Algebraic K-theory and configuration spaces

Algebraic K - Theory and Configuration Spaces

Its importance lies in the fact that the right hand side of the above equivalence (which wewill briefly refer to as a configuration space model) is a much smaller space than the function space on the left, and thus is much easier to deal with. The second author noted that in the configuration space model, one can replace the X" by spaces like BGln(R). For there is an obvious ~n action on BGln(R...

The Spaces That Define Algebraic K-theory

We characterize spaces W such that the W -nullification functor PW , applied to any BGL(R), gives BGL(R) . Let X be a pointed space and X the Quillen plus-construction on X with respect to the maximal perfect subgroup of π1X. If W is a pointed CW-complex, let PW (X) denote the W -nullification of X. (Recall that PWX is up to homotopy the initial space Y under X which is W -null in the sense tha...

Relative Hyperbolicity, Classifying Spaces, and Lower Algebraic K-theory

For Γ a relatively hyperbolic group, we construct a model for the universal space among Γ-spaces with isotropy on the family VC of virtually cyclic subgroups of Γ. We provide a recipe for identifying the maximal infinite virtually cyclic subgroups of Coxeter groups which are lattices in O+(n, 1) = Isom(Hn). We use the information we obtain to explicitly compute the lower algebraic K-theory of t...

Algebraic K-theory of spaces stratified fibered over hyperbolic orbifolds.

Among other results, we rationally calculate the algebraic K-theory of any discrete cocompact subgroup of a Lie group G, where G is either O(n, 1), U(n, 1), Sp(n, 1), or F(4), in terms of the homology of the double coset space Gamma\G/K, where K is a maximal cocompact subgroup of G. We obtain the formula K(n)(ZGamma) [unk] [unk] congruent with [unk](i=0) (infinity)H(i)(Gamma\G/K; [unk](n-i)), w...

Configuration Spaces in Algebraic Topology: Lecture 2