An analogue of the Novikov Conjecture in complex algebraic geometry

We introduce an analogue of the Novikov Conjecture on higher signatures in the context of the algebraic geometry of (nonsingular) complex projective varieties. This conjecture asserts that certain ``higher Todd genera' are birational invariants. This implies birational invariance of certain extra combinations of Chern classes (beyond just the classical Todd genus) in the case of varieties with large fundamental group (in the topological sense). We prove the conjecture under the assumption of the ``strong Novikov Conjecture' for the fundamental group, which is known to be correct for many groups of geometric interest. We also show that, in a certain sense, our conjecture is best possible.

     

The Fibered Isomorphism Conjecture for Complex Manifolds

In this paper we show that the Fibered Isomorphism Conjecture of Farrell and Jones, corresponding to the stable topological pseudoisotopy functor, is true for the fundamental groups of a class of complex manifolds. A consequence of this result is that the Whitehead group, reduced projective class groups and the negative K-groups of the fundamental groups of these manifolds vanish whenever the fundamental group is torsion free. We also prove the same results for a class of real manifolds including a large class of 3-manifolds which has a finite sheeted cover fibering over the circle.     

Vanishing conditions for the simplicial volume of compact complex varieties

Gromov has defined a notion of simplicial volume: it is a topological invariant for compact manifolds which is closely related to the fundamental group. We investigate here the relevance of this notion in the realm of complex varieties.

     

On the fundamental group of Riemannian manifolds with nonnegative Ricci curvature

In 1968 Milnor conjectured that the fundamental group of any complete Riemannian manifold with nonnegative Ricci curvature is finitely generated. In this paper we obtain two results concerning Milnor’s conjecture. We first prove that the generators of fundamental group can be chosen so that it has at most logarithmic growth. Secondly we prove that the conjecture is true if additional the volume growth satisfies certain condition.     

Aspherical manifolds with relatively hyperbolic fundamental groups

We show that the aspherical manifolds produced via the relative strict hyperboli- zation of polyhedra enjoy many group-theoretic and topological properties of open finite volume negatively pinched manifolds, including relative hyperbolicity, nonvanishing of simplicial volume, co-Hopf property, finiteness of outer automorphism group, absence of splitting over elementary subgroups, and acylindricity. In fact, some of these properties hold for any compact aspherical manifold with incompressible aspherical boundary components, provided the fundamental group is hyperbolic relative to fundamental groups of boundary components. We also show that no manifold obtained via the relative strict hyperbolization can be embedded into a compact Kähler manifold of the same dimension, except when the dimension is two.     

Subgroup Separability and Linearity of Certain Graphs of Groups

We show that if the fundamental group of a finite graph of groups with finitely generated Abelian vertex groups is subgroup separable, then it is linear.     

Solvable Fundamental Groups of Algebraic Varieties and Käler Manifolds

It is shown that if the fundamental group of a normal algebraic variety, respectively Zariski open subset of a compact Kähler manifold, is solvable with a faithful linear representation over Q, respectively polycyclic, then it is virtually nilpotent.     

Almost all minimal idempotent varieties are congruence modular

We show that, up to term equivalence, the only minimal idempotent varieties that are not congruence modular are the variety of sets and the variety of semilattices. From this it follows that a minimal idempotent variety that is not congruence distributive is term equivalent to the variety of sets, the variety of semilattices, or a variety of affine modules over a simple ring. Received March 29, 1999; accepted in final form February 8, 2000.     

On L-spaces and left-orderable fundamental groups

Examples suggest that there is a correspondence between L-spaces and three-manifolds whose fundamental groups cannot be left-ordered. In this paper we establish the equivalence of these conditions for several large classes of manifolds. In particular, we prove that they are equivalent for any closed, connected, orientable, geometric three-manifold that is non-hyperbolic, a family which includes all closed, connected, orientable Seifert fibred spaces. We also show that they are equivalent for the twofold branched covers of non-split alternating links. To do this we prove that the fundamental group of the twofold branched cover of an alternating link is left-orderable if and only if it is a trivial link with two or more components. We also show that this places strong restrictions on the representations of the fundamental group of an alternating knot complement with values in $\text{ Homeo}_+(S^1)$ .     

ON CLASS PAIRS AND RADICALS

ABSTRACT

We show that, for a class of schurian algebras, which we call schurian almost triangular, the fundamental group of the algebra is isomorphic to the fundamental group of an associated simplicial complex. Moreover, we obtain a simple presentation of this group in terms of generators and relations. Finally we use it to obtain easy short proofs of some known facts on the simple connectedness of incidence algebras.     

Homology equivalences inducing an epimorphism on the fundamental group and Quillen's plus construction

Quillen's plus construction is a topological construction that kills the maximal perfect subgroup of the fundamental group of a space without changing the integral homology of the space. In this paper we show that there is a topological construction that, while leaving the integral homology of a space unaltered, kills even the intersection of the transfinite lower central series of its fundamental group. Moreover, we show that this is the maximal subgroup that can be factored out of the fundamental group without changing the integral homology of a space.

