Buhstaber's work on two-valued formal groups

In this note we outline a connection between the generalized cohomology theories of unoriented cobordism and (weakly-) complex cobordism and the theory of formal commutative groups of one variable [4], [5]. This connection allows us to apply Carrier's theory of typical group laws to obtain an explicit decomposition of complex cobordism theory localized at a prime p into a sum of Brown-Peterson cohomology theories [ l ] and to determine the algebra of cohomology operations in the latter theory.

This survey is devoted to a detailed exposition of the relations between algebraic topology and the theory of formal Lie groups. The focus of attention is the theory of characteristic classes of vector bundles lying at the basis of these relations.

Hopf algebras of set systems

Let L/K be a Galois extension of fields which are extensions of finite degree over the field Qp , and let Lab be the maximal abelian extension of L. This article constructs the Safarevic map from the Galois group of the extension Lab/K to a certain concrete skew field arising in the theory of formal groups. Bibliography: 5 items.

We define the class of multivariate group entropies as a novel set of information-theoretical measures, which extends significantly the family of group entropies. We propose new examples related to the "super-exponential" universality class of complex systems; in particular, we introduce a general entropy, representing a suitable information measure for this class. We also show that the group-theoretical structure associated with our multivariate entropies can be used to define a large family of exactly solvable discrete dynamical models. The natural mathematical framework allowing us to formulate this correspondence is offered by the theory of formal groups and rings.

Motivated in part by recent works on the genus of classifying spaces of compact Lie groups, here we study the set of filtered λ‐ring structures over a filtered ring from a purely algebraic point of view. From a global perspective, we first show that this set has a canonical topology compatible with the filtration on the given filtered ring. For power series rings R[[x]], where R is between ℤ and ℚ, with the x‐adic filtration, we mimic the construction of the Lazard ring in formal group theory and show that the set of filtered λ‐ring structures over R[[x]] is canonically isomorphic to the set of ring maps from some “universal” ring U to R. From a local perspective, we demonstrate the existence of uncountably many mutually nonisomorphic filtered λ‐ring structures over some filtered rings, including rings of dual numbers over binomial domains, (truncated) polynomial, and power series rings over ℚ‐algebras.

The paper is devoted to problems at the intersection of formal group theory, the theory of Hirzebruch genera, and the theory of elliptic functions. In the focus of our interest are Tate formal groups corresponding to the general five-parametric model of the elliptic curve as well as formal groups corresponding to the general four-parametric Krichever genus. We describe coefficient rings of formal groups whose exponentials are determined by elliptic functions of levels 2 and 3.

Brief notes on further applications of formal group (law) theory

Let h* be a complex (or real) oriented cohomology theory. It has recently been observed that the of h* plays a very important role in such a theory. Above all, D. Quillen [6] showed that the formal group FU of the complex cobordism theory MU* ( ) is isomorphic to the Lazard universal formal group, and its coefficients generate the ground ring MU* (pt). Unfortunately, in the case of quaternionic oriented cohomology theory (see § 2), there exists no such formal group, since the tensor product of quaternionic line bundles does not yield a quaternionic bundle. This situation makes it difficult, for example, to produce enough generators of MSp*(pt). However there are some substitutes for the formal group. In particular, using the total Pontrjagin class of a certain quaternionic vector bundle 2C (see § 3), N. Ja. Gozman [5] defined a subring A of MU* (pt) which is contained in the image of the forgetful homomorphism (p: MSp* (pt}-*MU*(pt). In this paper, we will generalize his approach to arbitrary quaternionic oriented cohomology theory 7i*, and define a subring Ah of h* (pt) which is generated by the coefficients of certain power series. Then it will be shown that

Algebraic topology of manifolds defined by simple polytopes V M Bukhshtaber and T E Panov Torus actions, combinatorial topology, and homological algebra V M Buchstaber and T E Panov Hirzebruch genera of manifolds with torus action T E Panov The index of an equivariant vector field and addition theorems for Pontryagin classes V M Bukhshtaber and K E Feldman Embedding of compacta, stable homotopy groups of spheres, and singularity theory P M Akhmet'ev Complex cobordism and formal groups Viktor M Buchstaber Massey Products and the Bukhshtaber spectral sequence I V Artel'nykh Toric manifolds and complex cobordisms

Our primary aim is to develop a theory of equivariant genera for stably complex manifolds equipped with compatible actions of a torus Tk. In the case of omnioriented quasitoric manifolds, we present computations that depend only on their defining combinatorial data; these draw inspiration from analogous calculations in toric geometry, which seek to express arithmetic, elliptic, and associated genera of toric varieties in terms only of their fans. Our theory focuses on the universal toric genus Φ, which was introduced independently by Krichever and Löffler in 1974, albeit from radically different viewpoints. In fact, Φ is a version of tom Dieck’s bundling transformation of 1970, defined on Tk-equivariant complex cobordism classes and taking values in the complex cobordism algebra of the classifying space. We proceed by combining the analytic, the formal group theoretic, and the homotopical approaches to genera and refer to the index-theoretic approach as a recurring source of insight and motivation. The resultant flexibility allows us to identify several distinct genera within our framework and to introduce parametrized versions that apply to bundles equipped with a stably complex structure on the tangents along their fibers. In the presence of isolated fixed points, we obtain universal localization formulae, whose applications include the identification of Krichever’s generalized elliptic genus as universal among the genera that are rigid on SU-manifolds. We follow the traditions of toric geometry by working with a variety of illustrative examples wherever possible. For background and prerequisites, we attempt to reconcile the literature of east and west, which developed independently for several decades after the 1960s.

Theoretical analysis of the new social class of the precariat has been carried out and its scientific status has been clarified. The analysis has been undertaken from the standpoint of basic class theories. Three main approaches to determining the status of a precariat have been considered: the precariat as part of the proletariat, the precariat as a stratum, and the precariat as a separate class. The perfomed analysis shows that the precariat does not fully correspond to any class theory (in all cases certain social groups are excluded from it). The specifics of the precariat, its place and role in modern society do not allow us make an unequivocal conclusion about its status and the people included in it. In this regard an example of the author’s definition of the precariat, which can be used to conduct empirical research on this issue has been given in the article.

The study of homology groups ${\mathfrak {B}_n}(\Pi ,A),\mathfrak {B}$ a variety, $\Pi$ a group in $\mathfrak {B}$, and $A$ a suitable $\Pi$-module, is continued. A ’Tor’ is constructed which gives a better (but imperfect) approximation to these groups than a Tor previously considered. ${\mathfrak {B}_2}(\Pi ,Z)$ is calculated for various varieties $\mathfrak {B}$ and groups $\Pi$.

The workshop aimed to study discrete and Lie groups and their actions using measure theoretic methods and their asymptotic invariants, such as \ell^2 -invariants, the rank gradient, cost, torsion growth, entropy-type invariants and invariants coming from random walks and percolation theory. The participants came from a wide range of mathematics: asymptotic group theory, geometric group theory, ergodic theory, \ell^2 -theory, graph convergence, representation theory, probability theory, descriptive set theory and algebraic topology.

The general subject of the conference was geometric group theory, that is the study of groups via suitable actions on spaces endowed with additional structures, for instance a distance, a measure (class), or both of them with natural compatibilities. The interactions with other mathematical fields such as algebraic topology (e.g. via homological methods), functional analysis (e.g. via the theory of operator algebras), probability theory (e.g. via random walks) or even representation theory and number theory, play an increasing role in the development of this part of group theory.