I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the set of equivalence classes of Poisson structures on X modulo diffeomorphisms. In fact, a more general statement is proven ("Formality conjecture"), relating the Lie superalgebra of polyvector fields on X and the Hochschild complex of the algebra of functions on X. Coefficients in explicit formulas for the deformed product can be interpreted as correlators in a topological open string theory, although I do not use explicitly the language of functional integrals. One of corollaries is a justification of the orbit method in the representation theory.

Deformation quantization of Poisson manifolds, I

'Operads are powerful tools, and this is the book in which to read about them' - ""Bulletin of the London Mathematical Society"". Operads are mathematical devices that describe algebraic structures of many varieties and in various categories. Operads are particularly important in categories with a good notion of 'homotopy', where they play a key role in organizing hierarchies of higher homotopies. Significant examples from algebraic topology first appeared in the sixties, although the formal definition and appropriate generality were not forged until the seventies. In the nineties, a renaissance and further development of the theory were inspired by the discovery of new relationships with graph cohomology, representation theory, algebraic geometry, derived categories, Morse theory, symplectic and contact geometry, combinatorics, knot theory, moduli spaces, cyclic cohomology, and, last but not least, theoretical physics, especially string field theory and deformation quantization. The book contains a detailed and comprehensive historical introduction describing the development of operad theory from the initial period when it was a rather specialized tool in homotopy theory to the present when operads have a wide range of applications in algebra, topology, and mathematical physics. Many results and applications currently scattered in the literature are brought together here along with new results and insights. The basic definitions and constructions are carefully explained and include many details not found in any of the standard literature.

Operads in algebra, topology, and physics

These notes provide an introduction to the theory of localization for triangulated categories Localization is a machinery to formally invert morphisms in a category We explain this formalism in some detail and we show how it is applied to triangulated categories

Localization theory for triangulated categories

In present paper we develop the deformation theory of operads and algebras over operads. Free resolutions (constructed via Boardman-Vogt approach) are used in order to describe formal moduli spaces of deformations. We apply the general theory to the proof of Deligne's conjecture. The latter says that the Hochschild complex of an associative algebra carries a canonical structure of a dg-algebra over the chain operad of the little discs operad. In the course of the proof we construct an operad of geometric nature which acts on the Hochschild complex. It seems to be different from the brace operad (the latter was used in the previous approaches to the Deligne's conjecture). It follows from our results that the Grothendieck-Teichm\"uller group acts (homotopically) on the moduli space of structures of 2-algebras on the Hochschild complex. In the Appendix we develop a theory of piecewise algebraic chains and forms. It is suitable for real semialgebraic manifolds with corners (like Fulton-Macpherson compactifications of the configuration spaces of points).

Deformations of algebras over operads and Deligne's conjecture

The purpose of this paper is to complete Getzler-Jones’ proof of Deligne’s Conjecture, thereby establishing an explicit relationship between the geometry of configurations of points in the plane and the Hochschild complex of an associative algebra. More concretely, it is shown that the B ∞-operad, which is generated by multilinear operations known to act on the Hochschild complex, is a quotient of a certain operad associated to the compactified configuration spaces. Different notions of homotopy Gerstenhaber algebras are discussed: One of them is a B ∞-algebra, another, called a homotopy G-algebra, is a particular case of a B ∞-algebra, the others, a G ∞-algebra, an Ē1-algebra, and a weak G ∞-algebra, arise from the geometry of configuration spaces. Corrections to the paper of Kimura, Zuckerman, and the author related to the use of a nonextant notion of a homotopy Gerstenhaber algebra are made.

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Homotopy Gerstenhaber algebras

This paper has two goals. The first (see Theorem 2.15 and Remark 2.16(a)) is to construct a small Em chain operad S which acts naturally on the normalized cochains S*X of a topological space X. This is of interest in view of a theorem of Mandell [15, page 44] which states that if (9 is any E, chain operad over F, (the algebraic closure of the field with p elements) which acts naturally on S*X 0 F,, then the homotopy category of connected p-complete nilpotent spaces of finite type

Multivariable cochain operations and little -cubes

In this paper we show that if a cosimplicial space has a certain kind of combinatorial structure then its total space has an action of an operad weakly equivalent to the little n -cubes operad. Our results are also valid for cosimplicial spectra.

COSIMPLICIAL OBJECTS AND LITTLE n-CUBES, I

This is a revision of a paper first posted June 4, 2001. It will appear in the Journal of the AMS. 
In this paper we construct a small $E_\infty$ chain operad $\S$ which acts naturally on the normalized cochains $S^*X$ of a topological space. We also construct, for each $n$, a suboperad $\S_n$ which is quasi-isomorphic to the normalized singular chains of the little $n$-cubes operad. The case $n=2$ leads to a substantial simplification of our earlier proof of Deligne's Hochschild cohomology conjecture.

Multivariable cochain operations and little n-cubes

In this paper we show that the cohomology of a connected CW-complex is periodic if and only if it is the base space of a spherical fibration with total space that is homotopically finite dimensional. As applications we characterize those discrete groups that act freely and properly on In x Sm; we construct nonstandard free actions of rank-two simple groups on finite complexes Y Sn x Sm; and we prove that a finite p-group P acts freely on such a complex if and only if it does not contain a subgroup isomorphic to (Z/p)3.

Periodic Complexes and Group Actions

In this paper we show that the cohomology of a connected CW complex is periodic if and only if it is the base space of an orientable spherical fibration with total space that is homotopically finite dimensional. As applications we characterize those discrete groups that act freely and properly on a cartesian product of euclidean space and a sphere; we construct non-standard free actions of rank two simple groups on finite complexes Y homotopy equivalent to a product of two spheres and we prove that a finite p-group P acts freely on such a complex if and only if it does not contain a subgroup isomorphic to Z/p X Z/p X Z/p.

Jeffrey H. Smith

Papers

Multivariable cochain operations and little -cubes

COSIMPLICIAL OBJECTS AND LITTLE n-CUBES, I

Multivariable cochain operations and little n-cubes

Periodic Complexes and Group Actions

Periodic Complexes and Group Actions