no. 5
The Hochschild cohomology ring of the singular cochain algebra of a space
Kuribayashi, Katsuhiko1
1 Shinshu University, Faculty of Science, Department of Mathematical Sciences, Matsumoto, Nagano 390-8621, Japan
Annales de l'Institut Fourier, Volume 61 (2011) no. 5, pp. 1779-1805.

We determine the algebra structure of the Hochschild cohomology of the singular cochain algebra with coefficients in a field on a space whose cohomology is a polynomial algebra. A spectral sequence calculation of the Hochschild cohomology is also described. In particular, when the underlying field is of characteristic two, we determine the associated bigraded Batalin-Vilkovisky algebra structure on the Hochschild cohomology of the singular cochain on a space whose cohomology is an exterior algebra.

Nous déterminons la structure d’algèbre sur la cohomologie de Hochschild des cochaînes singulières à coefficients dans un corps d’un espace dont la cohomologie est une algèbre polynômiale. Un calcul de cohomologie de Hochschild à l’aide d’une suite spectrale est aussi décrit. En particulier, quand le corps sous-jacent est de caractéristique deux, nous déterminons la structure d’algèbre de Batalin-Vilkovisky bigraduée associée à la cohomologie de Hochschild des cochaînes singulières d’un espace dont la cohomologie est une algèbre extérieure.

DOI: 10.5802/aif.2658
Classification: 16E40, 16E45, 55P35
Keywords: Hochschild cohomology, singular cochain algebra, Batalin-Vilkovisky algebra, Koszul-Tate resolution.
Mot clés : Cohomologie de Hochschild, cochaînes singulières, algèbre de Batalin-Vilkovisky, résolution de Koszul-Tate.
Kuribayashi, Katsuhiko 1

1 Shinshu University, Faculty of Science, Department of Mathematical Sciences, Matsumoto, Nagano 390-8621, Japan 
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Kuribayashi, Katsuhiko. The Hochschild cohomology ring of the singular cochain algebra of a space. Annales de l'Institut Fourier, Volume 61 (2011) no. 5, pp. 1779-1805. doi : 10.5802/aif.2658. http://www.numdam.org/articles/10.5802/aif.2658/

[1] Avramov, L. L.; Buchweitz, R. -O.; Iyengar, S. B.; Miller, C. Homology of perfect complexes, Adv. Math., Volume 223 (2010), pp. 1731-1781 (arXiv: math.AC/0609008v2) | DOI | MR | Zbl

[2] Benson, D.; Iyengar, S. B.; Krause, H. Local cohomology and support for triangulated categories, Ann. Sci. Éc. Norm. Supér, Volume 41 (2008), pp. 573-619 | Numdam | MR | Zbl

[3] Benson, S. B. D.and Iyengar; Krause, H. Stratifying triangulated categories, preprint, (2009)

[4] Buchweitz, R. -O.; Flenner, H. Global Hochschild (co-)homology of singular spaces, Advances in Math., Volume 217 (2008), pp. 205-242 | DOI | MR | Zbl

[5] Buchweitz, R. -O.; Green, E. L.; Snashell, N.; Solberg, Ø. Multiplicative structures for Koszul algebras, preprint (2005) (arXiv.org/abs/math/0508177)

[6] Chas, M.; Sullivan, D. String topology, preprint (1999) (arXiv.org/abs/math/9911159, to appear in Ann. of Math)

[7] Cibils, C.; Solotar A. The Buenos Aires Cyclic Homology Group Cyclic homology of algebras with one generator, K-theory, Volume 5 (1991), pp. 51-69 | DOI | MR | Zbl

[8] Cohen, R. Multiplicative properties of Atiyah duality, Homology Homotopy Appl., Volume 6 (2004), pp. 269-281 | MR | Zbl

[9] Cohen, R. L.; Jones, J. D. S. A homotopy theoretic realization of string topology, Math. Ann., Volume 324 (2001), pp. 773-798 | DOI | MR | Zbl

[10] Félix, Y.; Halperin, S.; Thomas, J. -C. Gorenstein spaces, Adv. in Math., Volume 71 (1988), pp. 92-112 | DOI | MR | Zbl

[11] Félix, Y.; Halperin, S.; Thomas, J. -C. Adams’ cobar equivalence, Trans. Amer. Math. Soc., Volume 329 (1992), pp. 531-549 | DOI | MR | Zbl

[12] Félix, Y.; Halperin, S.; Thomas, J. -C. Differential graded algebras in topology, I.M. James (Ed.) (Handbook of Algebraic Topology), Elsevier, Amsterdam, 1995, pp. 829-865 | MR | Zbl

[13] Félix, Y.; Halperin, S.; Thomas, J. -C. Rational Homotopy Theory, Graduate Texts in Mathematics, 205, Springer-Verlag, 2011 | MR | Zbl

[14] Félix, Y.; Menichi, L.; Thomas, J. -C. Gerstenhaber duality in Hochschild cohomology, J. Pure Appl. Algebra, Volume 199 (2005), pp. 43-59 | DOI | MR | Zbl

[15] Félix, Y.; Thomas, J. -C. Rational BV-algebra in string topology, Bull. Soc. Math. France, Volume 136 (2008), pp. 311-327 | Numdam | MR | Zbl

[16] Félix, Y.; Thomas, J. -C. String topology on Gorenstein spaces, Math. Ann., Volume 345 (2009), pp. 417-452 | DOI | MR | Zbl

[17] Félix, Y.; Thomas, J. -C.; Vigué-Poirrier, M. The Hochschild cohomology of a closed manifold, Publ. Math. Inst. Hautes Études Sci., Volume 99 (2004), pp. 235-252 | Numdam | MR | Zbl

