## Algebraic & Geometric Topology

### Iterated bar complexes of $E$–infinity algebras and homology theories

Benoit Fresse

#### Abstract

We proved in a previous article that the bar complex of an $E∞$–algebra inherits a natural $E∞$–algebra structure. As a consequence, a well-defined iterated bar construction $Bn(A)$ can be associated to any algebra over an $E∞$–operad. In the case of a commutative algebra $A$, our iterated bar construction reduces to the standard iterated bar complex of $A$.

The first purpose of this paper is to give a direct effective definition of the iterated bar complexes of $E∞$–algebras. We use this effective definition to prove that the $n$–fold bar construction admits an extension to categories of algebras over $En$–operads.

Then we prove that the $n$–fold bar complex determines the homology theory associated to the category of algebras over an $En$–operad. In the case $n=∞$, we obtain an isomorphism between the homology of an infinite bar construction and the usual $Γ$–homology with trivial coefficients.

#### Article information

Source
Algebr. Geom. Topol., Volume 11, Number 2 (2011), 747-838.

Dates
Accepted: 17 December 2010
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715210

Digital Object Identifier
doi:10.2140/agt.2011.11.747

Mathematical Reviews number (MathSciNet)
MR2782544

Zentralblatt MATH identifier
1238.57034

#### Citation

Fresse, Benoit. Iterated bar complexes of $E$–infinity algebras and homology theories. Algebr. Geom. Topol. 11 (2011), no. 2, 747--838. doi:10.2140/agt.2011.11.747. https://projecteuclid.org/euclid.agt/1513715210

