Cobordisme complexe des espaces profinis et foncteur T de Lannes
This volume represents a continuation of the paper F.-X. Dehon, J. Lannes, Sur les espaces fonctionnels dont la source est le classifiant d’un groupe de Lie compact commutative, Publ. Math. Inst. Hautes Ėtudes Sci. 89 (1999), 127-177, and also of the paper N. Kuhn, M. Winstead, On the torsion in the cohomology of certain mapping spaces, Topology 39 (1996), 875-881. Its aim is the following: let p be a prime, let π be a compact commutative Lie group, and let MU denote the spectrum representing the complex cobordism. We take a space X such that its cohomology with coefficients in the ring of p-adic integers has no torsion.
The authors consider the functional space consisting of continuous mappings whose source is the classifying space Bπ of the group π and the target is the profinite p-completion of the space X. They prove that the continuous MU-cohomology completed at the prime p of this functional space can be identified with the image under a functor TBπ of the p-completed MU-cohomology of the target space. The functor TBπ is an analogy of the Lannes’ functor T, where π is a cyclic group of order p. Let us mention that the authors work in the simplicial setting. This highly interesting text is intended primarily for specialists. In order to make the approach to the results easier, the authors attached an appendix with some explanations.