×

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

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
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] H J Baues, The cobar construction as a Hopf algebra, Invent. Math. 132 (1998) 467 · Zbl 0912.55015
[2] F R Cohen, J C Moore, J A Neisendorfer, Torsion in homotopy groups, Ann. of Math. \((2)\) 109 (1979) 121 · Zbl 0405.55018
[3] S Eilenberg, J C Moore, Homology and fibrations. I. Coalgebras, cotensor product and its derived functors, Comment. Math. Helv. 40 (1966) 199 · Zbl 0148.43203
[4] Y Félix, S Halperin, J C Thomas, Differential graded algebras in topology, North-Holland (1995) 829 · Zbl 0868.55016
[5] V K A M Gugenheim, H J Munkholm, On the extended functoriality of Tor and Cotor, J. Pure Appl. Algebra 4 (1974) 9 · Zbl 0358.18015
[6] K Hess, P E Parent, J Scott, Co-rings over operads characterize morphisms
[7] K Hess, P E Parent, J Scott, A chain coalgebra model for the James map, Homology, Homotopy Appl. 9 (2007) · Zbl 1131.55002
[8] K Hess, P E Parent, J Scott, A Tonks, A canonical enriched Adams-Hilton model for simplicial sets, Adv. Math. 207 (2006) 847 · Zbl 1112.55010
[9] M Markl, Operads and PROPs · Zbl 1211.18007
[10] M Markl, S Shnider, J Stasheff, Operads in algebra, topology and physics, Mathematical Surveys and Monographs 96, American Mathematical Society (2002) · Zbl 1017.18001
[11] R J Milgram, Iterated loop spaces, Ann. of Math. \((2)\) 84 (1966) 386 · Zbl 0145.19901
[12] H R Miller, A localization theorem in homological algebra, Math. Proc. Cambridge Philos. Soc. 84 (1978) 73 · Zbl 0388.55013
[13] J W Milnor, J C Moore, On the structure of Hopf algebras, Ann. of Math. \((2)\) 81 (1965) 211 · Zbl 0163.28202
[14] B Ndombol, J C Thomas, On the cohomology algebra of free loop spaces, Topology 41 (2002) 85 · Zbl 1011.16008
[15] J Stasheff, S Halperin, Differential algebra in its own rite, Mat. Inst. (1970) · Zbl 0224.55027
[16] R H Szczarba, The homology of twisted cartesian products, Trans. Amer. Math. Soc. 100 (1961) 197 · Zbl 0108.35901
[17] D Tanré, Homotopie rationnelle: modèles de Chen, Quillen, Sullivan, Lecture Notes in Mathematics 1025, Springer (1983) · Zbl 0539.55001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.