arXiv subjects

Nikolai V. Ivanov

Publications and source records attributed to Nikolai V. Ivanov.

At least 19 recordsLinked to original sources

arXiv2025

The unitary group in the strong topology and a construction of Dixmier-Douady

By a theorem of Dixmier-Douady the unitary group of an infinite-dimensional separable Hilbert space $H$ in the strong operator topology is contractible. The Dixmier-Douady proof is based on an explicit construction of families of subspaces and operators in $H$ with rather special properties. Unfortunately, this proof leaves hidden the geometric meaning of the theorem. The first goal of this note is to give a direct geometric proof of this theorem. The second goal is to provide a geometic analogue of Dixmier-Douady construction.

math.FA↗

arXiv2023

Non-abelian cohomology and Seifert-Van Kampen theorem

The first goal of the present paper it to present a simple and elementary proof of the standard Seifert-van Kampen theorem based on ideas of P. Olum. The key tool is the singular cohomology theory with non-abelian coefficients in dimensions 0 and 1. After this we apply non-abelian cohomology to prove Crowell-Fox version of Seifert-van Kampent theorem and its improvement die to Brown-Salleh (this proof shows that the Lebesgue dimension of the disc is irrelevant for this improvement). Finally, we apply non-abelian cohomology to prove some theorem of van Kampen, which are more general than the standard version (and, in particular, include the computation of the fundamental group of the circle).

math.AT↗

arXiv2023

Group actions on complexes, Kozsul models, presentations, and a theorem of Coxeter

By a well known theorem of K.S. Brown an action of a discrete group on a simply-connected complex allows to construct a presentation of this group modulo the stabilizers of vertices. The main goal of the present paper is to provide a new proof of this theorem based on ideas of J.-L. Kozsul. In contrast with Brown proof, we start with a redundant but fairly canonical presentation and then simplify it. We illustrate this procedure by the examples of fundamental groups of CW-complexes, the symmetric groups, the group of rotations of the regular dodecahedron, and the binary icosahedral group, and also discuss in details a remarkable theorem of Coxeter about the presentations of the latter group.

math.GR↗

arXiv2023

The virtual cohomology dimension of Teichmüller modular groups: the first results and a road not taken

The paper is devoted to a detailed exposition of results outlined in author's 1983 note "On the virtual cohomology dimension of the Teichmüller modular group". The paper includes the full proofs and a discussion of the context which motivated both the results and the methods used. The reminiscences included into the last section are an integral part of this discussion. A key idea is to use the simply-connectedness of the Hatcher-Thurston complex. This idea was hardly developed further, but still keeps a promise.

math.GT↗

arXiv2023

The geometric meaning of the complex dilatation

The paper is devoted to an approach to the notion of the complex dilatation based on the following observations. (1) A natural measure of the distortion of the conformal structure by a real linear automorphism of the complex plane is the pull-back of the standard conformal structure of complex plane. (2) The set of all conformal structures on the complex plane carries a canonical structure of a model of the hyperbolic plane and can be naturally identified with the unit disc together with its structure of the Klein model of the hyperbolic plane. (3) The standard isomorphism of the Klein model with the Poincaré unit disc model transforms this measure of distortion into the classical complex dilatation. In version 2 this approach is related to Arnold's proof of the hyperbolic altitudes theorem.

math.CV↗

arXiv2023

Boundary triplets and the index of families of self-adjoint elliptic boundary problems

The paper is devoted to an abstract axiomatic version of a construction of boundary triplets implicit in the works of M.I. Vishik and G. Grubb and its applications to the index of families of self-adjoint elliptic differential boundary problems of order one. This leads to an analytic proof of the index theorem for Dirac-like self-adjoint boundary problems from arXiv:2207.09574, and to an Agranovich-Dynin type theorem computing the difference of indices of families of self-adjoint boundary problems differing only by the boundary conditions.

