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

Hess, Kathryn; Levi, Ran

doi:10.2140/agt.2007.7.1699

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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

DOI: 10.2140/agt.2007.7.1699

arXiv: math.AT/0410503

Abstract

Let $F$ denote the homotopy fiber of a map $f : K \to L$ 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 $G F$ , 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 $D C S H$ 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.

Keywords

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

References

Publication

Received: 26 April 2007

Accepted: 2 October 2007

Published: 18 December 2007

Authors

Kathryn Hess
Institut de géométrie, algèbre et topologie (IGAT) École Polytechnique Fédérale de Lausanne CH-1015 Lausanne Switzerland
http://sma.epfl.ch/~hessbell/
Ran Levi
Department of Mathematical Sciences King’s College University of Aberdeen Aberdeen AB24 3UE UK
http://www.maths.abdn.ac.uk/~ran/

Volume 7, issue 4 (2007)