Algebra/Topology Seminar: On the E-theory of Configuration Spaces

Speaker: Lukas Brantner (University of Oxford)

Abstract: Given natural numbers n and h, one can investigate the Morava K- and E-theory of n-fold loop spaces at height h. Partial computations have been carried out by Langsetmo, Ravenel, Tamaki, and Yamaguchi, but their techniques either rely on phenomena specific to height h=1 or become increasingly intractable as the number n of loops grows large.

In joint work with Knudsen and Hahn, we introduce a new computational technique whose difficulty is uniform in n. More precisely, we exhibit a spectral sequence converging to the E-theory of configuration spaces in n-manifolds and, in good cases, identify its E_2 page as the purely algebraic Chevalley-Eilenberg complex of a Hecke Lie algebra. 

We illustrate the tractability of our approach by computing several new E-, K-, and F_p-homology groups of configuration spaces.