math.DG↗

arXiv2023

The index of self-adjoint Shapiro-Lopatinskii boundary problems of order one

The paper is devoted to an analogue of Atiyah-Bott-Singer index theorem for families of self-adjoint elliptic (i.e. satisfying the Shapiro-Lopatinskii condition) local boundary problems of order 1. The proofs are based on classical topological and pseudo-differential methods, but in the self-adjoint case one encounters some new phenomena. The topological index is defined following Atiyah-Bott, but in the self-adjoint case one encounters an obstruction not present in the classical situation. The analytical index is defined with the help of author's approach arXiv:2111.15081, which generalized the one of Atiyah-Singer. On the analytic index side one encounters an obstruction to the realization of symbols by self-adjoint boundary problems, similar to the obstruction to defining the topological index. As an application, we generalize results of Gorokhovsky and Lesch arXiv:1310.0210. In the first version of this paper the index theorem was proved only under an additional technical assumption. A theory of multiplicative properties of symbols and operators, developed in the second version, allows to remove this assumption.

math.DG↗

arXiv2023

Topological categories related to Fredholm operators: II. The analytical index

Naively, the analytic index of a family of self-adjoint Fredholm operators ought to be (an equivalence class of) the family of the kernels of these operators. The present paper is devoted to a rigorous version of this idea based on ideas of Segal as developed by the author in arXiv:2111.14313 [math.KT]. The resulting new definition of the analytic index makes sense under much weaker continuity assumptions than the Atiyah-Singer one and can be easily adjusted to families of operators in fibers of a Hilbert bundle. We prove the correctness of the new definition and show that it agrees with the Atiyah-Singer one when the latter applies. As an illustration, these results are used to clarify some subtle aspects of the notion of spectral sections introduced by Melrose and Piazza. The necessary definitions and results from arXiv:2111.14313 [math.KT] are repeated or reviewed in order to make this paper independent to the extent possible.

math.DG↗

arXiv2023

Topological categories related to Fredholm operators: I. Classifying spaces

In 1970s Segal outlined proofs of two theorems relating spaces of Fredholm and self-adjoint Fredholm operators with Quillen's constructions used to define higher algebraic K-theory. In the present paper we provide detailed proofs of these theorems of Segal (different from the ones outlined by Segal) and then transplant some further ideas of Quillen to the real of Fredholm and self-adjoint Fredholm operators. The main results may be considered as partial analogues of the equivalence of Quillen's two definition of higher algebraic K-theory. Along the way we relate spaces of Fredholm and self-adjoint Fredholm operators with more concrete classifying spaces. These results are partially motivated by applications to the index theory in a related paper.

math.KT↗

arXiv2022

Beyond Sperner's lemma

In 1967 Herbert Scarf suggested a new proof of Brouwer's fixed point theorem based on a combinatorial analogue of Sperner's lemma. Scarf presented his arguments in very geometric language, even purely combinatorial ones. Recently H. Petri and M. Voorneveld published an almost geometry-free version of Scarf's proof. Their version eliminated even only implicitly geometric aspects of Scarf's proof, namely, the structure of an abstract simplicial complex behind the combinatorial arguments. The present paper is devoted to a proof of Scarf's analogue of Sperner's lemma in the abstract setting of a collection of linear orders on a finite set. This proof partially follows the proof by Petri and Voorneveld, but restores the implicit geometry to its rightful place. We also deduce Brouwer's fixed point theorem from this analogue and discuss various versions of Scarf's proof.

math.AT↗

arXiv2022

Scarf's theorems, simplices, and oriented matroids

In 1967 Herbert Scarf suggested a new proof of Brouwer fixed point theorem based on a surprising analogue of Sperner's lemma. This analogue was motivated by Scarf's work in game theory and mathematical economics. Moreover, Scarf proved a much general version of Sperner's lemma dealing with colorings by vectors. The present paper begins by revisiting Scarf's ideas from the point of view of the basic theory of simplicial cochains in the spirit of author's papers arXiv:1909.00940 and arXiv:2012.13104. After this we get to the main new results of the paper, namely, to a generalization of Scarf results to colorings with colors belonging to an oriented matroid. No knowledge of the theory of oriented matroids is assumed. In the last section we return to the original Scarf theorem and reprove it using even more classical methods of the combinatorial topology of Euclidean spaces. Also, we generalize a theorem of Kannai.

