I'm writing a thesis that runs an AHSS for twisted K-theory over an Eilenberg-MacLane space, namely an infinite-dimensional CW-complex.
In Atiyahs works Twisted K-Theory and Twisted K-Theory and cohomology, I could not find any reference on how the twisted AHSS would behave for infinite-dimensional CW-complexes. Neither could I find any reference in his foundational work Vector bundles and homogeneous spaces, where Atiyah introduces the untwisted version of the AHSS. Is there any published material defining a twisted AHSS infinite-dimensional CW-complex? I do not need convergence of the sequence (that might be affected), only that the calculations of the successive pages work.
I've also tried theorem 15.7. of Switzers Algebraic Topology, but this theorem gives the sequence for generalized homology theories, not generalized cohomology theories, such as twisted K-Theory. At the moment, I'm trying to extract the information from Whitehead's Elements of Homotopy Theory, Theorem 3.6 of chapter XIII, but not with too much success.