×

Hochschild cohomology of a strongly homotopy commutative algebra. (English) Zbl 1336.55004

The Hochschild cohomology of a DG algebra \(A\) with coefficients in itself is, up to a suspension of degrees, a graded Lie algebra. Let \(A = (\{A^i\}_{i\in\mathbb Z},d)\) be a DG algebra with homology \(H(A)\). There is a canonical homomorphism of graded algebras \(q_A: HH^*(A,A)\rightarrow H(A)\) which is up to a suspension a homomorphism of graded Lie algebras. When \(A\) is graded commutative, the homomorphism \(q_A\) admits a canonical linear retraction. When \(A\) is a strongly homotopy commutative algebra over a principal ideal domain \(\mathbb F\) (\(1\)-connected), the main theorem of the paper gives an explicit description of the kernel of \(s\circ q_A\circ s^{-1}: sHH^*(A,A)\rightarrow sH(A)\) as the homology of a DG Lie algebra of derivations of \(T\), where \(T\) is a cofibrant replacement of \(A\) (i.e. a semifree model). This has applications to the computation of the relative groups \(H_*(LX,X;\mathbb F)\) where \(LX\) is the free loop space, or to the computation of \(H_{*\geq n}(LX,\mathbb F)\) if \(X\) is a simply connected Poincaré duality space of formal dimension \(n\). The second main result of this paper gives a way to compute this kernel in cases when \(T = (T^aV,d)\) is such that \(V=\{V^i\}_{i\geq 2}\) and \(H^i(T) = 0\) for \(i>n\) and \(i=n-1\), and \(H^n(T) = \mathbb F\omega\).
We recall that a strongly homotopy commutative algebra is an augmented DG algebra \(A\) together with a homomorphism of DG coalgebras \(BA\otimes BA\rightarrow BA\) that is associative, commutative up to homotopy and admitting a unit. In the special case \(A\) is commutative, \(BA\) is known to be a DG Hopf algebra.

MSC:

