An algebraic model for the loop space homology of a homotopy fiber

Hess, Kathryn¹ ; Levi, Ran²

¹Institut de géométrie, algèbre et topologie (IGAT), École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland
²Department of Mathematical Sciences, King’s College, University of Aberdeen, Aberdeen AB24 3UE, UK

Algebraic and Geometric Topology, Tome 7 (2007) no. 4, pp. 1699-1765

Résumé

Let F denote the homotopy fiber of a map f : K → L of 2–reduced simplicial sets. Using as input data the strongly homotopy coalgebra structure of the chain complexes of K and L, we construct a small, explicit chain algebra, the homology of which is isomorphic as a graded algebra to the homology of GF, the simplicial (Kan) loop group on F. To construct this model, we develop machinery for modeling the homotopy fiber of a morphism of chain Hopf algebras.

Essential to our construction is a generalization of the operadic description of the category DCSH of chain coalgebras and of strongly homotopy coalgebra maps given by Hess, Parent and Scott [Co-rings over operads characterize morphisms arxiv:math.AT/0505559] to strongly homotopy morphisms of comodules over Hopf algebras. This operadic description is expressed in terms of a general theory of monoidal structures in categories with morphism sets parametrized by co-rings, which we elaborate here.

Détail

DOI : 10.2140/agt.2007.7.1699

Keywords: Double loop space, homotopy fiber, cobar construction, Adams–Hilton model, strongly homotopy coalgebra, operad, co-ring

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