Space Of Cocycles Research Articles

Horofunctions and symbolic dynamics on Gromov hyperbolic groups

Let X be a proper geodesic metric space which is \delta-hyperbolic in the sense of Gromov. We study a class of functions on X, called horofunctions, which generalize Busemann functions. To each horofunction is associated a point in the boundary at infinity of X. Horofunctions are used to give a description of the boundary. In the case where X is the Cayley graph of a hyperbolic group \Gamma, we show, following ideas of Gromov sketched in his paper Hyperbolic groups, that the space of cocycles associated to horofunctions which take integral values on the vertices is a one-sided subshift of finite type.

The primary approximation to the cohomology of the moduli space of curves and cocycles for the stable characteristic classes

The primary approximation to the cohomology of the moduli space of curves and cocycles for the stable characteristic classes

An algebraic characterization of planar graphs

AbstractA cycle in a graph is a set of edges that covers each vertex an even number of times. A cocycle is a collection of edges that intersects each cycle in an even number of edges. A bicycle is a collection of edges that is both a cycle and a cocycle. The cycles, cocycles, and bicycles each form a vector space over the integers modulo two when addition is defined as symmetric difference of sets. In this paper we examine the relationship between the left‐right paths in a planar graph and the cycle space, cocylce space, and bicycle space. We show that planar graphs are characterized by the existence of a diagonal—a double cover by tours that interacts with the cycle space, cocycle space, and bicycle space in a special manner. This generalizes a result of Rosenstiehl and Read that characterized those planar graphs with no nonempty bicycles. © 1995 John Wiley & Sons, Inc.

A Theory of Algebraic Cocycles

We introduce the notion of an algebraic cocycle as the algebraic analogue of a map to an Eilenberg-MacLane space. Using these cocycles we develop a “cohomology theory” for complex algebraic varieties. The theory is bigraded, functorial, and admits Gysin maps. It carries a natural cup product and a pairing to L-homology. Chern classes of algebraic bundles are defined in the theory. There is a natural transformation to (singular) integral cohomology theory that preserves cup products. Computations in special cases are carried out. On a smooth variety it is proved that there are algebraic cocycles in each algebraic rational (p, p)-cohomology class. In this announcement we present the outlines of a cohomology theory for algebraic varieties based on a new concept of an algebraic cocycle. Details will appear in [FL]. Our cohomology is a companion to the L-homology theory recently studied in [L, F, L-F1, L-F2, FM]. This homology is a bigraded theory based directly on the structure of the space of algebraic cycles. It admits a natural transformation to integral homology that generalizes the usual map taking a cycle to its homology class. Our new cohomology theory is similarly bigraded and based on the structure of the space of algebraic cocycles. It carries a ring structure coming from the complex join (an elementary construction of projective geometry), and it admits a natural transformation Φ to integral cohomology. Chern classes are defined in the theory and transform under Φ to the usual ones. Our definition of cohomology is very far from a duality construction on L-homology. Nonetheless, there is a natural and geometrically defined Kronecker pairing between our “morphic cohomology” and L-homology. The foundation stone of our theory is the notion of an effective algebraic cocycle, which is of some independent interest. Roughly speaking, such a cocycle on a variety X , with values in a projective variety Y , is a morphism from X to the space of cycles on Y . When X is normal, this is equivalent (by “graphing”) to a cycle on X × Y with equidimensional fibres over X . Such cocycles abound in algebraic geometry and arise naturally in many circumstances. The simplest perhaps is that of a flat morphism f : X → Y whose corresponding cocycle associates to x ∈ X , the pullback cycle f({x}). Many more arise naturally from synthetic constructions in projective geometry. We show that every variety is rich in cocycles. Indeed if 1991 Mathematics Subject Classification. Primary 14F99, 14C05.

A theory of algebraic cocycles

We introduce the notion of an algebraic cocycle as the algebraic analogue of a map to an Eilenberg-MacLane space. Using these cocycles we develop a “cohomology theory” for complex algebraic varieties. The theory is bigraded, functorial, and admits Gysin maps. It carries a natural cup product and a pairing to L-homology. Chern classes of algebraic bundles are defined in the theory. There is a natural transformation to (singular) integral cohomology theory that preserves cup products. Computations in special cases are carried out. On a smooth variety it is proved that there are algebraic cocycles in each algebraic rational (p, p)-cohomology class. In this announcement we present the outlines of a cohomology theory for algebraic varieties based on a new concept of an algebraic cocycle. Details will appear in [FL]. Our cohomology is a companion to the L-homology theory recently studied in [L, F, L-F1, L-F2, FM]. This homology is a bigraded theory based directly on the structure of the space of algebraic cycles. It admits a natural transformation to integral homology that generalizes the usual map taking a cycle to its homology class. Our new cohomology theory is similarly bigraded and based on the structure of the space of algebraic cocycles. It carries a ring structure coming from the complex join (an elementary construction of projective geometry), and it admits a natural transformation Φ to integral cohomology. Chern classes are defined in the theory and transform under Φ to the usual ones. Our definition of cohomology is very far from a duality construction on L-homology. Nonetheless, there is a natural and geometrically defined Kronecker pairing between our “morphic cohomology” and L-homology. The foundation stone of our theory is the notion of an effective algebraic cocycle, which is of some independent interest. Roughly speaking, such a cocycle on a variety X , with values in a projective variety Y , is a morphism from X to the space of cycles on Y . When X is normal, this is equivalent (by “graphing”) to a cycle on X × Y with equidimensional fibres over X . Such cocycles abound in algebraic geometry and arise naturally in many circumstances. The simplest perhaps is that of a flat morphism f : X → Y whose corresponding cocycle associates to x ∈ X , the pullback cycle f({x}). Many more arise naturally from synthetic constructions in projective geometry. We show that every variety is rich in cocycles. Indeed if 1991 Mathematics Subject Classification. Primary 14F99, 14C05.

Matrix Generalizations of Some Theorems on Trees, Cycles and Cocycles in Graphs

We extend to arbitrary matrices four theorems of graph theory, one about projections onto the cycle and cocycle spaces, one about the intersection of these spaces, and two matrix-tree theorems. The squares of certain determinants, not apparent for graphs, appear in the extensions.