55P35 Loop spaces
53D55 Deformation quantization, star products
13D10 Deformations and infinitesimal methods in commutative ring theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adams J.F.: On the cobar construction. Proc. Natl. Acad. Sci. USA 42, 409-412 (1956) · Zbl 0071.16404 · doi:10.1073/pnas.42.7.409
[2] Baues H.: The double bar and cobar construction. Compos. Math. 43, 331-341 (1981) · Zbl 0478.57027
[3] Ndombol B., Thomas J.-C.: On the cohomology algebra of free loop spaces. Topology 41, 85-106 (2002) · Zbl 1011.16008 · doi:10.1016/S0040-9383(00)00022-7
[4] Chas, M., Sullivan, D.: String topology. Preprint math.GT/0107187 · Zbl 1185.55013
[5] Cibils C., Solotar A.: Hochschild cohomology algebra of abelian groups. Arch. Math. (Basel) 68, 17-21 (1997) · Zbl 0872.16004 · doi:10.1007/PL00000389
[6] Félix Y., Halperin S., Thomas J.-C.: Adams’ cobar construction. Trans. Am. Math. 329, 531-548 (1992) · Zbl 0765.55005
[7] Félix, Y., Halperin, S., Thomas, J-C.: Differential graded algebras in topology. In: James, I. (ed.) Handbook of Algebraic Topology, Chapter 16. Elsevier, Amsterdam, pp. 829-865 (1995) · Zbl 0868.55016
[8] Félix Y., Menichi L., Thomas J.-C.: Gerstenhaber duality in Hochschild cohomology. J. Pure Appl. Algebra 199, 43-59 (2005) · Zbl 1076.55003 · doi:10.1016/j.jpaa.2004.11.004
[9] Félix, Y., Halperin, S., Thomas, J.-C.: Rational Homotopy Theory. Graduate Texts in Mathematics, vol. 205. Springer, Berlin (2000) · Zbl 0374.57002
[10] Félix, Y.; Thomas, J.-C., No article title, Bull. Soc. Math. France, 136, 311-327 (2008) · Zbl 1160.55006
[11] Félix Y., Thomas J.-C., Vigué M.: The Hochschild cohomology of a closed manifold. Publ. Math. IHES 99, 235-252 (2005) · Zbl 1060.57019 · doi:10.1007/s10240-004-0021-y
[12] Gerstenhaber M.: The cohomology structure of an associative ring. Ann. Math. 78, 59-103 (1963) · Zbl 0131.27302 · doi:10.2307/1970343
[13] Getzler E., Jones J.C.: A∞ algebra and the cyclic bar complex. Ill. J. of Math. 34, 126-159 (1990)
[14] Getzler, E., Jones, J.C.: Operads, homotopy algebra and iterated integrals for double loopspaces. Preprint hep-th/9403055 (1994)
[15] Ginot, G.: On the Hochschild and Harrison (Co)homology of C∞ Algebras and Applications to String Topology. Deformation Spaces, Aspects in Mathematics, vol. E40, pp. 1-51. Vieweg + Teubner, Wiesbaden (2010) · Zbl 1227.16010
[16] Gromoll D., Meyer W.: Periodic geodesic on compact Riemannian manifold. J. Differ. Geom. 3, 493-510 (1969) · Zbl 0203.54401
[17] Halperin S., Vigué-Poirrier M.: The homology of a free loop space. Pac. J. Math. 147, 311-324 (1991) · Zbl 0666.55011 · doi:10.2140/pjm.1991.147.311
[18] Hamilton, A., Lazarev, A.: Symplectic C∞ algebras. Preprint QA/0707.4003 · Zbl 1160.13008
[19] Husemoller D., Moore J.C., Stasheff J.: Differential homological algebra and homogeneous spaces. J. Pure Appl. Algebra 5, 113-185 (1974) · Zbl 0364.18008 · doi:10.1016/0022-4049(74)90045-0
[20] Jones J.D.S.: Cyclic homology and equivariant homology. Invent. math. 87, 403-423 (1987) · Zbl 0644.55005 · doi:10.1007/BF01389424
[21] Kadeshvili T.V.: On the homology theory of fiber spaces. Russ. Math. Surv. 35(3), 231-238 (1980) · Zbl 0521.55015 · doi:10.1070/RM1980v035n03ABEH001842
[22] Lambrechts P.: The Betti numbers of the free loop space of a connected sum. J. Lond. Math Soc. 64, 205-228 (2001) · Zbl 1018.55006 · doi:10.1017/S0024610701002198
[23] Mac Cleary J., Ziller W.: On the free loop space of homogeneous spaces. Am. J. Math. 109, 765-781 (1987) · Zbl 0635.57026 · doi:10.2307/2374612
[24] Mac Lane, S.: Homology. Grundlehren der Mathematischen Wissenscaften, vol. 114. Springer, Berlin (1975) · Zbl 0689.57026
[25] Menichi L.: The cohomology ring of free loop spaces. Homol. Homotopy Appl. 3, 193-224 (2001) · Zbl 0974.55005 · doi:10.4310/HHA.2001.v3.n1.a9
[26] Menichi L.: Batalin-Vilkovisky algebra structures on Hochschild cohomology. Bull. Soc. Math. France 137, 277-295 (2009) · Zbl 1180.16007
[27] Menichi L.: String topology for spheres. Comment. Math. Helv. 84, 135-157 (2009) · Zbl 1159.55004 · doi:10.4171/CMH/155
[28] Milgram R.J.: The bar construction and abelian H-spaces. Ill. J. Math. 11, 242-250 (1967) · Zbl 0152.40502
[29] Moore, J.: Algèbre homologique et homologie des espaces classifiants. In: “Séminaire Cartan 1959/1960” exposé 7 · Zbl 0115.17205
[30] Munkholm H.J.: The Eilenberg-Moore spectral sequence and strongly homotopy multiplicative maps. J. Pure Appl. Algebra 5, 1-50 (1974) · Zbl 0294.55011 · doi:10.1016/0022-4049(74)90002-4
[31] Munkolm H.J.: DGA-algebras as Quillen model category. J. Pure Appl. Algebra 13, 221-232 (1978) · Zbl 0409.55018 · doi:10.1016/0022-4049(78)90009-9
[32] Quillen D.: Algebra cochains and cyclic cohomology. Publ. Math. IHES 68, 139-174 (1988) · Zbl 0689.57026 · doi:10.1007/BF02698546
[33] Roos, J.E.: Homology of Free Loop Spaces, Cyclic Homology and Rational Poincaré-Betti Series. Preprint series, vol. 39. University of Stockholm, Stockholm (1987) · Zbl 0662.55004
[34] Sánchez-Flores, S.: The Lie structure on the Hochschild cohomology of a modular group algebra. arXiv:1103.3218v1 [math.RA] · Zbl 1279.16010
[35] Smith L.: The EMSS and the mod 2 cohomology of certain free loop spaces. Ill. J. Math. 28, 516-522 (1984) · Zbl 0538.57025
[36] Stasheff J.D.: The intrinsic bracket on the deformation complex of an associative algebra. J. Pure Appl. Algebra 89, 231-335 (1993) · Zbl 0786.57017 · doi:10.1016/0022-4049(93)90096-C
[37] Sullivan D.: Infinitesimal computations in topology. Publ. Math. Inst. Hautes Études Scientifiques 47, 269-331 (1977) · Zbl 0374.57002 · doi:10.1007/BF02684341
[38] Tamarkin D.: Another proof M. Kontsevich’ formality theorem. Preprint QA/9803.025 · Zbl 0478.57027
[39] Tradler, T., Zeinalian, M.: Algebraic string operations. Preprint arXiv.math/0605770 (2006) · Zbl 1144.55012
[40] Vigué-Poirrier M., Sullivan D.: The homology theory of the closed geodesic problem. J. Differ. Geom. 11, 633-644 (1976) · Zbl 0361.53058
[41] Ziller W.: The free loop space on globally symmetric spaces. Invent. Math. 41, 1-22 (1977) · Zbl 0338.58007 · doi:10.1007/BF01390161
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.