The Hochschild cohomology ring of the singular cochain algebra of a space

Kuribayashi, Katsuhiko

Annales de l'Institut Fourier, Tome 61 (2011) no. 5, pp. 1779-1805.

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.

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.

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.

[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. $A_{\infty }$-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 $k\left[X\right]/\left(f\right)$, 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)