|Last updated: 29 November 2012||Our Invitation to Toric Topology||Construction of the School: Nick Higham's photos||Return to Nige Ray's homepage|
2011 was a busy year. Successful events included
In 2010, Toric Topology was brought to the masses via the Malaga Spring School, where 7 lectures were presented for beginners in the subject.
The highlight of 2009 was the joint Manchester-Moscow meeting of the TTT in Manchester, attended by seven Royal Society/RFBR-funded participants from Moscow State University. Natalia Dobrinskaya also participated, as our first Adams Visitor. Several of the talks featured aspects or applications of toric topology.
During 2008 we organised:
The very first meeting at which we believe Toric Topology was identified took place in Manchester on 10 February 1999. This was the Transpennine Topology Triangle Toric Think Tank, known more concisely as the Manchester TTTTTT.
Subsequent meetings included:
The n--dimensional torus T n is the product of n circles, and is the archetypal compact abelian Lie group. Topological spaces with torus actions are particularly interesting objects, which have been studied for over a century as an important sub-branch of equivariant topology. Restriction to the particular cases of Toric Topology has its origins in algebraic geometry, where torus embeddings were first conceived. The classic model is provided by a nonsingular projective toric variety V n, whose quotient space is a simple convex n--polytope P that is nicely embedded in Euclidean space. The advantage of this situation lies in the geometrical simplicity of the polytope, and the possibility of recovering the original variety, together with its torus action, from combinatorial information that is encoded in the embedding.
In principle, something similar is true for arbitrary T n actions; in practice, toric varieties allow genuine computation. For example, several invariants (both geometrical and topological) of V n may be calculated in terms of P and its associated data. This part of the theory is under rapid current development, and ensures that algebraic geometry continues to exert a decisive influence on toric topology.
In order to generalise the situation, it is helpful to consider the polar polytope of P, which is necessarily simplicial. Its boundary is a simplicial complex K, and P decomposes as the cone on the barycentric subdivision of K; the isotropy subgroups of the action are recorded by assigning certain combinatorial data to the vertices of K. Analogous versions of this procedure may then be repeated for arbitrary simplicial complexes K, leading to beautiful and far-reaching generalisations of the original concept. The underlying principle is that many aspects of algebraic topology may be studied in terms of spaces that decompose as a union of simpler subspaces, indexed by the faces of K.
In 1997, Victor Buchstaber and Nigel Ray 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 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. Taras now visits Manchester regularly, funded by EPSRC, the London Mathematical Society, and MIMS, amongst others. In September 2005 Victor Buchstaber was appointed to a half-time Professorship of Mathematics, and worked in Manchester until his retirement in 2010. He is currently Professor Emeritus, and returned for a week in this capacity during October 2012.
Since 2006, a steady stream of visitors with interests in Toric Topology has included Tony Bahri, Natalia Dobrinskaya, and Matthias Franz. In January 2007 Jelena Grbić took up a permanent position in Manchester, and is now bringing her special expertise in homotopy theory to bear on the subject.
Several postgraduate students have now completed PhD thesis in related areas, including David Carter, Adrian Dobson, Goran Dubajic, Craig Laughton, and Andrew Fenn; other students are at various stages of their projects.
In May 2008, Manchester and Moscow were awarded a Royal Society/RFBR grant for collaborative research into Toric Topology. This supported further exchanges of staff and postgraduate students, and ensured a stimulating environment for the subsequent two years!
In September 2008, Gareth Williams of the Open University joined our team. His knowledge of the techniques of general equivariant topology are most valuable, and are contributing to our understanding of equvariant Ktheory and non-commuting symmetries of quasitoric manifolds.
During the academic year 2009-10, Gery Debongnie (of Louvain-la-Neuve) visited Manchester on a Belgian postdoctoral grant. His interests centre on rational homotopy theory and combinatorial aspects of subspace arrangements.
In October 2010, Michael Wiemeler arrived from Fribourg to take up a 1 year postdoctoral position, funded by the Swiss National Science Foundation. He had previously completed a fascinating PhD thesis at Fribourg, on the extension of torus actions to non-abelian Lie groups; this made good contact with Gareth Williams's work on the discrete case.
In the first half of 2012 Shintaro Kuroki and Soumen Sarkar made separate visits to Manchester en route from KAIST to elsewhere, and generated excellent discussions with Gareth, Nige, and local PhD students.
As the subject has grown, research teams have developed in an increasing number of locations. These include the following ... with apologies to those that I have inadvertantly omitted. Please email me to include your group!