#### References

• S T Ahearn, N J Kuhn, Product and other fine structure in polynomial resolutions of mapping spaces, Algebr. Geom. Topol. 2 (2002) 591–647
• G Arone, M Kankaanrinta, The homology of certain subgroups of the symmetric group with coefficients in $\mathscr L\mathit{ie}(n)$, J. Pure Appl. Algebra 127 (1998) 1–14
• G Arone, M Mahowald, The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres, Invent. Math. 135 (1999) 743–788
• M G Barratt, P J Eccles, $\Gamma \sp{+}$–structures. I. A free group functor for stable homotopy theory, Topology 13 (1974) 25–45
• M Basterra, M A Mandell, Homology of $E_n$ ring spectra and iterated $\mathit{THH}$
• M A Batanin, Symmetrisation of $n$–operads and compactification of real configuration spaces, Adv. Math. 211 (2007) 684–725
• M A Batanin, The Eckmann–Hilton argument and higher operads, Adv. Math. 217 (2008) 334–385
• C Berger, Opérades cellulaires et espaces de lacets itérés, Ann. Inst. Fourier $($Grenoble$)$ 46 (1996) 1125–1157
• C Berger, B Fresse, Combinatorial operad actions on cochains, Math. Proc. Cambridge Philos. Soc. 137 (2004) 135–174
• S Betley, J Słomińska, New approach to the groups $H\sb *(\Sigma\sb n,{\rm Lie}\sb n)$ by the homology theory of the category of functors, J. Pure Appl. Algebra 161 (2001) 31–43
• J M Boardman, R M Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Math. 347, Springer, Berlin (1973)
• N Bourbaki, Éléments de mathématique. Fasc. XXXVII. Groupes et algèbres de Lie. Chapitre II: Algèbres de Lie libres. Chapitre III: Groupes de Lie, Act. Sci. et Ind. 1349, Hermann, Paris (1972)
• H Cartan, J C Moore, R Thom, J-P Serre, Algèbres d'Eilenberg–Mac Lane et homotopie, 2ème éd edition, Séminaire Henri Cartan de l'Ecole Normale Supérieure, 1954/1955, Secrétariat math. (1956)
• D Chataur, M Livernet, Adem–Cartan operads, Comm. Algebra 33 (2005) 4337–4360
• F R Cohen, T J Lada, J P May, The homology of iterated loop spaces, Lecture Notes in Math. 533, Springer, Berlin (1976)
• B Fresse, Koszul duality complexes for the cohomology of iterated loop spaces of spheres
• B Fresse, Koszul duality of $E_n$–operads, to appear in Selecta Math.
• B Fresse, La catégorie des arbres élagués de Batanin est de Koszul
• B Fresse, On the homotopy of simplicial algebras over an operad, Trans. Amer. Math. Soc. 352 (2000) 4113–4141
• B Fresse, Koszul duality of operads and homology of partition posets, from: “Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic $K$–theory”, (P Goerss, S Priddy, editors), Contemp. Math. 346, Amer. Math. Soc. (2004) 115–215
• B Fresse, Théorie des opérades de Koszul et homologie des algèbres de Poisson, Ann. Math. Blaise Pascal 13 (2006) 237–312
• B Fresse, Iterated bar complexes and the poset of pruned trees, Preprint (2008) Available at \setbox0\makeatletter\@url http://math.univ-lille1.fr/~fresse/IteratedBarAppendix.html {\unhbox0
• B Fresse, Modules over operads and functors, Lecture Notes in Math. 1967, Springer, Berlin (2009)
• B Fresse, The bar complex of an $E$–infinity algebra, Adv. Math. 223 (2010) 2049–2096
• E Getzler, J Jones, Operads, homotopy algebra and iterated integrals for double loop spaces
• G Ginot, Homologie et modèle minimal des algèbres de Gerstenhaber, Ann. Math. Blaise Pascal 11 (2004) 95–127
• V Ginzburg, M Kapranov, Koszul duality for operads, Duke Math. J. 76 (1994) 203–272
• D K Harrison, Commutative algebras and cohomology, Trans. Amer. Math. Soc. 104 (1962) 191–204
• V A Hinich, V V Schechtman, On homotopy limit of homotopy algebras, from: “$K$–theory, arithmetic and geometry (Moscow, 1984–1986)”, (Y I Manin, editor), Lecture Notes in Math. 1289, Springer, Berlin (1987) 240–264
• M Hovey, Model categories, Math. Surveys and Monogr. 63, Amer. Math. Soc. (1999)
• M Kontsevich, Operads and motives in deformation quantization, Lett. Math. Phys. 48 (1999) 35–72 Moshé Flato (1937–1998)
• M Livernet, B Richter, An interpretation of $E_n$–homology as functor homology, to appear in Math. Z.
• J-L Loday, Generalized bialgebras and triples of operads, Astérisque 320, Soc. Math. France (2008)
• S Mac Lane, Homology, Grund. der math. Wissenschaften 114, Academic Press, New York (1963)
• M A Mandell, Topological André–Quillen cohomology and $E\sb \infty$ André–Quillen cohomology, Adv. Math. 177 (2003) 227–279
• M Markl, Distributive laws and Koszulness, Ann. Inst. Fourier $($Grenoble$)$ 46 (1996) 307–323
• M Markl, Models for operads, Comm. Algebra 24 (1996) 1471–1500
• M Markl, S Shnider, J Stasheff, Operads in algebra, topology and physics, Math. Surveys and Monogr. 96, Amer. Math. Soc. (2002)
• J P May, The geometry of iterated loop spaces, Lectures Notes in Math. 271, Springer, Berlin (1972)
• J E McClure, J H Smith, Multivariable cochain operations and little $n$–cubes, J. Amer. Math. Soc. 16 (2003) 681–704
• C Reutenauer, Free Lie algebras, London Math. Soc. Monogr., New Series 7, Oxford Science Publ., The Clarendon Press, Oxford Univ. Press, New York (1993)
• B Richter, A lower bound for coherences on the Brown–Peterson spectrum, Algebr. Geom. Topol. 6 (2006) 287–308
• A Robinson, Gamma homology, Lie representations and $E\sb \infty$ multiplications, Invent. Math. 152 (2003) 331–348
• V A Smirnov, On the chain complex of an iterated loop space, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989) 1108–1119, 1135–1136
• V A Smirnov, The homology of iterated loop spaces, Forum Math. 14 (2002) 345–381 With an appendix by F Sergeraert
• C R Stover, The equivalence of certain categories of twisted Lie and Hopf algebras over a commutative ring, J. Pure Appl. Algebra 86 (1993) 289–326