Fibrewise completion and unstable Adams spectral sequences
Let R denote the field with p elements, and let \(R_{\infty}X\) denote the R-completion of a space X, constructed cosimplicially. The fibrewise completion of a fibration \(E\to B\) is a space \(BR_{\infty}E\), also constructed cosimplicially, which is a pullback of B and \(R_{\infty}E\) over \(R_{\infty}B\). For each space Y over B, a relative unstable Adams spectral sequence is constructed which has homologically identifiable \(E_ 2\)-term and converges to the homotopy groups of the space \(\Gamma (Y,BR_{\infty}E)\) of maps from Y to \(BR_{\infty}E\) over B. This leads to a new proof of the generalized Sullivan conjecture, which states that there is a homotopy equivalence \(R_{\infty}(X^{\pi})\to \Gamma (B\pi,E_{\pi}(R_{\infty}X))\), where X is a finite CW-complex acted on by the group \(\pi =Z/p\), with fixed point set \(X^{\pi}\), and \(E_{\pi}(-)\) is the Borel construction.
scientific article; zbMATH DE number 4124606
scientific article