Modern topology originates from Poincaré, who already understood the importance of group actions and their associated symmetries. The study of such actions forms the substance of equivariant algebraic topology, which has developed rapidly since the 1950s, and has always recognised the significance of torus actions and their quotient spaces. Only in the late 1980s, however, did topologists began to appreciate the richness of the theory associated to a special class of torus actions, which may be developed by analogy with the toric varieties of algebraic geometry.
The ground-breaking paper was Mike Davis and Tadeusz Januszkiewicz's Convex polytopes, Coxeter orbifolds and torus actions, in the Duke Mathematical Journal 62:417--451 (1991); their work drew on several external sources, but was the first to present results in an overtly topological setting. Thus was created the fourth vertex T=Topology of the Toric Tetrahedron TT!The Toric Tetrahedron has emerged from the deep since 1970, when the initial vertex A=Algebraic Geometry first broke through the surface via work of Michel Demazure. The second vertex C=Combinatorial Geometry was brought forth in 1980 by Richard Stanley, and the third vertex S=Symplectic Geometry a couple of years later, by Victor Guillemin and Shlomo Sternberg, and independently by Michael Atiyah. Every one of the six edges and four facets is now above the surface, and enjoys its own history and literature; current work is even beginning to colonise the interior. Time is a Morse function, which allows the flow of primeval liquid back along the 1--skeleton to A, whence it returns to the depths!
Toric Topology came of age in 2006, when the Osaka Conference established its status as as an internationally recognized subdiscipline of algebraic topology. The conference combined lectures on cobordism theory, equivariant topology, homotopy theory, and simplicial topology (amongst other classical areas of algebraic topology) with talks on related disciplines such as algebraic geometry, the theory of arrangements, convex geometry, model category theory, and symplectic geometry. Almost every talk acquired a location within the Toric Tetrahedron, whose barycentric coordinates measure the input from each vertex discipline. For a proper introduction to TT, see our Invitation to Toric Topology, which prefaces the proceedings of the Osaka conference.
Even more recently, the suspicion has grown that a fifth vertex lies hidden in the future. It is being uncovered by theoretical P=Physicists, whose interests in mirror symmetry, toric orbifolds and weighted projective spaces are starting to have a major impact. Roll on the Toric 4-simplex ...!!
This will not be a complete archive --- just a pointer to some influential literature, and papers with Mancunian connections (in reverse chronological order).
To every finite simplicial complex K there is associated a moment-angle complex, which may be interpreted as the complement of a coordinate subspace arrangement; this paper applies techniques of classical homotopy theory to prove that it is homotopy equivalent to a wedge of spheres for shifted complexes K and many of their coproducts. This result had eluded expectant combinatorialists, who did not have access to the machinery of Toric Topology.
This article appear in the Mathematical Proceedings of the Cambridge Philosophical Society, 146(02):395--405 (2009). It computes the equivariant cohomology of a weighted projective space in terms of piecewise polynomials, and by generators and relations. Two applications are given: the first confirms that the equivariant cohomology ring suffices to distinguish between weighted projective spaces, and the second proves a Chern class formula for weighted projective bundles.
This article appeared in the Moscow Mathematical Journal 7 (2007), and a version is lodged at arXiv as math.AT/0609346. It applies geometric methods to the interpretation of quasitoric manifolds in terms of combinatorial data, and discusses the relationship with non-singular projective toric varieties. In particular, it applies Khovanskii's theory of analogous polytopes, and constructs representatives for every complex cobordism class as the quotient of a free torus action on a real quadratic complete intersection. The opportunity is taken to simplify and correct certain proofs of the first and third author (from article 6 below) by modifying their definitions of omniorientation and connected sum of polytopes.
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 appears in Algebraic Topology: Categorical Decomposition Techniques (2003), 261-291, which is Volume 215 of the Birkhauser Progress in Mathematics series. It describes the construction of 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 proves 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.
This book has become the classic reference for background, history, and further developments associated to Davis and Januszkiewicz's original work in the subject!
This article purports to answer the "Hirzebruch question" raised by the paper below, by 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. [Errors pointed out by Neil Strickland and Kostya Feldman have now been corrected in article 3 above]
The translation of this article into English appears in Russian Mathematical Surveys 53 (1998), 371-373 pdf
This is the paper that began the subject, by introducing topological analogues of non-singular toric varieties. Condition (*) is crucial!
This section provides links to appropriate colleagues' homepages. I have probably offended several of my friends by not yet having had time to add them to the list; so please email me immediately if that includes you!
Here are a few resources that may be useful for students who are attempting to learn some of the basics of toric topology.
Return to Nige Ray's homepage.