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 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 formality of equivariant classifying spaces

Partially supported by NSF MCS 7701623     

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 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.     

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.     

Rational equivalences between classifying spaces

The paper contains a homotopy classification of rational equivalences between classifying spaces of compact connected Lie groups with an application to genus sets of such spaces. Received: 22 June 1992; in final form: 30 August 1993/ Published online: 6 August 2002     

Kernels of maps between classifying spaces

For homomorphisms between groups, one can divide out the kernel to get an injection. Here, we develop a notion of kernels for maps between classifying spaces of compact Lie groups. We show that the kernel is a normal subgroup in a modified sense and prove a generalization of a theorem of Quillen, namely, a mapf:BG→BH _p ^∧ is injective, iff the induced map in mod-p cohomology is finite. Moreover, for compact connected Lie groups, every mapf:BG→BH _p ^∧ factors over a quotient ofG in a modified sense and this factorisation is an injection.     

The homotopy Lie algebra of classifying spaces

Let X be a 1-connected CW-complex of finite type and L_x its rational homotopy Lie algebra. In this work, we show that there is a spectral sequence whose E² term is the Lie algebra Ext_ULx(Q, L_x), and which converges to the homotopy Lie algebra of the classifying space B autX. Moreover, some terms of this spectral sequence are related to derivations of L_x and to the Gottlieb group of X.     

Maps between classifying spaces of Kac-Moody groups

We classify up to homotopy the self-maps of the classifying space of any non-affine Kac-Moody group of rank two.