱[X, Y] never vanishes and H is a contact structure. If additionally, M has a finite‐fundamental group, then 𝒱[X, Y] never vanishes on M, by the above together with a theorem of Sullivan (1979). In this case H is always a contact structure. We conclude with some simple examples.

THE CULLER-SHALEN SEMINORMS OF THE (-2, 3, n) PRETZEL KNOT

We show that the [Formula: see text]-character variety of the (-2, 3, n) pretzel knot consists of two (respectively three) algebraic curves when 3 ∤ n (respectively 3 | n) and given an explicit calculation of the Culler-Shalem seminorms of these curves. Using this calculation, we describe the fundamental polygon and Newton polygon for these knots and give a list of Dehn surgerise yielding a manifold with finite or cyclic fundamental group. This constitutes a new proof of property P for these knots.

The Yamabe Invariant for Non-Simply Connected Manifolds

The Yamabe invariant is an invariant of a closed smooth manifold defined using conformal geometry and the scalar curvature. Recently, Petean showed that the Yamabe invariant is nonnegative for all closed simply connected manifolds of dimension ≥ 5. We extend this to show that Yamabe invariant is nonnegative for all closed manifolds of dimension ≥ 5 with fundamental group of odd order having all Sylow subgroups abelian. The main new geometric input is a way of studying the Yamabe invariant on Toda brackets. A similar method of proof shows that all closed oriented manifolds of dimension ≥ 5 with non-spin universal cover, with finite fundamental group having all Sylow subgroups elementary abelian, admit metrics of positive scalar curvature, once one restricts to the complement of manifolds whose homology classes are toral. The exceptional toral homology classes only exist in dimensions not exceeding the rank of the fundamental group, so this proves important cases of the Gromov-Lawson-Rosenberg Conjecture once the dimension is sufficiently large.

Cohomogeneity one manifolds with positive Ricci curvature

One of the central problems in Riemannian geometry is to determine how large the classes of manifolds with positive/nonnegative sectional -, Ricci or scalar curvature are (see [Gr]). For scalar curvature the situation is fairly well understood by comparison. Special surgery constructions as in [SY, Wr] and bundle constructions as in [Na] have resulted in a large number of interesting manifolds with positive Ricci curvature. So far the only known obstructions to have positive Ricci curvature come from obstructions to have positive scalar curvature, (see [Li] and [RS]), and from the classical Bonnet-Myers Theorem, which implies that a closed manifold with positive Ricci curvature must have finite fundamental group. It is well known that among homogeneous manifolds G/H this is also a sufficient condition (see e.g. the proof of Corollary 3.5 or [Br]). In this paper we prove that this is true as well when the manifold admits an action by a compact Lie group G with orbits of codimension one.

CYCLIC AND FINITE SURGERIES ON PRETZEL KNOTS

We classify Dehn surgeries on (p,q,r) pretzel knots resulting in a manifold M(α) having cyclic fundamental group and analyze those leading to a finite fundamental group. The proof uses the theory of cyclic and finite surgeries developed by Culler, Shalen, Boyer, and Zhang. In particular, Culler-Shalen seminorms play a central role.

Fundamental groups of open K3 surfaces, Enriques surfaces and Fano 3-folds

We investigate when the fundamental group of the smooth part of a K3 surface or Enriques surface with Du Val singularities, is finite. As a corollary we give an effective upper bound for the order of the fundamental group of the smooth part of a certain Fano 3-fold. This result supports Conjecture A below, while Conjecture A (or alternatively the rational-connectedness conjecture in Kollar et al. (J. Algebra Geom. 1 (1992) 429) which is still open when the dimension is at least 4) would imply that every log terminal Fano variety has a finite fundamental group.

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.

Contact structures, equivariant spin bordism, and periodic fundamental groups

Using contact surgery and equivariant bordism theory, we prove the existence of contact structures on all 5-dimensional spin manifolds with certain finite fundamental groups.

Homogeneous symplectic manifolds with Ricci-type curvature

We consider invariant symplectic connections ∇ on homogeneous symplectic manifolds ( M, ω) with curvature of Ricci type. Such connections are solutions of a variational problem studied by Bourgeois and Cahen, and provide an integrable almost complex structure on the bundle of almost complex structures compatible with the symplectic structure. If M is compact with finite fundamental group then ( M, ω) is symplectomorphic to P n( C) with a multiple of its Kähler form and ∇ is affinely equivalent to the Levi-Civita connection.

The pq-condition for $3$-manifold groups

We give an elementary, topological proof of the fact that any sugroup of order pq of a finite 3-manifold group is cyclic if p and q are dis- tinct odd primes. This condition, together with related results of Milnor and Reidemeister, implies that such a group acts orthogonally and without fixed points on some sphere. Our aim here is to understand the restrictions on finite groups that are fundamen- tal groups of 3-manifolds. Two necessary conditions are well known; namely that if G is a such a group, then its subgroups of order 2p, for p a prime, are cyclic(2)(the 2p-condition) and so are its subgroups of order p 2 (3)(4) (the p 2 -condition). The latter is equivalent to the statement that all abelian subgroups of G are cyclic. The p 2 -condition for 3-manifold groups is an immediate consequence of the fact that (Z/pZ) 2 does not have a balanced presentation. The above conditions together with the pq-condition characterise when solvable finite groups act orthogonally on some sphere (5) with fixed points. The pq-condition is the condition that every subgroup of order pq of G, where p and q are distinct, odd primes, is abelian (hence cyclic). Our main result is that finite fundamental groups of 3-manifolds satisfy this condition. This was previously known, but proofs in the literature (for example in (2)) involve using the fact that G has cohomology with periodicity 4 together with some deep group theory to restrict the class of finite fundamental groups, and then observing that this condition is satisfied. It seems desirable to deduce this result using a direct topological argument, especially given the fundamental role of this condition in characterising orthogonal free actions. The proof given here is elementary and topological. Theorem 0.1. Suppose the finite group G is the fundamental group of a 3-manifold M, and p and q are distinct, odd primes. Then every subgroup of G of order pq is cyclic.

