W∞W_{\infty}--Geometry and Associated Continuous Toda System

We discuss an infinite--dimensional k\"ahlerian manifold associated with the area--preserving diffeomorphisms on two--dimensional torus, and, correspondingly, with a continuous limit of the $A_r$--Toda system. In particular, a continuous limit of the $A_r$--Grassmannians and a related Pl\"ucker type formula are introduced as relevant notions for $W_{\infty}$--geometry of the self--dual Einstein space with the rotational Killing vector.Comment: 6 pages, no figure report\# ETH-TH/93-2

A Lefschetz type coincidence theorem

A Lefschetz-type coincidence theorem for two maps f,g:X->Y from an arbitrary topological space X to a manifold Y is given: I(f,g)=L(f,g), the coincidence index is equal to the Lefschetz number. It follows that if L(f,g) is not equal to zero then there is an x in X such that f(x)=g(x). In particular, the theorem contains some well-known coincidence results for (i) X,Y manifolds and (ii) f with acyclic fibers.Comment: The final version, 23 pages, to appear in Fund. Mat

Higher order Nielsen numbers

Suppose X,Y are manifolds, f,g:X->Y are maps. The well-known Coincidence Problem studies the coincidence set C={x:f(x)=g(x)}. The number m=dimX-dimY is called the codimension of the problem. More general is the Preimage Problem. For a map f:X->Z and a submanifold Y of Z, it studies the preimage set C={x:f(x) in Y}, and the codimension is m=dimX+dimY-dimZ. In case of codimension 0, the classical Nielsen number N(f,Y) is a lower estimate of the number of points in C changing under homotopies of f, and for an arbitrary codimension, of the number of components of C. We extend this theory to take into account other topological characteristics of C. The goal is to find a "lower estimate" of the bordism group Omega_{p}(C) of C. The answer is the Nielsen group S_{p}(f,Y) defined as follows. In the classical definition the Nielsen equivalence of points of C based on paths is replaced with an equivalence of singular submanifolds of C based on bordisms. We let S_{p}^{prime}(f,Y) be the quotient group of Omega_{p}(C) with respect to this equivalence relation, then the Nielsen group of order p is the part of this group preserved under homotopies of f. The Nielsen number N_{p}(f,Y) of order p is the rank of this group (then N(f,Y)=N_{0}(f,Y)). These numbers are new obstructions to removability of coincidences and preimages. Some examples and computations are provided.Comment: New version, 18 pages. Minor revisions throughout the pape

Ultrafilter extensions of linear orders

It was recently shown that arbitrary first-order models canonically extend to models (of the same language) consisting of ultrafilters. The main precursor of this construction was the extension of semigroups to semigroups of ultrafilters, a technique allowing to obtain significant results in algebra and dynamics. Here we consider another particular case where the models are linearly ordered sets. We explicitly calculate the extensions of a given linear order and the corresponding operations of minimum and maximum on a set. We show that the extended relation is not more an order however is close to the natural linear ordering of nonempty half-cuts of the set and that the two extended operations define a skew lattice structure on the set of ultrafilters