Completion of Dominant K-theory
Cattell, Stephen E.
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Nitu Kitchloo generalized equivariant K-theory to include non-compact Kac-Moody groups, calling the new theory Dominant K-theory. For a non-compact Kac-Moody group there are no non-trivial finite dimensional dominant representations, so there is no notion of a augmentation ideal, and the spaces we can work with have to have compact isotropy groups. To resolve these we complete locally, at the compact subgroups. We show that there is a 1 dimensional representation in the dominant representation ring such that when inverted we recover the regular representation ring. This shows that if H is a compact subgroup of a Kac-Moody group K(A), the completion of the Dominant K-theory of a H-space X is identical to the equivariant K-theory completed at the augmentation ideal. This is the local information. To glue this together we find a new spectrum whose cohomology theory is isomorphic to K ∗ (X × K EK). This enables us to use compute K ∗ (X × K EK) using a skeletal filtration as we now know the E 1 page of this spectral sequence is formed out of known algebras.