A short proof of an index theorem

We give a -theoretical proof of an index theorem for Dirac-Schrödinger operators on a noncompact manifold.

     

A proof of a partition theorem for

In this note we give a proof of Devlin's theorem via Milliken's theorem about weakly embedded subtrees of the complete binary tree . Unlike the original proof which is (still unpublished) long and uses the language of category theory, our proof is short and uses direct combinatorial reasoning.

     

Scott's rigidity theorem for Seifert fibered spaces; revisited

We will present a new proof of the rigidity theorem for Seifert fibered spaces of infinite by Scott (1983) in the case when the base of the fibration is a hyperbolic triangle 2-orbifold. Our proof is based on arguments in the rigidity theorem for hyperbolic 3-manifolds by Gabai (1997).

     

Cotlar-Stein lemma and the theorem

In this note we give a generalization of the Cotlar-Stein lemma and using this lemma we give a new proof of a special case of the theorem which, in general, was proved by David, Journé and Semmes.

     

A lower bound for the Bloch radius of -quasiregular mappings

We give a quantitative proof to Eremenko's theorem (2000), which extends Bloch's classical theorem to the class of -dimensional -quasiregular mappings.

     

An elementary proof of the triangle inequality for the Wasserstein metric

We give an elementary proof for the triangle inequality of the -Wasserstein metric for probability measures on separable metric spaces. Unlike known approaches, our proof does not rely on the disintegration theorem in its full generality; therefore the additional assumption that the underlying space is Radon can be omitted. We also supply a proof, not depending on disintegration, that the Wasserstein metric is complete on Polish spaces.

     

Bundles of -correspondences over directed graphs and a theorem of Ionescu

We give a short proof of a recent theorem of Ionescu which shows that the Cuntz-Pimsner -algebra of a certain correspondence associated to a Mauldin-Williams graph is isomorphic to the graph algebra.

     

Superrigidity for irreducible lattices and geometric splitting

We prove general superrigidity results for actions of irreducible lattices on CAT spaces, first in terms of the ideal boundary, and then for the intrinsic geometry (also for infinite-dimensional spaces). In particular, one obtains a new and self-contained proof of Margulis' superrigidity theorem for uniform irreducible lattices in non-simple groups. The proofs rely on simple geometric arguments, including a splitting theorem which can be viewed as an infinite-dimensional (and singular) generalization of the Lawson-Yau/Gromoll-Wolf theorem. Appendix A gives a very elementary proof of commensurator superrigidity; Appendix B proves that all our results also hold for certain non-uniform lattices.

     

A generalization of Banchoff's triple point theorem

Consider an immersion of a surface into . Banchoff's theorem states that the parity of the number of triple points and the parity of the Euler characteristic of the surface coincide. Here we generalize this theorem to codimension 1 immersions of arbitrary even dimensional manifolds in spheres. The proof is an analogue of a proof of Banchoff's theorem circulated in preprint form due to R. Fenn and P. Taylor in 1977.

     

Evaluations of a class of double -values

An analytic proof of an evaluation theorem for the ``convolution"-type double -values of non-principal characters is given. Along the way, Dirichlet character analogues of generalized single and double polylogarithms are defined. The monodromies of these functions play a pivotal role.

     

Ideals without ccc and without property

We prove a strong version of a theorem of Balcerzak-Roslanowski-Shelah by showing, in ZFC, that there exists a simply definable Borel -ideal for which both the ccc and property (M) fail. The proof involves Polish group actions.

     

On hypersphericity of manifolds with finite asymptotic dimension

We prove the following embedding theorems in the coarse geometry:



The Corollary is used in the proof of the following.

Theorem B together with a theorem of Gromov-Lawson implies the result, previously proven by G. Yu (1998), which states that an aspherical manifold whose fundamental group has a finite asymptotic dimension cannot carry a metric of positive scalar curvature.

We also prove that if a uniformly contractible manifold of bounded geometry is large scale uniformly embeddable into a Hilbert space, then is stably integrally hyperspherical.

     

A new proof of proximinality for -ideals

We give a new and a simple proof of proximinality for -ideals. Unlike the known proofs, our proof derives proximinality of -ideals directly from the definition of an -ideal, using the Bishop-Phelps theorem.

     

Stanley's theorem on codimension 3 Gorenstein -vectors

In this note we supply an elementary proof of the following well-known theorem of R. Stanley: the -vectors of Gorenstein algebras of codimension 3 are SI-sequences, i.e. are symmetric and the first difference of their first half is an -sequence.

     

No embeddings of solenoids into surfaces

A quick proof of Bing's theorem indicated by the title is given. Indeed the inverse limit of a sequence of degree maps between closed oriented -manifolds can never be embedded into any closed orientable -manifold. The proof also concludes Gumerov's result on the covering of solenoids.

     

Algebraic groups over finite fields, a quick proof of Lang's theorem

We give an easy proof of Lang's theorem about the surjectivity of the Lang map on a linear algebraic group defined over a finite field, where is a Frobenius endomorphism.

     

Morse-Palais lemma for nonsmooth functionals on normed spaces

Using elementary differential calculus we get a version of the Morse-Palais lemma. Since we do not use powerful tools in functional analysis such as the implicit theorem or flows and deformations in Banach spaces, our result does not require the -smoothness of functions nor the completeness of spaces. Therefore it is stronger than the classical one but its proof is very simple.

     

A note on the cohomological below -pinching theorem

By making use of a theorem of Toda, we establish a sharper version of the below -pinching theorem of Abresch and Meyer.

     

Degree-one maps between hyperbolic 3-manifolds with the same volume limit

Suppose that are degree-one maps between closed hyperbolic 3-manifolds with

Then, our main theorem, Theorem 2, shows that, for all but finitely many , is homotopic to an isometry. A special case of our argument gives a new proof of Gromov-Thurston's rigidity theorem for hyperbolic 3-manifolds without invoking any ergodic theory. An example in §3 implies that, if the degree of these maps is greater than 1, the assertion corresponding to our theorem does not hold.

     

Trace methods in twisted group algebras, II

In this note, we continue our discussion of trace methods in twisted group algebras. Specifically, we obtain the twisted analog of Bass' theorem on the traces of idempotents in ordinary group algebras. Indeed, we show that with suitable normalization, the characteristic trace values of an idempotent are all contained in a cyclotomic field. The proof is a variant of the original argument combined with a reduction to finitely presented groups.