Last updated: 20 August 2007 Construction of the new School: Nick Higham's photos Return to Nige Ray's homepage


In 1997, Nigel Ray and Victor Buchstaber (of Moscow State University) were awarded a Royal Society grant for a two-year exchange of academics between Manchester and Moscow, in order to carry out a programme of research in algebraic topology. The other staff involved were Peter Eccles of Manchester, and Pyotr Akhmet'ev and Vladimir Smirnov of Moscow. The participants are extremely grateful to the Royal Society for enabling this research to take place, thereby strengthening academic links between the two cities.

The collaboration led directly to the further award of a Royal Society/NATO Postdoctoral Fellowship to Taras Panov of Moscow State University, who spent the period February 01 - March 02 in Manchester. Since then, Taras has visited Manchester regularly, funded by EPSRC and the London Mathematical Society, and we thank them both. From September 2005 onwards, Victor Buchstaber is working for six months of the year in Manchester as Professor of Mathematics.

We therefore look forward to increasing cooperation between Moscow State University, the Steklov Mathematical Institute and the University of Manchester.

Part of the original grant included provision for IT equipment to be installed at the Steklov Institute, on which this site will be mirrored as soon as the circumstances of the local web allow.

Toric Topology

One of the great successes of the Exchange has been its role in stimulating the development of Toric Topology at Manchester. This recently identified area of algebraic topology involves interaction with algebraic geometry, combinatorics, symplectic geometry, and theoretical physics, and has become one of the most exciting and fastest growing subjects over the last five years. We are currently constructing The Manchester Toric Topology Page to provide information about these developments.


The following preprints and offprints are available online, and have resulted from the collaboration; others are in the pipeline.

PM Akhmet'ev and PJ Eccles, "A geometrical proof of Browder's result on the vanishing of the Kervaire invariant" ps

This paper is due to appear in the Transactions of the Steklov Mathematical Institute. It uses the study of multiple points of immersed manifolds to provide a new proof of Browder's famous result that the Kervaire invariant of a (4k-2)-dimensional framed manifold is zero unless k is a power of 2.

Victor Buchstaber and Nigel Ray, "Operations and quantum doubles in complex oriented cohomology theory" HHA Volume 1

This paper appeared in Homology, Homotopy and Applications 1 (1999), 169-185. It develops geometrical background to Victor's quantum algebraic description of the algebra of cohomology operations in double complex cobordism, and interprets the constructions in the context of Boardman's eightfold way.

Victor Buchstaber and Nigel Ray, "Flag Manifolds and the Landweber-Novikov Algebra" G & T Volume 2

This paper appeared in Geometry and Topology 2 (1998), 79-101. It introduces the complex bounded flag manifolds (as toric varieties, even), and discusses how their remarkably rich combinatorial and geometrical structure is related to the dual of the Landweber-Novikov algebra.

Victor Buchstaber and Nigel Ray, "Tangential structures on toric manifolds, and connected sums of polytopes" dvi ps pdf

This article appeared in the International Mathematics Research Nortices 4 (2001), 193-219. It answers the "Hirzebruch question" raised by the paper below, proving that every complex cobordism class contains a toric manifold, which is necessarily connected; in addition, it confirms that the appropriate stably complex structure is associated with the torus action, and preserved by it. The concept of omniorientation is defined for the purpose, and an operation of connected sum is developed for simple convex polytopes endowed with specific combinatorial data.

Taras Panov, Nigel Ray, and Rainer Vogt, "Colimits, Stanley-Reisner algebras and loop spaces" dvi ps pdf

This article grew out of a talk at the International Conference on Algebraic Topology, held on the Isle of Skye in June 2001, and is lodged at arXiv as math.AT/0202081. It constructs models for loops on Davis and Januszkiewicz's spaces DJ(K) (defined for finite simplicial complexes K), which rely on the development of homotopy colimits in the category of topological monoids and a proof that they commute with the classifying space functor. Whenever K is flag, the construction reduces to a standard colimit, which is a right-angled Artin or Coxeter group in the real or exterior case; in the complex case, it is a continuous analogue, which we call a circulation group.

The following articles have also appeared as a result of our collaboration, but are not currently available in electronic format.

VM Buchstaber and N Ray, "Toric varieties and complex cobordisms", Uspekhi Mat Nauk 53 (1998), 139-140
Translated in Russian Mathematical Surveys 53 (1998), 371-373.

VM Buchstaber and TE Panov, "Algebraic topology of manifolds defined by simple polytopes", preprint, Moscow State University, submitted to Uspekhi Mat Nauk.

VA Smirnov, "Homotopy type and A-infinity group structure", Mat Sbornik 189 (1998), 135-144.

The notion of an A-infinity group structure is defined here, and it is shown that such a structure exists on the homotopy groups of a topological space X; moreover, the structure determines the homotopy type of X.

VA Smirnov, "Bioperads and Hopf bialgebras in Cobordism theory", Mat Zametki 65 (1999), 270-279.

The notions of bioperad and a bialgebra are defined, and shown to be applicable to cobordism theory.

VA Smirnov, "The Dyer-Lashoff and Steenrod algebras for generalized homology and cohomology", Mat Sbornik 190 (1999), 93-128.

The notions of Dyer-Lashof and Steenrod algebra are defined for generalised homology and cohomology theories, and some calculations are presented for cobordism theory.

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