An algebraic model for the loop space homology of a homotopy fiber. (English) Zbl 1182.55008

Hess, Kathryn; Levi, Ran

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

Summary: Let \(F\) denote the homotopy fiber of a map \(f:K\rightarrow L\) 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 \(\mathbf{DCSH}\) of chain coalgebras and of strongly homotopy coalgebra maps given by K. Hess, P.-E. Parent and J. 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.

MSC:

55P35	Loop spaces
55U35	Abstract and axiomatic homotopy theory in algebraic topology
57T30	Bar and cobar constructions
16T05	Hopf algebras and their applications
18D50	Operads (MSC2010)
55U10	Simplicial sets and complexes in algebraic topology
57T35	Applications of Eilenberg-Moore spectral sequences

Keywords:

double loop space; homotopy fiber; cobar construction; Adams-Hilton model; strongly homotopy coalgebra; operad; co-ring

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