For an arbitrary simplicial complex K , Davis and Januszkiewicz have defined a family of homotopy equivalent CW-complexes whose integral cohomology rings are isomorphic to the Stanley-Reisner algebra of K . Subsequently, Buchstaber and Panov gave an alternative construction (here called c(K)), which they showed to be homotopy equivalent to Davis and Januszkiewicz's examples. It is therefore natural to investigate the extent to which the homotopy type of a space is determined by having such a cohomology ring. We begin this study here, in the context of model category theory. In particular, we extend work of Franz by showing that the singular cochain algebra of c(K) is formal as a differential graded noncommutative algebra. We specialise to the rationals by proving the corresponding result for Sullivan's commutative cochain algebra, and deduce that the rationalisation of c(K) is unique for a special family of complexes K . In a sequel, we will consider the uniqueness of c(K) at each prime separately, and apply Sullivan's arithmetic square to produce global results for this family.

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We describe the integral equivariant cohomology ring of a weighted projective space in terms of piecewise polynomials, and thence by generators and relations. We deduce that the ring is a perfect invariant, and prove a Chern class formula for weighted projective bundles.

Our primary aim is to develop a theory of equivariant genera for stably complex manifolds equipped with compatible actions of a torus T k . 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 T k -equivariant complex cobordism classes and taking values in the complex cobordism algebra Ω * U (BT k + ) 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 parametrised versions that apply to bundles equipped with a stably complex structure on the tangents along their fibres. In the presence of isolated fixed points, we obtain universal localisation formulae, whose applications include the identification of Krichever's generalised elliptic genus as universal amongst 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.

Motivation. Sometime around the turn of the recent millennium, those of us in Manchester and Moscow who had been collaborating since the mid-1990s began using the term toric topology to describe our widening interests in certain well-behaved actions of the torus. Little did we realise that, within seven years, a significant international conference would be planned with the subject as its theme, and delightful Japanese hospitality at its heart.When first asked to prepare this article, we fantasised about an authoritative and comprehensive survey; one that would lead readers carefully through the foothills above which the subject rises, and provide techniques for gaining sufficient height to glimpse its extensive mathematical vistas. All this, and more, would be illuminated by references to the wonderful Osaka lectures! Soon afterwards, however, reality took hold, and we began to appreciate that such a task could not be completed to our satisfaction within the timescale available. Simultaneously, we understood that at least as valuable a service could be rendered to conference participants by an invitation to a wider mathematical audience -an invitation to savour the atmosphere and texture of the subject, to consider its geology and history in terms of selected examples and representative literature, to glimpse its exciting future through ongoing projects; and perhaps to locate favourite Osaka lectures within a novel conceptual framework. Thus was born the Toric Tetrahedron TT , which identifies aspects of algebraic, combinatorial, and symplectic geometry as the precursors of toric topology, and symbolises the powerful mathematical bonds between all four areas.The Tetrahedron is the convex hull of these vertex disciplines, and every point has barycentric coordinates that measure the extent of their respective contributions. We introduce the vertices in chronological order (a mere two years separates