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

Kathryn Hess and Ran Levi

Algebraic & Geometric Topology 7 (2007) 1699–1765

arXiv: math.AT/0410503


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.

Double loop space, homotopy fiber, cobar construction, Adams–Hilton model, strongly homotopy coalgebra, operad, co-ring
Mathematical Subject Classification 2000
Primary: 55P35, 16W30
Secondary: 18D50, 18G55, 55U10, 57T05, 57T25
Received: 26 April 2007
Accepted: 2 October 2007
Published: 18 December 2007
Kathryn Hess
Institut de géométrie, algèbre et topologie (IGAT)
École Polytechnique Fédérale de Lausanne
CH-1015 Lausanne
Ran Levi
Department of Mathematical Sciences
King’s College
University of Aberdeen
Aberdeen AB24 3UE