Fibrewise completion and unstable Adams spectral sequences
type: Item
instance of: scholarly article
title: Fibrewise completion and unstable Adams spectral sequences
zbMATH Open document ID: 0686.55014
DOI: 10.1007/BF02765891
author: William Dwyer
author: Haynes Miller
author: Joseph Neisendorfer
published in: Israel Journal of Mathematics
publication date: 1989-01-01T00:00:00Z
review text: 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.
reviewed by: Donald M. Davis
Mathematics Subject Classification ID: 55T15
Mathematics Subject Classification ID: 55P60
zbMATH DE Number: 4124606
zbMATH Keywords: R-completion
zbMATH Keywords: homotopy groups
zbMATH Keywords: generalized Sullivan conjecture
zbMATH Keywords: Borel construction
MaRDI profile type: Publication
cites work: Homotopy spectral sequences and obstructions
cites work: Homotopy limits, completions and localizations
cites work: Q3790321
cites work: Q3800748
cites work: The Sullivan conjecture on maps from classifying spaces
cites work: Cosimplicial Homotopies
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description: scientific article; zbMATH DE number 4124606