A generalization of the non-triviality theorem of Serre

We generalize the classical theorem of Serre on the non-triviality of infinitely many homotopy groups of -connected finite CW-complexes to CW-complexes where the cohomology groups either grow too fast or do not grow faster than a certain rate given by connectivity. For example, this result can be applied to iterated suspensions of finite Postnikov systems and certain spaces with finitely generated cohomology ring. In particular, we obtain an independent, short proof of a theorem of R. Levi on the non-triviality of -invariants associated to finite perfect groups. Another application concerns spaces where the cohomology grows like a polynomial algebra on generators in dimension , , for a fixed number . We also consider spectra where we prove a non-triviality result in the case of fast growing cohomology groups.

     

A theorem on analytic strong multiplicity one

Let K be an algebraic number field, and π=⊗πv an irreducible, automorphic, cuspidal representation of GLm(AK) with analytic conductor C(π). The theorem on analytic strong multiplicity one established in this note states, essentially, that there exists a positive constant c depending on ε>0,m, and K only, such that π can be decided completely by its local components πv with norm N(v)<c⋅C(π)^2m+ε.     

A set intersection theorem and applications

Using Scarf's algorithm for “computing” a fixed point of a continuous mapping, the following is proved: LetM ₁ ? M _n be closed sets inR ⁿ which cover the standard simplexS, so thatM _i coversS _i , the face ofS opposite vertexi. We say a point inS iscompletely labeled if it belongs to everyM _i andk-almost-completely labeled if it belongs to all butM _k . Then there exists a closed setT ofk-almost-completely labeled points which connects vertexk with some completely labeled point. This result is used to prove Browder's theorem (a parametric fixed-point theorem) inR ⁿ . It is also used to generate “algorithms” for the nonlinear complementarity problem which are analogous to the Lemke—Howson algorithm and the Cottle—Dantzig algorithm, respectively, for the linear complementarity problem.     

A separation theorem and Serre duality for the Dolbeault cohomology

LetX be a complex manifold with finitely many ends such that each end is eitherq-concave or (n−q)-convex. If , then we prove thatH ^pn−q (X) is Hausdorff for allp. This is not true in general if (Rossi’s example withn=2 andq=1). If all ends areq-concave, then this is the classical Andreotti-Vesentini separation theorem (and holds also for ). Moreover the result was already known in the case when theq-concave ends can be ‘filled in’ (again also for ). To prove the result we first have to study Serre duality for the case of more general families of supports (instead of the family of all closed sets and the family of all compact sets) which is the main part of the paper. At the end we give an application to the extensibility of CR-forms of bidegree (p, q) from (n−q)-convex boundaries, . This research was partially supported by TMR Research Network ERBFMRXCT 98063.     

Chow groups and Borel-Moore schemes

Summary In this paper we study the Chow groups of schemes for which the class map to Borel-Moore homology is an isomorphism. Then we determine the Chow groups of the scheme Cop^k P ⁿ parametrizing finite coplanary subschemes of lenght k ofP ⁿ and of the variety of «complete S-tuples» of Le Barz.The authors were partially supported by the DGICYT.     

A density theorem for (multiplicity,rank) pairs

For measure-preserving dynamical systems, we investigate the possible values of the pair (m, r), wherer is the maximal number of Rokhlin towers approximating the system andm is the maximal number of cyclic spaces approximating the associated unitary operator. We prove that in this way we can get every pair (d, n) for everyn≥3, and every divisord of some arithmetic function ψ(n) related to the Euler function. We show that for fixedn, the set of possibled is of density one.     

A multiplicity theorem for the Neumann problem

Here is a particular case of the main result of this paper: Let be a bounded domain, with a boundary of class , and let be two continuous functions, , with 0$">, , with n$">. If

and if the set of all global minima of the function has at least connected components, then, for each 0$"> small enough, the Neumann problem

admits at least strong solutions in .

     

A multiplicity theorem for p-superlinear p-Laplacian equations using critical groups and morse theory

We study a nonlinear elliptic equation driven by the Dirichlet p-Laplacian and with a Carathéodory nonlinearity. We assume that the nonlinearity exhibits a p-superlinear growth near infinity but need not satisfy the Ambrosetti–Rabinowitz condition. Using truncation techniques, minimax methods and Morse theory, we show that the problem admits at least three nontrivial solutions, two of which have constant sign (one positive, the other negative).     

Milnor K-theory of rings,higher Chow groups and applications

If R is a smooth semi-local algebra of geometric type over an infinite field, we prove that the Milnor K-group K ^M _n (R) surjects onto the higher Chow group CH ⁿ (R , n) for all n≥0. Our proof shows moreover that there is an algorithmic way to represent any admissible cycle in CH ⁿ (R , n) modulo equivalence as a linear combination of “symbolic elements” defined as graphs of units in R. As a byproduct we get a new and entirely geometric proof of results of Gabber, Kato and Rost, related to the Gersten resolution for the Milnor K-sheaf. Furthermore it is also shown that in the semi-local PID case we have, under some mild assumptions, an isomorphism. Some applications are also given. Oblatum 17-XII-1998 & 1-X-2001?Published online: 18 January 2002     

Generic points of quadrics and Chow groups

In this article we study the sufficient conditions for the k̅-defined element of the Chow group of a smooth variety to be k-rational (defined over k). For 0-cycles this question was addressed earlier. Our methods work for cycles of arbitrary dimension. We show that it is sufficient to check this property over the generic point of a quadric of sufficiently large dimension. Among the applications one should mention the uniform construction of fields with all known u-invariants.     

Chow groups are finite dimensional,in some sense

When S is a surface with p_g(S)>0, Mumford proved that its Chow group A_*S is not finite dimensional in some sense. In this paper, we propose another definition of finite dimensionality for the Chow groups. Using this new definition, at least the Chow group of some surface S with p_g(S)>0 (for example, the product of two curves) becomes finite dimensional. The finite dimensionality of the Chow groups follows from the finite dimensionality of the Chow motives. It turns out that the finite dimensionality of the Chow motives is a very strong property. For example, we can prove Blochs conjecture (representability of the Chow groups of surfaces with p_g(S)=0) under the assumption that the Chow motive of S is finite dimensional.Mathematics Subject Classification (2000): 14C     

Oriented Chow groups, Hermitian K-theory and the Gersten conjecture

We show that the oriented Chow groups of Barge–Morel appear in the E ₂-term of the coniveau spectral sequence for Hermitian K-theory. This includes a localization theorem and the Gersten conjecture (over infinite base fields) for Hermitian K-theory. We also discuss the conjectural relationship between oriented and higher oriented Chow groups and Levine’s homotopy coniveau spectral sequence when applied to Hermitian K-theory.     

A multiplicity theorem for Neumann problems with asymmetric nonlinearity

We consider a nonlinear Neumann problem with a reaction term which exhibits an asymmetric behavior near +∞ and near −∞. Namely, it is asymptotically superlinear at +∞ and linear at −∞. Using variational methods based on critical point theory, together with truncation techniques and Morse theory, we show that the problem has at least three nontrivial smooth solutions, two of which have constant sign (one positive and the other negative).