Abstract
It is shown that some classical local geometries are of infinity origin, i.e. their smooth formal germs are (homotopy) representations of cofibrant (di) operads in spaces concentrated in degree zero. In particular, they admit natural infinity generalizations when one considers homotopy representations of the (di) operads in generic differential graded spaces. Poisson geometry provides us with a simplest manifestation of this phenomenon.
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References
Adams, J.F.: Infinite loop spaces. Princeton NJ: Princeton University Press, 1978
Gan, W.L.: Koszul duality for dioperads. Math. Res. Lett. 10, 109–124 (2003)
Getzler, E., Jones, J.D.S.: Operads, homotopy algebra, and iterated integrals for double loop spaces. http://arxiv.org/list/hep-th/9403055, 1994
Ginzburg, V., Kapranov, M.: Koszul duality for operads. Duke Math. J. 76, 203–272 (1994)
Hertling, C.: Frobenius manifolds and moduli spaces for singularities. Cambridge: Cambridge University Press, 2002
Hertling, C., Manin, Yu.I.: Weak Frobenius manifolds. Intern. Math. Res. Notices 6, 277–286 (1999)
Lyubashenko, V.: Private communication
Kontsevich, M.: Deformation quantization of Poisson manifolds I. Lett. Math.Phys. 66, 157–216 (2003)
Kontsevich, M.: Topics in algebra-deformation theory. Berkeley Lectures 1995 (Unpublished notes by A. Weinstein)
Markl, M.: Distributive laws and Koszulness. Ann. Inst. Fourier, Grenoble 46, 307–323 (1996)
Markl, M.: Homotopy algebras are homotopy algebras. http://arxiv.org/list/math.AT/9907138, 1999
Markl, M., Shnider, S., Stasheff, J.D.: Operads in Algebra, Topology and Physics. Providence, RI: AMS, 2002
Markl, M., Voronov, A.A.: PROPped up graph cohomology. http://arxiv.org/list/math.QA/ 0307081, 2003
Merkulov, S.A.: Operads, deformation theory and F-manifolds. In: Frobenius manifolds, quantum cohomology, and singularities, eds. C. Hertling, M. Marcolli, Wiesbaden: Vieweg 2004
Merkulov, S.A.: Nijenhuis infinity and contractable dg manifolds. http://arxiv.org/list/ math.AG/040324, 2004 to appear in Compositio Math
Stasheff, J.D.: On the homotopy associativity of H-spaces, I II. Trans. Amer. Math. Soc. 108, 272–292 & 293–312 (1963)
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Communicated by A. Connes
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Merkulov, S. PROP Profile of Poisson Geometry. Commun. Math. Phys. 262, 117–135 (2006). https://doi.org/10.1007/s00220-005-1385-7
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DOI: https://doi.org/10.1007/s00220-005-1385-7