Koszul duality complexes for the cohomology of iterated loop spaces of spheres

by Benoit Fresse
Preprint (2010).


Abstract :

The goal of this article is to make explicit a structured complex computing the cohomology of a profinite completion of the n-fold loop space of a sphere of dimension d less than n. Our construction involves: the free complete algebra in one variable associated to any fixed E_n-operad; an element in this free complete algebra determined by a morphism from the operad of L-infinity algebras to an operadic suspension of our E_n-operad. As such, our complex is defined purely algebraically in terms of characteristic structures of E_n-operads. We deduce our main theorem from several results obtained in a previous series of articles - namely: a connection between the cohomology of iterated loop spaces and the cohomology of algebras over E_n-operads, and a Koszul duality result for E_n-operads. We use notably that the Koszul duality of E_n-operads makes appear structure maps of the cochain algebra of spheres.

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(22/1/2010)