A remark on maps between classifying spaces of compact Lie groups (original) (raw)

HOMOTOPY THEORY OF CLASSIFYING SPACES OF COMPACT LIE GROUPS

1994

The basic problem of homotopy theory is to classify spaces and maps between spaces, up to homotopy, by means of invariants like cohomology. In the last decade some striking progress has been made with this problem when the spaces involved are classifying spaces of compact Lie groups. For example, it has been shown, for G connected and simple, that if two self maps of BG agree in rational cohomology then they are homotopic. It has also been shown that if a space X has the same mod p cohomology, cup product, and Steenrod operations as a classifying space BG then (at least if p is odd and G is a classical group) X is actually homotopy equivalent to BG after mod p completion. Similar methods have also been used to obtain new results on Steenrod's problem of constructing spaces with a given polynomial cohomology ring. The aim of this paper is to describe these results and the methods used to prove them.

On the Homomorphisms of the Lie Groups and

Abstract and Applied Analysis, 2013

We first construct all the homomorphisms from the Heisenberg group to the 3-sphere. Also, defining a topology on these homomorphisms, we regard the set of these homomorphisms as a topological space. Next, using the kernels of homomorphisms, we define an equivalence relation on this topological space. We finally show that the quotient space is a topological group which is isomorphic to S 1 .

Classifying spaces for 1–truncated compact Lie groups

Algebraic & Geometric Topology, 2018

A 1-truncated compact Lie group is any extension of a finite group by a torus. In this note we compute the homotopy types of Map * (BG, BH), Map(BG, BH), and Map(EG, BGH) for compact Lie groups G and H with H 1-truncated, showing that they are computed entirely in terms of spaces of homomorphisms from G to H. These results generalize the well-known case when H is finite, and the case of H compact abelian due to Lashof, May, and Segal.

An algebraic description of the elliptic cohomology of classifying spaces

Journal of Pure and Applied Algebra, 1998

Let G be a finite group of order (GI odd and let 6Y~*(-)~?@[l/lGl] denote elliptic cohomology tensored by Z[l/lGl]. Then we give a description of &Y*(E(N,G) x ,vX) @ Z[l/lGl], where N is a normal subgroup of G, E(N, G) is the universal N-free G space and X is any finite G-CW complex where N acts freely. We explain how some of the results of Hopkins-Kuhn-Ravenel can be recovered for our results.

A 333-dimensional nonabelian cohomology of groups with applications to homotopy classification of continuous maps

Canadian Journal of Mathematics, 1991

The general problem of what should be a non-abelian cohomology, what is it supposed to do, and what should be the coefficients, form a set of interesting questions which has been around for a long time. In the particular setting of groups, a comprehensible and well motivated cohomology theory has been so far stated in dimensions ≤ 2, the coefficients for being crossed modules. The main effort to define an appropriate for groups has been done by Dedecker [16] and Van Deuren [40]; they studied the obstruction to lifting non-abelian 2-cocycles and concluded with first approach for , which requires “super crossed groups” as coefficients. However, as Dedecker said “some polishing work remains necessary” for his cohomology.

Topological K–(co)homology of classifying spaces of discrete groups

Algebraic & Geometric Topology, 2013

Let G be a discrete group. We give methods to compute for a generalized (co-)homology theory its values on the Borel construction EG × G X of a proper G-CW -complex X satisfying certain finiteness conditions. In particular we give formulas computing the topological K-(co)homology K * (BG) and K * (BG) up to finite abelian torsion groups. They apply for instance to arithmetic groups, word hyperbolic groups, mapping class groups and discrete cocompact subgroups of almost connected Lie groups. For finite groups G these formulas are sharp. The main new tools we use for the K-theory calculation are a Cocompletion Theorem and Equivariant Universal Coefficient Theorems which are of independent interest. In the case where G is a finite group these theorems reduce to well-known results of Greenlees and Bökstedt.

On the Homomorphisms of the Lie Groups SU(2)SU(2)SU(2) and S3{S}^{3}S3

Abstract and Applied Analysis, 2013

We first construct all the homomorphisms from the Heisenberg group to the 3-sphere. Also, defining a topology on these homomorphisms, we regard the set of these homomorphisms as a topological space. Next, using the kernels of homomorphisms, we define an equivalence relation on this topological space. We finally show that the quotient space is a topological group which is isomorphic to S 1 .

Homotopy classification of maps into homogeneous spaces

2008

We give an alternative to Postnikov's homotopy classification of maps from 3-dimensional CW-complexes to homogeneous spaces G/H of Lie groups. It describes homotopy classes in terms of lifts to the group G and is suitable for extending the notion of homotopy to Sobolev maps. This is required for applications to variational problems of mathematical physics.

Cohomology of Lie 222-groups

L’Enseignement Mathématique, 2009

We study the cohomology of (strict) Lie 2-groups. We obtain an explicit Bott-Shulman type map in the case of a Lie 2-group corresponding to the crossed module A 1. The cohomology of the Lie 2-groups corresponding to the universal crossed modules G Aut(G) and G Aut (G) is the abutment of a spectral sequence involving the cohomology of GL(n Z) and SL(n Z). When the dimension of the center of G is less than 3, we compute these cohomology groups explicitly. We also compute the cohomology of the Lie 2-group corresponding to a crossed module G i H for which Ker(i) is compact and Coker(i) is connected, simply connected and compact, and we apply the result to the string 2-group.