[18] Félix, Y.; Thomas, J. -C.; Vigué-Poirrier, M. Rational string topology, J. Eur. Math. Soc. (JEMS), Volume 9 (2007), pp. 123-156 | MR | Zbl

[19] Getzler, E.; Jones, J. D. S. A ∞ -algebras and the cyclic bar complex, J. Eur. Math. Soc. (JEMS), Volume 34 (1990), pp. 256-283 | MR | Zbl

[20] Halperin, S.; Lemaire, J. -M. Notions of category in differential algebra, Algebraic Topology: Rational Homotopy (Springer Lecture Notes in Math.), Volume 1318, Springer, Berlin, New York, 1988, pp. 138-154 | MR | Zbl

[21] Han, Y.; Xu, Y. Hochschild (co)homology of exterior algebras, Comm. Algebra, Volume 35 (2007), pp. 115-131 | MR | Zbl

[22] Holm, T. Hochschild cohomology rings of algebras k[X]/(f), Beiträge Algebra Geom., Volume 41 (2000), pp. 291-301 | MR | Zbl

[23] Jones, J. D. S. Cyclic homology and equivariant homology, Invent. Math., Volume 87 (1987), pp. 403-423 | DOI | MR | Zbl

[24] Jørgensen, P. Auslander-Reiten theory over topological spaces, Comment. Math. Helv., Volume 79 (2004), pp. 160-182 | DOI | MR | Zbl

[25] Jørgensen, P. The Auslander-Reiten quiver of a Poincaré duality space, Algebr. Represent. Theory, Volume 9 (2006), pp. 323-336 | DOI | MR | Zbl

[26] Kaufmann, R. M. A proof of a cyclic version of Deligne’s conjecture via cacti, Math. Res. Lett., Volume 15 (2008), pp. 901-921 | MR | Zbl

[27] Kaufmann, R. M. Moduli space actions on the Hochschild co-chains of a Frobenius algebra. II. Correlators, J. Noncommut. Geom., Volume 2 (2008), pp. 283-332 | DOI | MR | Zbl

[28] Kraines, D.; Schochet, C. Differentials in the Eilenberg-Moore spectral sequence, J. Pure Appl. Algebra, Volume 2 (1972), pp. 131-148 | DOI | MR | Zbl

[29] Kuribayashi, K. On the mod p cohomology of the spaces of free loops on the Grassmann and Stiefel manifolds, J. Math. Soc. Japan, Volume 43 (1991), pp. 331-346 | DOI | MR | Zbl

[30] Kuribayashi, K. On the levels of spaces and topological realization of objects in a triangulated category, preprint (2010)

[31] Kuribayashi, K. Upper and lower bounds of the (co)chain type level of a space, preprint (2010) (arXiv: math.AT/1006.2669)

[32] Linckelmann, M. On the graded centers and block cohomology, Proc. Edinburgh Math. Soc, Volume 52 (2009), pp. 489-514 | DOI | MR | Zbl

[33] Menichi, L. Batalin-Vilkovisky algebra structures on Hochschild cohomology, Bull. Soc. Math. France, Volume 137 (2009), pp. 277-295 | Numdam | MR | Zbl

[34] Menichi, L. String topology for spheres, With an appendix by Gerald Gaudens and Menichi, Comment. Math. Helv., Volume 84 (2009), pp. 135-157 | MR | Zbl

[35] Merkulov, S. A. De Rham model for string topology, Int. Math. Res. Not. (2004) no. 55, pp. 2955-2981 | DOI | MR | Zbl

[36] Mimura, M.; Toda, H. Topology of Lie groups I, Translations of Mathematical Monographs, 91, American Mathematical Society, Providence, RI, 1991 | MR | Zbl

[37] Munkholm, H. J. The Eilenberg-Moore spectral sequence and strongly homotopy multiplicative maps, J. Pure Appl. Algebra, Volume 5 (1974), pp. 1-50 | DOI | MR | Zbl

[38] Rouquier, R. Dimensions of triangulated categories, J. K-Theory, Volume 1 (2008), pp. 193-256 | DOI | MR | Zbl

[39] Sanada, K. On the Hochschild cohomology of crossed products, Comm. Algebra, Volume 21 (1993), pp. 2727-2748 | DOI | MR | Zbl

[40] Schmidt, K. Families of Auslander-Reiten components for simply connected differential graded algebras, Math. Z., Volume 426 (2010), pp. 43-62 | DOI | MR

[41] Siegel, S. F.; Witherspoon, S. J. The Hochschild cohomology ring of a group algebra, Proc. London Math. Soc. (3), Volume 79 (1999), pp. 131-157 | DOI | MR | Zbl

[42] Smith, L. On the characteristic zero cohomology of the free loop space, Amer. J. Math., Volume 103 (1981), pp. 887-910 | DOI | MR | Zbl

[43] Smith, L. The Eilenberg-Moore spectral sequence and mod 2 cohomology of certain free loop spaces, Illinois J. Math., Volume 28 (1984), pp. 516-522 | MR | Zbl

[44] Snashall, N.; Solberg, Ø. Support varieties and Hochschild cohomology ring, Proc. London Math. Soc. (3), Volume 88 (2004), pp. 705-732 | DOI | MR | Zbl

[45] Thomas, T.; Zeinalian, M. Infinity structure of Poincaré duality spaces, Appendix A by Dennis Sullivan, Algebr. Geom. Topol., Volume 7 (2007), pp. 233-260 | MR | Zbl

[46] Tradler, T. The Batalin-Vilkovisky algebra on Hochschild cohomology induced by infinity inner products, Ann. Inst. Fourier, Volume 58 (2008), pp. 2351-2379 | DOI | Numdam | MR | Zbl

[47] Yang, T. A Batalin-Vilkovisky algebra suructure on the Hochschild cohomology of truncated polynomials, preprint, (2007)

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