Algebraic & Geometric Topology

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

Kathryn Hess and Ran Levi

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Let F denote the homotopy fiber of a map f:KL 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.

Article information

Algebr. Geom. Topol., Volume 7, Number 4 (2007), 1699-1765.

Received: 26 April 2007
Accepted: 2 October 2007
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55P35: Loop spaces 16W30
Secondary: 18D50: Operads [See also 55P48] 18G55: Homotopical algebra 55U10: Simplicial sets and complexes 57T05: Hopf algebras [See also 16T05] 57T25: Homology and cohomology of H-spaces

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


Hess, Kathryn; Levi, Ran. An algebraic model for the loop space homology of a homotopy fiber. Algebr. Geom. Topol. 7 (2007), no. 4, 1699--1765. doi:10.2140/agt.2007.7.1699.

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