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Homotopy algebras are homotopy algebras. (English) Zbl 1067.55011
The concept of homotopy invariant algebraic structures for topological spaces formulated by J. M. Boardman and R. M. Vogt [Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Mathematics 347. Berlin-Heidelberg-New York: Springer-Verlag. X, 257 p. DM 22.00; $ 09.10 (1973; Zbl 0285.55012)] is transferred to the algebra of chain complexes. It is shown that a certain concept of strongly homotopy algebra which is broad enough to include both classical examples such a strongly homotopy associative algebras, strongly homotopy associative commutative algebras and more recent concepts arising in quantum field theory is homotopy invariant.

55U35 Abstract and axiomatic homotopy theory in algebraic topology
55U15 Chain complexes in algebraic topology
18G55 Nonabelian homotopical algebra (MSC2010)
Zbl 0285.55012
Full Text: DOI
[1] Baez J. C., Adv. in Math. 135 pp 145– (1998)
[2] Berger C. and Moerdijk I.: Axiomatic homotopy theory for operads. Preprint math.AT/ 0206094, June 2002 · Zbl 1041.18011
[3] Boardman J. M. and Vogt R. M.: Homotopy Invariant Algebraic Structures on Topological Spaces. Springer-Verlag, 1973 · Zbl 0285.55012
[4] Clark A., Pacific J. Math. 15 pp 65– (1965)
[5] Getzler E. and Jones J. D. S.: Operads, homotopy algebra, and iterated integrals for double loop spaces. Preprint hep-th/9403055, March 1994
[6] Ginzburg V., Duke Math. J. 76 pp 203– (1994)
[7] Gugenheim V. K. A. M., Bull. Soc. Math. Belgique 38 pp 237– (1986)
[8] Halperin S.: Lectures on Minimal Models. Volume 261 of Memoirs Soc. Math. France, Nouv. Ser. 9-10. Soc. Math. France, 1983
[9] Hess K.: Perturbation and transfer of generic algebraic structures. In: Higher Homotopy Structures in Topology and Mathematical Physics (ed. by J. McCleary). Volume 227 of Contemp. Mathematics, pages 103-143. American Mathematical Society, 1999 · Zbl 0936.16026
[10] Hilton P. J. and Stammbach U.: A Course in Homological Algebra. Volume 4 of Graduate Texts in Mathematics. Springer-Verlag, 1971 · Zbl 0238.18006
[11] Hinich V., Comm. Algebra 25 pp 3291– (1997)
[12] Hinich V.: Tamarkin’s proof of Kontsevich formality theorem. Preprint math.QA/ 0003052, March 2000 · Zbl 1081.16014
[13] Huebschmann J., Math. Z. 207 pp 245– (1991)
[14] Johansson L. and Lambe L.: Transfering algebra structures up to homology equivalence. To appear in Math. Scand. November 1996 · Zbl 1023.16010
[15] Kadeishvili T. V., Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 77 pp 50– (1985)
[16] Kontsevich M.: Operads and motives in deformation quantization. Preprint math.QA/ 9904055, April 1999 · Zbl 0945.18008
[17] Lada T.: Strong homotopy algebras over monads. In: The Homology of Iterated Loop Spaces (ed. by F. R. Cohen, T. Lada, and P. May). Volume 533 of Lecture Notes in Mathematics, pages 399-479. Springer, 1976
[18] Lada T., Comm. Algebra 23 pp 2147– (1995)
[19] Lada T., Internat. J. Theoret. Phys. 32 pp 1087– (1993)
[20] Lambe L., Manuscripta Math. 58 pp 363– (1987)
[21] Mac Lane S., Rice Univ. Stud. 49 pp 28– (1963)
[22] Markl M., J. Pure Appl. Algebra 83 pp 141– (1992)
[23] Markl M., Comm. Algebra 24 pp 1471– (1996)
[24] Markl M.: Homotopy algebras via resolutions of operads. In: Proceedings of the 19th Winter School “Geometry and physics”, Srni?, Czech Republic.January9-15, 1999. Volume 63 of Supplem. ai Rend. Circ. Matem. Palermo, Ser. II, pages 157-164, 2000
[25] Markl M.: Homotopy diagrams of algebras. Preprint math.AT/0103052. To appear in the Proceedings of the Winter School “Geometry and Physics”, March 2001
[26] Markl M., Comm. Algebra 29 pp 5209– (2001)
[27] Markl M., Shnider S., and Stashe J. D.: Operads in Algebra, Topology and Physics. Volume 96 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, Rhode Island, 2002
[28] Stashe J. D., Trans. Amer. Math. Soc. 108 pp 275– (1963)
[29] Sugawara M., Ser. A Math. 33 pp 257–
[30] Sullivan D.: Infinitesimal computations in topology. Publ. Math. Inst. Hautes Eatudes Sci. 47 (1977), 269-331 · Zbl 0374.57002
[31] Vogt R. M., Preprint E 99 pp 81–
[32] Voronov A. A.: Homotopy Gerstenhaber algebras. Preprint math.QA/9908040, August 1999 · Zbl 0974.16005
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