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Abstract. We discuss a notion of integration with respect to the Euler characteristic in the projectivization $\P{\cal O}_{\C^n,0}$ of the ring ${\cal O}_{\C^n,0}$ of germs of functions on C^n$ and show that the Alexander polynomial and the zeta-function of a plane curve singularity can be expressed as certain integrals over $\P{\cal O}_{\C^2,0}$ with respect to the Euler characteristic.

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Abstract. The paper surveys some new results and open problems connected with such fundamental combinatorial concepts as polytopes, simplicial complexes, cubical complexes, and subspace arrangements. Particular attention is paid to the case of simplicial and cubical subdivisions of manifolds and, especially, spheres. We describe important constructions which allow to study all these combinatorial objects by means of methods of commutative and homological algebra. The proposed approach to combinatorial problems relies on the theory of moment-angle complexes, currently being developed by the authors. The theory centres around the construction that assigns to each simplicial complex $K$ with $m$ vertices a $T^m$-space $\zk$ with a special bigraded cellular decomposition. In the framework of this theory, the well-known non-singular toric varieties arise as orbit spaces of maximally free actions of subtori on moment-angle complexes corresponding to simplicial spheres. We express different invariants of simplicial complexes and related combinatorial-geometrical objects in terms of the bigraded cohomology rings of the corresponding moment-angle complexes. Finally, we show that the new relationships between combinatorics, geometry and topology result in solutions to some well-known topological problems.

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Abstract. We extend work of Davis and Januszkiewicz by considering omnioriented toric manifolds, whose canonical codimension-2 submanifolds are independently oriented. We show that each omniorientation induces a canonical stably complex structure, which is respected by the torus action and so defines an element of an equivariant cobordism ring. As an application, we compute the complex bordism groups and cobordism ring of an arbitrary omnioriented toric manifold. We consider a family of examples B_i,j, which are toric manifolds over products of simplices, and verify that their natural stably complex structure is induced by an omniorientation. Studying connected sums of products of the B_i,j allows us to deduce that every complex cobordism class of dimension >2 contains a toric manifold, necessarily connected, and so provides a positive answer to the toric analogue of Hirzebruch's famous question for algebraic varieties. In previous work, we dealt only with disjoint unions, and ignored the relationship between the stably complex structure and the action of the torus. In passing, we introduce a notion of connected sum # for simple n-dimensional polytopes; when Pⁿ is a product of simplices, we describe Pⁿ#Qⁿ by applying an appropriate sequence of pruning operators, or hyperplane cuts, to Qⁿ.

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Abstract. We consider nonlocal field-theoretical Poisson brackets containing the operator of integration in the nonlocal part. The main attention is given to the nonlocal brackets of Hydrodynamic Type for which we introduce the Physical and Canonical forms. We use the Canonical form of these brackets for the investigation of a Poisson structure on the loop spaces defined by the corresponding pseudo-differential expression.

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Abstract. A quasitoric manifold is a smooth 2n-manifold M²ⁿ with an action of the compact torus Tⁿ such that the action is locally isomorphic to the standard action of Tⁿ on Cⁿ and the orbit space is diffeomorphic, as manifold with corners, to a simple polytope Pⁿ. The name refers to the fact that topological and combinatorial properties of quasitoric manifolds are similar to that of non-singular algebraic toric varieties (or toric manifolds). Unlike toric varieties, quasitoric manifolds may fail to be complex; however, they always admit a stably (or weakly almost) complex structure, and their cobordism classes generate the complex cobordism ring. As it have been recently shown by Buchstaber and Ray, a stably complex structure on a quasitoric manifold is defined in purely combinatorial terms, namely, by anorientation of the polytope and a function from the set of codimension-one faces of the polytope to primitive vectors of an integer lattice. We calculate the x_y-genus of a quasitoric manifold with fixed stably complex structure in terms of the corresponding combinatorial data. In particular, this gives explicit formulae for the classical Todd genus and signature. We also relate our results with well-known facts in the theory of toric varieties.

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Abstract. An n-dimensional polytope Pⁿ is called simple if exactly n codimension-one faces meet at each vertex. The lattice of faces of a simple polytope Pⁿ with m codimension-one faces defines an arrangement of coordinate subspaces in C^m. The group R^m-n acts on the complement of this arrangement by dilations. The corresponding quotient is a smooth manifold Z_P invested with a canonical action of the compact torus T^m with the orbit space Pⁿ. For each smooth projective toric variety M²ⁿ defined by a simple polytope Pⁿ with the given lattice of faces there exists a subgroup T^m-n\subset T^m acting freely on Z_P such that Z_P/T^m-n=M²ⁿ. We calculate the cohomology ring of Z_P and show that it is isomorphic to the cohomology of the Stanley-Reisner ring of Pⁿ regarded as a module over the polynomial ring. In this way the cohomology of Z_P acquires a bigraded algebra structure, and the additional grading allows to catch combinatorial invariants of the polytope. At the same time this gives an example of explicit calculation of the cohomology ring for the complement of a subspace arrangement defined by simple polytope, which is of independent interest.