     

Difference Forms

Currently, there is much interest in the development of geometric integrators, which retain analogues of geometric properties of an approximated system. This paper provides a means of ensuring that finite difference schemes accurately mirror global properties of approximated systems. To this end, we introduce a cohomology theory for lattice varieties, on which finite difference schemes and other difference equations are defined. We do not assume that there is any continuous space, or that a scheme or difference equation has a continuum limit. This distinguishes our approach from theories of “discrete differential forms” built on simplicial approximations and Whitney forms, and from cohomology theories built on cubical complexes. Indeed, whereas cochains on cubical complexes can be mapped injectively to our difference forms, a bijection may not exist. Thus our approach generalizes what can be achieved with cubical cohomology. The fundamental property that we use to prove our results is the natural ordering on the integers. We show that our cohomology can be calculated from a good cover, just as de Rham cohomology can. We postulate that the dimension of solution space of a globally defined linear recurrence relation equals the analogue of the Euler characteristic for the lattice variety. Most of our exposition deals with forward differences, but little modification is needed to treat other finite difference schemes, including Gauss-Legendre and Preissmann schemes. Dedicated to Professor Arieh Iserles on the Occasion of his Sixtieth Birthday.     

Radicals of 0-regular algebras

Summary We consider a generalisation of the Kurosh--Amitsur radical theory for rings (and more generally multi-operator groups) which applies to 0-regular varieties in which all operations preserve 0. We obtain results for subvarieties, quasivarieties and element-wise equationally defined classes. A number of examples of radical and semisimple classes in particular varieties are given, including hoops, loops and similar structures. In the first section, we introduce 0-normal varieties (0-regular varieties in which all operations preserve 0), and show that a key isomorphism theorem holds in a 0-normal variety if it is subtractive, a property more general than congruence permutability. We then define our notion of a radical class in the second section. A number of basic results and characterisations of radical and semisimple classes are then obtained, largely based on the more general categorical framework of L. M\'arki, R. Mlitz and R. Wiegandt as in [13]. We consider the subtractive case separately. In the third section, we obtain results concerning subvarieties and quasivarieties based on the results of the previous section, and also generalise to subtractive varieties some results for multi-operator group radicals defined by simple equational rules. Several examples of radical and semisimple classes are given for a range of fairly natural 0-normal varieties of algebras, most of which are subtractive.     

Hyperbolic cone-manifold structures with prescribed holonomy I: punctured tori

We consider the relationship between hyperbolic cone-manifold structures on surfaces, and algebraic representations of the fundamental group into a group of isometries. A hyperbolic cone-manifold structure on a surface, with all interior cone angles being integer multiples of 2π, determines a holonomy representation of the fundamental group. We ask, conversely, when a representation of the fundamental group is the holonomy of a hyperbolic cone-manifold structure. In this paper we prove results for the punctured torus; in the sequel, for higher genus surfaces. We show that a representation of the fundamental group of a punctured torus is a holonomy representation of a hyperbolic cone-manifold structure with no interior cone points and a single corner point if and only if it is not virtually abelian. We construct a pentagonal fundamental domain for hyperbolic structures, from the geometry of a representation. Our techniques involve the universal covering group [(PSL₂\mathbb R)\tilde]{\widetilde{{\it PSL}_2{\mathbb R}}} of the group of orientation-preserving isometries of \mathbb H²{{\mathbb H}^2} and Markoff moves arising from the action of the mapping class group on the character variety.     

Hodge Structures Associated to SU(p, 1)

Let A be an abelian variety over ? such that the semisimple part of the Hodge group of A is a product of copies of SU(p, 1) for some p > 1. We show that any effective Tate twist of a Hodge structure occurring in the cohomology of A is isomorphic to a Hodge structure in the cohomology of some abelian variety.     

Lie group symmetries as integral transforms of fundamental solutions

We obtain fundamental solutions for PDEs of the form ut=σxγuxx+f(x)ux−μxru by showing that if the symmetry group of the PDE is nontrivial, it contains a standard integral transform of the fundamental solution. We show that in this case, the problem of finding a fundamental solution can be reduced to inverting a Laplace transform or some other classical transform.     

The word problem for some uncountable groups given by countable words

We investigate the fundamental group of Griffiths? space, and the first singular homology group of this space and of the Hawaiian Earring by using (countable) reduced tame words. We prove that two such words represent the same element in the corresponding group if and only if they can be carried to the same tame word by a finite number of word transformations from a given list. This enables us to construct elements with special properties in these groups. By applying this method we prove that the two homology groups contain uncountably many different elements that can be represented by infinite concatenations of countably many commutators of loops. As another application we give a short proof that these homology groups contain the direct sum of ℵ₀² copies of Q. Finally, we show that the fundamental group of Griffiths? space contains Q.     

On fundamental groups of compact Hausdorff spaces

We discuss which groups can be realized as the fundamental groups of compact Hausdorff spaces. In particular, we prove that the claim ``every group can be realized as the fundamental group of a compact Hausdorff space' is consistent with the Zermelo-Fraenkel-Choice set theory.

     

The Orevkov invariant of an affine plane curve

We show that although the fundamental group of the complement of an algebraic affine plane curve is not easy to compute, it possesses a more accessible quotient, which we call the Orevkov invariant.

     

On Constantive Simple and Order-Primal Algebras

A finite algebra is said to be order-primal if its clone of all term operations is the set of all operations defined on A which preserve a given partial order ≤ on A. In this paper we study algebraic properties of order-primal algebras for connected ordered sets (A; ≤). Such order-primal algebras are constantive, simple and have no non-identical automorphisms. We show that in this case cannot have only unary fundamental operations or only one at least binary fundamental operation. We prove several properties of the varieties and the quasi-varieties generated by constantive and simple algebras and apply these properties to order-primal algebras. Further, we use the properties of order-primal algebras to formulate new primality criteria for finite algebras.* Research supported by the Hungarian research grant No. TO34137 and by the János Bolyai grant.** Research supported by the Thailand Research Fund.