Splitting homomorphisms and the Geometrization Conjecture

This paper gives an algebraic conjecture which is shown to be equivalent to Thurston's Geometrization Conjecture for closed, orientable 3-manifolds. It generalizes the Stallings-Jaco theorem which established a similar result for the Poincare Conjecture. The paper also gives two other algebraic conjectures; one is equivalent to the finite fundamental group case of the Geometrization Conjecture, and the other is equivalent to the union of the Geometrization Conjecture and Thurston's Virtual Bundle Conjecture.

Manifolds with finite cyclic fundamental groups and codimension 2 fibrators

A closed connected n-manifold N is called a codimension 2 fibrator (codimension 2 orientable fibrator, respectively) if each proper map p:M→B on an (orientable, respectively) (n+2)-manifold M each fiber of which is shape equivalent to N is an approximate fibration. Let r be a nonnegative integer and let N be a closed n-manifold whose fundamental group is isomorphic to H 1×H 2 , where H 1 is a group whose order is odd and H 2 is a finite direct product of cyclic groups of order 2 r . Let q 1:N 1→N be the covering associated with H 1 . The main purpose of this paper shows that if N 1 is a codimension 2 orientable fibrator, then N is a codimension 2 fibrator.

FIXED POINT FREE CIRCLE ACTIONS AND FINITENESS THEOREMS

We prove a vanishing theorem of certain cohomology classes for an 2n-manifold of finite fundamental group which admits a fixed point free circle action. In particular, it implies that any Tk-action on a compact symplectic manifold of finite fundamental group has a non-empty fixed point set. The vanishing theorem is used to prove two finiteness results in which no lower bound on volume is assumed. (i) The set of symplectic n-manifolds of finite fundamental groups with curvature, λ ≤ sec ≤ Λ, and diameter, diam ; ≤ d, contains only finitely many diffeomorphism types depending only on n, λ, Λ and d. (ii) The set of simply connected n-manifolds (n ≤ 6) with λ ≤ sec ≤ Λ and diam ≤ d contains only finitely many diffeomorphism types depending only on n, λ, Λ and d.

Finite group actions on 4-manifolds

AbstractLet X be a closed, oriented, smooth 4-manifold with a finite fundamental group and with a non-vanishing Seiberg-Witten invariant. Let G be a finite group. If G acts smoothly and freely on X, then the quotient X/G cannot be decomposed as X1#X2 with (Xi) > 0, i = 1, 2. In addition let X be symplectic and c1(X)2 > 0 and b+2(X) > 3. If σ is a free anti-symplectic involution on X then the Seiberg-Witten invariants on X/σ vanish for all spinc structures on X/σ, and if η is a free symplectic involution on X then the quotients X/σ and X/η are not diffeomorphic to each other.

A Note on Finite Dehn Fillings

AbstractLet M be a compact, connected, orientable 3-manifold whose boundary is a torus and whose interior admits a complete hyperbolic metric of finite volume. In this paper we show that if theminimal Culler-Shalen norm of a non-zero class in H1(∂M) is larger than 8, then the finite surgery conjecture holds for M. This means that there are at most 5 Dehn fillings of M which can yieldmanifolds having cyclic or finite fundamental groups and the distance between any slopes yielding such manifolds is at most 3.

On Culler-Shalen Seminorms and Dehn Filling

Consider the exterior M of a hyperbolic knot lying in a closed, connected, orientable 3-manifold. Culler and Shalen defined norm on H_1(dM;R) using the SL(2,C) character variety of pi_1(M). The Culler-Shalen norm encodes many topological properties of M; in particular it provides information about Dehn fillings of M. Their construction may be applied to arbitrary curves in the SL_2(C)-character variety of a connected, compact, orientable, irreducible 3-manifold whose boundary is a torus, though in this generality one can only guarantee that it will define a seminorm. The first half of this paper is devoted to the development of the general theory of Culler-Shalen seminorms defined for curves of PSL_2(C)-characters. By working over PSL_2(C) we obtain a theory that is more generally applicable than its SL_2(C) counterpart, while being only mildly more difficult to set up. In the second half of this paper we apply the theory of Culler-Shalen seminorms to study the Dehn filling operation. In particular we examine the relationship between fillings which yield manifolds having a positive dimensional PSL_2(C)-character variety with those that yield manifolds having a finite or cyclic fundamental group. In one interesting application of this work we show that manifolds resulting from a nonintegral surgery on a knot in the 3-sphere tend to have a zero-dimensional PSL_2(C)-character variety. As a consequence we obtain an infinite family of closed, orientable, hyperbolic Haken manifolds which have zero-dimensional PSL_2(C)-character varieties.