Covariant and equivariant formality theorems

We give a proof of Kontsevich's formality theorem for a general manifold using Fedosov resolutions of algebras of polydifferential operators and polyvector fields. The main advantage of our construction of the formality quasi-isomorphism is that it is based on the use of covariant tensors unlike Kontsevich's original proof, which is based on ∞-jets of polydifferential operators and polyvector fields. Using our construction we prove that if a group G acts smoothly on a manifold M and M admits a G-invariant affine connection then there exists a G-equivariant quasi-isomorphism of formality. This result implies that if a manifold M is equipped with a smooth action of a finite or compact group G or equipped with a free action of a Lie group G then M admits a G-equivariant formality quasi-isomorphism. In particular, this gives a solution of the deformation quantization problem for an arbitrary Poisson orbifold.     

On the equivariant formality of Kähler manifolds with torus group actions

We consider simply connected compact Kähler manifolds which have a holomorphic action of a torus group. We use the existing equivariant models for rational homotopy to show that these spaces satisfy an equivariant formality condition over the complex numbers.     

On spaces of functions between classifying spaces

In this paper the results of Dwyer and Zabrodsky [DZ] are extended by showing that ifL is a compact Lie group andG is either ap-group or a torus, then every mapf:BG →BL is homotopic to one induced by a homomorphismφ :G →L, and two such induced maps are homotopic if and only if the corresponding homomophisms are conjugate. Several other results related to maps between classifying spaces, completions, and fibrations are also deduced.     

On the homotopy type of classifying spaces

Let βC (resp. BC) be the Milnor (resp. Milgram) classifying space of a topological category C as defined by G. Segal [13]. We show that βC and BC are homotopy equivalent if the inclusion of the degenerate simplices into the space of all simplices is a cofibration.     

On string topology of classifying spaces

Let G be a compact Lie group. By work of Chataur and Menichi, the homology of the space of free loops in the classifying space of G is known to be the value on the circle in a homological conformal field theory. This means in particular that it admits operations parameterized by homology classes of classifying spaces of diffeomorphism groups of surfaces. Here we present a radical extension of this result, giving a new construction in which diffeomorphisms are replaced with homotopy equivalences, and surfaces with boundary are replaced with arbitrary spaces homotopy equivalent to finite graphs. The result is a novel kind of field theory which is related to both the diffeomorphism groups of surfaces and the automorphism groups of free groups with boundaries. Our work shows that the algebraic structures in string topology of classifying spaces can be brought into line with, and in fact far exceed, those available in string topology of manifolds. For simplicity, we restrict to the characteristic 2 case. The generalization to arbitrary characteristic will be addressed in a subsequent paper.     

On the homotopy groups of p-completed classifying spaces

Among the generalizations of Serre's theorem on the homotopy groups of a finite complex we isolate the one proposed by Dwyer and Wilkerson. Even though the spaces they consider must be 2-connected, we show that it can be used to both recover known results and obtain new theorems about p-completed classifying spaces. All three authors are partially supported by MEC grant MTM2004-06686. The third author is supported by the program Ramón y Cajal, MEC, Spain, and thanks the CIB (Centre Interfacultaire Bernoulli), EPFL, Lausanne for its hospitality.     

Perturbation of equivariant moduli spaces

Partially supported by NSERC Grant A4000 and the Max Planck Institut für Mathematik     

On a theorem of Jaworowski on locally equivariant contractible spaces

Ancel's method of fiberwise trivial relations is applied to the problem of characterization of absolute equivariant extensors. We obtain a generalization of Jaworowski's theorem on characterization of equivariant extensors lying in to the case when the space is infinite-dimensional, has infinitely many orbit types and the acting compact group is not necessarily a Lie group.

     

Constructing modular classifying spaces

Using the homotopy limit construction over a certain small category, we construct spaces whose modp cohomology algebras are the rings of invariants of some unitary reflection groups of order divisible byp.     

Trunks and classifying spaces

Trunks are objects loosely analogous to categories. Like a category, a trunk has vertices and edges (analogous to objects and morphisms), but instead of composition (which can be regarded as given by preferred triangles of morphisms) it has preferred squares of edges. A trunk has a natural cubical nerve, analogous to the simplicial nerve of a category. The classifying space of the trunk is the realisation of this nerve. Trunks are important in the theory of racks [8]. A rackX gives rise to a trunkT (X) which has a single vertex and the setX as set of edges. Therack space BX ofX is the realisation of the nerveNT (X) ofT(X). The connection between the nerve of a trunk and the usual (cubical) nerve of a category determines in particular a natural mapBX BAs(X) whereBAs(X) is the classifying space of the associated group ofX. There is an extension to give a classifying space for an augmented rack, which has a natural map to the loop space of the Brown-Higgins classifying space of the associated crossed module [8, Section 2] and [3].The theory can be used to define invariants of knots and links since any invariant of the rack space of the fundamental rack of a knot or link is ipso facto an invariant of the knot or link.