math.CO↗

arXiv2022

Spectral sections: two proofs of a theorem of Melrose-Piazza

Spectral sections of families of self-adjoint Fredholm operators were introduced by Melrose and Piazza for the needs of index theory. The basic result about spectral sections is a theorem of Melrose and Piazza to the effect that a family admits a spectral section if and only if its analytic index vanishes. The present paper is devoted to two proofs of this theorem. These proofs allow to generalize this theorem and to clarify some subtle aspects related to the definition of the analytic index and trivializations of Hilbert bundles. It is based on ideas of author's paper arXiv:2111.15081, but is largely independent from it.

math.DG↗

arXiv2020

Cubes and cubical chains and cochains in combinatorial topology

The present paper is a continuation of author's paper arXiv:1909.00940 [math.AT] devoted to the lemmas of Alexander and Sperner, but is independent from it. We begin by a step back from Alexander and Sperner to Lebesgue work on the invariance of the dimension. In contrast with almost everybody else, Lebesgue worked with cubes rather than with simplices. His methods were developed by Hurewicz and Lusternik-Schnirelmann and then forgotten. In the present paper these methods are recast in the language of cubical chains and cochains. After this, we present a new approach to Lebesgue and Lusternik-Schnirelmann theorems which is both conceptual and elementary. It is based on adaptation of Serre's definition of products of singular cubical cochains to discrete setting. The main results are new purely combinatorial "cubical lemmas". This approach also clarifies the cubical versions of Sperner lemma of Kuhn and Ky Fan. In particular, Ky Fan's lemma can be understood as a natural strengthening of Lebesgue or Kuhn's results under a transversality assumption. The exposition does not assume any knowledge of algebraic topology.

math.CO↗

arXiv2020

Notes on the bounded cohomology theory

The paper is devoted to a generalized and improved version of author's approach to Gromov bounded cohomology theory. In particular, the awkward countability assumption is removed and the aspects related to homological algebra are clarified. The exposition is largely self-contained.

math.AT↗

arXiv2020

Leray theorems for $l_1$-norms of infinite chains

The paper is devoted to an adaptation of author's approach to Leray theorems in bounded cohomology theory to infinite chains. The main results are a stronger and more general form of Gromov's Vanishing-finiteness theorem and a generalization of the first part of his Cutting-of theorem. The proofs are elementary.

math.AT↗

arXiv2020

Leray theorems in bounded cohomology theory

The paper is devoted to a generalized and simplified version of author's approach to covering theorems in bounded cohomology theory. The amenability assumptions are replaced by weaker and more natural acyclicity assumprions. In the case of open coverings the paracompactness assumption is removed. It is shown that for paracompact spaces the case of closed coverings can be reduced to the case of open coverings if spaces and subspaces in question behave nicely with respect to fundamental groups and covering spaces. Another covering theorem for closed coverings assumes nice behavior with respect to singular homology; now only its proof uses the sheaf theory. The methods apply also to $l_1$-homology. The exposition is largely self-contained. In particular, the required results from the theory of paracompact spaces are presented with full proofs.

math.AT↗

arXiv2020

Simplicial sets, Postnikov systems, and bounded cohomology

The paper is devoted to an approach to the bounded cohomology theory based on the theories of simplicial sets and Postnikov systems. In particular, the main results of the bounded cohomology theory of topological spaces are extended to arbitrary Kan (fibrant) simplicial sets.

math.AT↗

arXiv2019

The lemmas of Alexander and Sperner

Alexander's lemma is a version of Sperner's lemma published by Alexander two years earlier than Sperner's paper. The present paper is devoted to a modern but elementary exposition of lemmas of Alexander and Sperner and their main topological applications: Brouwer's theorems about the topological invariance of dimension and of domains (here we follow Lebesgue ideas in the form given to them by Sperner), Brouwer's fixed-point theorem, and Alexander's theorem about the topological invariance of homology groups. Along the way we relate the Knaster-Kuratowski-Mazurkiewich argument with the notion of simplicial approximations, provide a cohomological interpretation of Sperner's lemma and of its combinatorial proof, and explain how classical proofs of Sperner's and Alexander's lemma lead to path-following algorithms. The exposition does not assume any knowledge of algebraic topology.

math.AT↗