Infinity structure of Poincaré duality spaces

Tradler, Thomas¹ ; Zeinalian, Mahmoud²

¹Department of Mathematics, College of Technology of the City University of New York, 300 Jay Street, Brooklyn NY 11201, USA
²Department of Mathematics, C W Post Campus of Long Island University, 720 Northern Boulevard, Brookville NY 11548, USA

Algebraic and Geometric Topology, Tome 7 (2007) no. 1, pp. 233-260

Résumé

We show that the complex C∙X of rational simplicial chains on a compact and triangulated Poincaré duality space X of dimension d is an A∞ coalgebra with ∞ duality. This is the structure required for an A∞ version of the cyclic Deligne conjecture. One corollary is that the shifted Hochschild cohomology HH∙+d(C∙X,C∙X) of the cochain algebra C∙X with values in C∙X has a BV structure. This implies, if X is moreover simply connected, that the shifted homology H∙+dLX of the free loop space admits a BV structure. An appendix by Dennis Sullivan gives a general local construction of ∞ structures.

Détail

DOI : 10.2140/agt.2007.7.233

Keywords: Poincaré duality space, local infinity structure

Bibliographie

[1] M Chas, D Sullivan, String topology

[2] M Chas, D Sullivan, Closed string operators in topology leading to Lie bialgebras and higher string algebra, from: "The legacy of Niels Henrik Abel", Springer (2004) 771

[3] D Chataur, A bordism approach to string topology, Int. Math. Res. Not. (2005) 2829

[4] R L Cohen, J D S Jones, A homotopy theoretic realization of string topology, Math. Ann. 324 (2002) 773

[5] R L Cohen, J Klein, D Sullivan, The homotopy invariance of the string topology loop product and string bracket

[6] D E Galewski, R J Stern, The relationship between homology and topological manifolds via homology transversality, Invent. Math. 39 (1977) 277

[7] A Gómez-Tato, S Halperin, D Tanré, Rational homotopy theory for non-simply connected spaces, Trans. Amer. Math. Soc. 352 (2000) 1493

[8] J D S Jones, Cyclic homology and equivariant homology, Invent. Math. 87 (1987) 403

[9] H Kajiura, Noncommutative homotopy algebras associated with open strings

[10] J R Klein, Fiber products, Poincaré duality and $A_\infty$-ring spectra, Proc. Amer. Math. Soc. 134 (2006) 1825

[11] M Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003) 157

[12] R Lawrence, D Sullivan, A free differential Lie algebra for the interval

[13] R Longoni, T Tradler, Infinity inner products for cyclic operads

[14] M Markl, A cohomology theory for $A(m)$-algebras and applications, J. Pure Appl. Algebra 83 (1992) 141

[15] M Markl, Homotopy algebras are homotopy algebras, Forum Math. 16 (2004) 129

[16] C Mccrory, A characterization of homology manifolds, J. London Math. Soc. $(2)$ 16 (1977) 149

[17] S A Merkulov, De Rham model for string topology, Int. Math. Res. Not. (2004) 2955

[18] A A Ranicki, Algebraic $L$-theory and topological manifolds, Cambridge Tracts in Mathematics 102, Cambridge University Press (1992)

[19] J D Stasheff, Homotopy associativity of $H$-spaces. I, II, Trans. Amer. Math. Soc. 108 $(1963)$, 275-292; ibid. 108 (1963) 293

[20] D Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. (1977)

[21] D Sullivan, Open and closed string field theory interpreted in classical algebraic topology, from: "Topology, geometry and quantum field theory", London Math. Soc. Lecture Note Ser. 308, Cambridge Univ. Press (2004) 344

[22] T Tradler, The BV algebra on Hochschild cohomology induced by infinity inner products

[23] T Tradler, $\infty$–inner products on $A_{\infty}$–Algebras

[24] T Tradler, M Zeinalian, Algebraic string operations

[25] T Tradler, M Zeinalian, On the cyclic Deligne conjecture, J. Pure Appl. Algebra 204 (2006) 280

[26] M V P Y. Felix J.-C. Thomas, Loop homology algebra of a closed manifold