Abstract
Let X be a simply connected CW-complex of finite type and \({\mathbb {K}}\) an arbitrary field. In this paper, we use the Eilenberg–Moore spectral sequence of \(C_*(\Omega (X), \mathbb K)\) to introduce a new homotopical invariant \(\textsc {r}(X, {\mathbb {K}})\). If X is a Gorenstein space with nonzero evaluation map, then \(\textsc {r}(X, {\mathbb {K}})\) turns out to interpolate \(\mathrm {depth}(H_*(\Omega (X), {\mathbb {K}}))\) and \(\mathrm {e}_{{\mathbb {K}}}(X)\). We also define for any minimal Sullivan algebra \((\Lambda V,d)\) a new spectral sequence and make use of it to associate to any 1-connected commutative differential graded algebra (A, d) a similar invariant \(\textsc {r}(A,d)\). When \((\Lambda V,d)\) is a minimal Sullivan model of X, this invariant fulfills the relation \(\textsc {r}(X, {\mathbb {K}}) = \textsc {r}(\Lambda V,d)\).
![](http://media.springernature.com/full/springer-static/image/art%3A10.1007%2Fs40065-017-0181-5/MediaObjects/40065_2017_181_Figa_HTML.gif)
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Anick, D.: Hopf algebras up to homotopy. J. Am. Math. Soc. 2, 417–453 (1989)
Bisiaux, L.: Depth and Toomer’s invariant. Topol. Appl. 97, 207–215 (1999)
Bogvad, R.; Halperin, S.: On a conjecture of Roos. Algebra Algebraic Topol. Intersect. LNM 1183, 120–127 (1986)
Félix, Y.; Halperin, S.: Rational LS category and its applications. Tran. Am. Math. Soc. 273, 1–38 (1982)
Félix, Y.; Halperin, S.; Lemaire, J.M.: The rational LS-category of products and Poincaré duality complexes. Topology 37(4), 749–756 (1998)
Félix, Y.; Halperin, S.; Lemaire, J.M.; Thomas, J.C.: Mod p loop space homology. Invent. Math. 95, 247–262 (1989)
Félix, Y.; Halperin, S.; Thomas, J.C.: Gorenstein spaces. Adv. Math. 71, 92–112 (1988)
Félix, Y.; Halperin, S.; Thomas, J.C.: Elliptic Hopf algebras. J. Lond. Math. Soc 2(43), 545–555 (1989)
Félix, Y.; Halperin, S.; Thomas, J.C.: Rational Homotopy Theory, G. T. M., p. 205. Springer, New York (2001)
Félix, Y.; Halperin, S.; Thomas, J.C.: Rational Homotopy Theory II. World Scientific, Singapore (2015)
Félix, Y.; Murillo, A.: Gorenstein graded algebras and the evaluation map. Can. Math. Bull. 41(1), 28–32 (1998)
Halperin, S.: Universal envelo** algebras and loop space homology. J. Pure Appl. Algebra 83, 237–282 (1992)
Halperin, S.: Finiteness in the minimal models of Sullivan. Trans. Am. Math. Soc. 230, 173–199 (1977)
Halperin, S.; Lemaire, J.M.: Notion of Category in Differential Algebra. In: Algebraic Topology-Rational Homotopy, Lecture Notes in Mathematics, vol. 1318(1988), pp. 138–154 (2007)
James, I.M.: On category, in the sense of Lusternik–Schnirelmann. Topology 17, 331–348 (1978)
Lechuga, L.; Murillo, A.: The fundamental class of a rational space, the graph collaring problem and other classical decision problems. Bull. Belg. Math. Soc. 8, 451–467 (2001)
Lechuga, L.; Murillo, A.: A formula for the rational LS-category of certain spaces. Ann. Inst. Fourier 52(5), 1585–1590 (2002)
Loday, J.L.; Valette, B.: Algebraic Operads. Springer, Berlin (2012)
Lusternik, L.; Schnirelmann, L.: Méthodes Topologiques dans les Problémes Variationnels. Hermann, Paris (1934)
McLeary, J.: A User’s Guide to Spectral Sequences. University Press, Cambridge (2001)
Murillo, A.: The evaluation map of some Gorenstein spaces. J. Pure Appl. Algebra 91, 209–218 (1994)
Sullivan, D.: Infinitesimal Computations in topology. Inst. Hautes Etudes Sci. Publ. Math. 47, 269–331 (1978)
Toomer, G.H.: Lusternik–Schnirelmann category and the Moore spectral sequence. Math. Zeit. 183, 123–143 (1974)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.