Toric Topology in the Work of Taras Panov

Since the 1970s, the study of torus actions has become increasingly important in various areas of pure mathematics, and has stimulated the formation of interdisciplinary links between algebraic geometry, combinatorial and convex geometry, commutative and homological algebra, differential topology and homotopy theory. As their net has spread wider, and the literature grown, a field of activity has emerged which merits the title toric topology. Toric topology is the study of algebraic, combinatorial, differential, geometric, and homotopy theoretic aspects of a particular class of torus actions, whose quotients are highly structured. A characteristic feature is the calculation of invariants in terms of combinatorial data associated to the quotients; a primary goal is to classify toric spaces by means of these invariants. The initial impetus for these developments was provided by the theory of toric varieties in algebraic geometry. This theory gives a bijection between, on one hand, complex algebraic varieties that are equipped with an action of a complex torus with a dense orbit, and, on the other hand, fans, which are combinatorial objects. The fan allows one to completely translate various algebraic-geometric notions into combinatorics. A valuable aspect of this theory is that it provides many explicit examples of algebraic varieties, with applications in deep subjects such as resolution of singularities and mirror symmetry. The quotient of a projective variety by the action of the compact torus Tn is a convex simple polytope P . The polar polytope of P is necessarily simplicial, and its boundary is a simplicial complex K. In symplectic geometry, since the early 1980s there has been much activity in the field of Hamiltonian group actions, largely following the Atiyah–Guillemin–Sternberg [6] convexity theorem and the Duistermaat–Heckman exact stationary phase formula [14]. Atiyah–Bott–Berline– Vergne exhibited the latter as a special case of localisation in equivariant cohomology, thus putting many of the activities on Hamiltonian group actions into the context of equivariant topology rather than symplectic geometry. Delzant [13], in 1988, showed that if the torus is of half the dimension of the manifold the moment map image determines the manifold up to equivariant symplectomorphism. In symplectic geometry, as in algebraic geometry, one translates various geometric constructions into the language of convex polytopes and combinatorics. There is a tight relationship between the algebraic and the symplectic pictures: a projective embedding of a toric manifold determines a symplectic form and a moment map. The image of the moment map is a convex polytope that is dual to the fan. In both the smooth algebraicgeometric and the symplectic situations, the compact torus action is locally isomorphic to the standard action of (S1)n on Cn by rotation of the coordinates. Thus the quotient of the manifold by this action is naturally a manifold with corners, stratified according to the dimension of the stabilisers, and each stratum can be equipped with data that encodes the isotropy torus action along that stratum. Not only does this structure of the quotient provide a powerful means of investigating the action, but some of its subtler combinatorial properties may also be illuminated by the topology of the manifold. Taking into account the topological nature of this feature, there is no surprise that since the beginning of the 1990s the ideas and methodology of toric varieties and Hamiltonian actions have started penetrating back into algebraic topology. Specifically, during the last two decades, the examples of smooth toric varieties and of symplectic toric manifolds have been generalised into several other classes of manifolds with torus action, mostly of purely topological nature. These more general manifolds are not necessarily algebraic or symplectic; thus there is more flexibility within these classes of spaces for topological or combinatorial applications, but on the other hand they still possess most of the important topological properties of their algebraic or symplectic predecessors. We now describe in more detail some of these generalisations.