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English:  tex      Russian:  ps

English:  tex  ps     Russian:  ps

Abstract. In this paper we construct a theory of indices of Morse-Bott vector fields on a manifold and use it to solve a famous localization problem for the transfer map. As a consequence, we obtain addition theorems for universal Pontryagin classes in cobordisms. This enables us to complete the construction of the theory of universal characteristic classes, which was begun more than twenty years ago.

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Abstract. We show that the cohomology algebra of the complement of a coordinate subspace arrangement in m-dimensional complex space is isomorphic to the cohomology algebra of Stanley-Reisner face ring of a certain simplicial complex on m vertices. (The face ring is regarded as a module over the polynomial ring on m generators.) Then we calculate the latter cohomology algebra by means of the standard Koszul resolution of polynomial ring. To prove these facts we construct an equivariant with respect to the torus action homotopy equivalence between the complement of a coordinate subspace arrangement and the moment-angle complex defined by the simplicial complex. The moment-angle complex is a certain subset of a unit poly-disk in m-dimensional complex space invariant with respect to the action of an m-dimensional torus. This complex is a smooth manifold provided that the simplicial complex is a simplicial sphere, but otherwise has more complicated structure. Then we investigate the equivariant topology of the moment-angle complex and apply the Eilenberg-Moore spectral sequence. We also relate our results with well known facts in the theory of toric varieties and symplectic geometry.

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Abstract.  Commutative rings of one-dimensional difference operators of rank l>1 and their deformations are effectively constructed. Our analytical constructions are based on the so-called ''Tyurin parameters'' for the stable framed holomorphic vector bundles over algebraic curves of the genus equal to g and Chern number equal to lg. These parameters were heavily used by the present authors already in 1978-80 for the differential operators. Their deformations in the discrete case are governed by the 2D Toda Lattice hierarhy instead of KP. New integrable systems appear here in the case l=2, g=1. The theory of higher rank difference operators is much more rich than the rank one case where only 2-point constructions on the spectral curve were used in the previous literature (i.e. number of 'infinite points'' is equal to 2). One-point constructions appear in this problem for every even rank l=2k. Only in this case commutative rings depend on the functional parameters. Two-point constructions will be studied in the next work: even for higher rank l>1 this case can be solved in Theta-functions. It is not so for one-point constructions with rank l>1.

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Abstract.  We review basic ideas and basic examples of the theory of the inverse spectral problems.

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Abstract. Let $A$ be a separable $C^*$-algebra and $B$ a stable $C^*$-algebra containing a strictly positive element. We show that the group $\Ext(SA,B)$ of unitary equivalence classes of extensions of $SA$ by $B$, modulo the extensions which are asymptotically split, coincides with the group of homotopy classes of such extensions. This is done by proving that the Connes-Higson construction gives rise to an isomorphism between $\Ext(SA,B)$ and the $E$-theory group $E(A,B)$ of homotopy classes of asymptotic homomorphisms from $S^2A$ to $B$.

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Abstract. We show that the E-theory of Connes and Higson can be formulated in terms of C*-extensions in a way quite similar to the way in which the KK-theory of Kasparov can. The essential difference is that the role played by split extensions should be taken by asymptotically split extensions. We call an extension of a C*-algebra A by a stable C*-algebra B asymptotically split if there exists an asymptotic homomorphism consisting of right inverses for the quotient map. An extension is called semi-invertible if it can be made asymptotically split by adding another extension to it. Our main result is that there exists a one-to-one correspondence between asymptotic homomorphisms from SA to B and homotopy classes of semi-invertible extensions of S²A by B.

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Abstract. Let $A$ be a separable $C^*$-algebra and let $B$ be a stable $C^*$-algebra with a strictly positive element. We consider the (semi)group $\Ext^{as}(A,B)$ (resp. $\Ext(A,B)$) of homotopy classes of asymptotic (resp. of genuine) homomorphisms from $A$ to the corona algebra $M(B)/B$ and the natural map $i:\Ext(A,B)\ar\Ext^{as}(A,B)$. We show that if $A$ is a suspension then $\Ext^{as}(A,B)$ coincides with $E$-theory of Connes and Higson and the map $i$ is surjective. In particular any asymptotic homomorphism from $SA$ to $M(B)/B$ is homotopic to some genuine homomorphism.

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