A propos de conjectures de Serre et Sullivan (Q1057539)

We first give a new proof of a conjecture of J.-P. Serre on the homotopy of finite complexes, which was recently proved by C. A. McGibbon and J. Neisendorfer. The emphasis is on a property of the mod 2 homology of certain spaces: their ''quasi-boundedness'' as right modules over the Steenrod algebra. This property is preserved when one goes from a simply connected space to its loop space and also when one takes a covering of an H-space. Then we show that this notion of quasi-boundedness simplifies H. Miller's proof of D. Sullivan's conjecture on the contractibility of the space of pointed maps from the classifying space of the group \({\mathbb{Z}}/2\) into a finite complex.