Abstract
In [HS] and [F1] Halperin, Stasheff, and Félix showed how an inductively-defined sequence of elements in the cohomology of a graded commutative algebra over the rationals can be used to distinguish among the homotopy types of all possible realizations, thus providing a collection of algebraic invariants for distinguishing among rational homotopy types of spaces. There is also a dual version, in the setting of graded Lie algebras (see [O]).
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
R. Adin & D. Blanc, “Resolutions of Associative and Lie algebras” Can. Math. Bull.43 (2000) No. 1, pp. 3–16.
C. Allday, “Rational Whitehead products and a spectral sequence of Quillen” Pac. J. Math.46 (1973) No. 2, pp. 313–323.
C. Allday, “Rational Whitehead products and a spectral sequence of Quillen, II” Houston J. Math.3 (1977), No. 3, pp. 301–308.
P.G. Andrews & M. Arkowitz, “Sullivan’s minimal models and higher order Whitehead products” Can. J. Math.30 (1978) No. 5, pp. 961–982.
H.J. Baues & J.-M. Lemaire. “Minimal models in homotopy theory” Math. Ann.225 (1977), pp. 219–242.
D. Blanc, “Derived functors of graded algebras” J. Pure Appl. Alg.64 (1990) No. 3, pp. 239–262.
D. Blanc, “Higher homotopy operations and the realizability of homotopy groups” Proc. Lond. Math. Soc. (3)70 (1995), pp. 214–240.
D. Blanc, “Homotopy operations and the obstructions to being an H-space” Manus. Math.88 (1995) No. 4, pp. 497–515.
D. Blanc, “New model categories from old” J. Pure Appl. Math.109 (1996) No. 1, pp. 37–60.
D. Blanc, “Algebraic invariants for homotopy types” Math. Proc. Camb. Phil. Soc.27 (1999), No. 3, pp. 497–523.
D. Blanc, “CW simplicial resolutions of spaces, with an application to loop spaces” Topology é9 Appl.100 (2000), No. 2–3, pp. 151–175.
D. Blanc, “Realizing coalgebras over the Steenrod algebra” Topology40 (2001), No. 5, pp. 993–1016.
D. Blanc, W.G. Dwyer, & P.G. Goerss, “The realization space of a II-algebra: a moduli problem in algebraic topology”, to appear in Topology.
D. Blanc & P.G. Goerss, “Realizing simplicial H-algebras”, preprint 2002.
D. Blanc &M. Markl, “Higher homotopy operations”, to appear in Math. Z..
D. Blanc & C.S. Stover, “A generalized Grothendieck spectral sequence”, in N. Ray & G. Walker, eds. Adams Memorial Symposium on Algebraic Topology Vol. 1Lond. Math. Soc. Lec. Notes Ser. 175, Cambridge U. Press, Cambridge, 1992, pp. 145–161.
J.M. Boardman & R.M. Vogt Homotopy Invariant Algebraic Structures on Topological SpacesSpringer-Verlag Lec. Notes Math.347, Berlin-New York, 1973.
A.K. Bousfield & E.M. Friedlander, “Homotopy theory of F-spaces, spectra, and bisimplicial sets”, in M.G. Barratt & M.E. Mahowald, eds. Geometric Applications of Homotopy Theory IISpringer-Verlag Lec. Notes Math.658, Berlin-New York, 1978, pp. 80–130.
D. Burghelea & M. Vigué-Poirrier, “Cyclic homology of commutative algebras, P”, in Y. Félix, ed. Algebraic Topology - Rational Homotopy 1986Springer-Verlag Lec. Notes Math.1318, Berlin-New York, 1987, pp. 51–72.
A. Dold, “Homology of symmetric products and other functors of complexes”, Ann. Math. (2)68 (1958), pp. 54–80.
W.G. Dwyer, D.M. Kan, & C.R. Stover, “An E2model category structure for pointed simplicial spaces” J. Pure i Appl. Alg.90 (1993) No. 2, pp. 137–152.
Y. Félix Dénombrement des types de k-homotopie: théorie de la déformationMem. Soc. Math. France 3, Paris, 1980.
Y. Félix, “Modèles bifiltrés: une plaque tournant en homotopie rationelle” Can. J. Math.33 (1981) No. 6, pp. 1448–1458.
A. Gómez-Tato, S. Halperin & D. Tanré, “Rational homotopy theory for non simply connected spaces” preprint1996.
R. Godement Topologie algébrique et théorie des faisceauxAct. Sci. & Ind. No. 1252, Publ. Inst. Math. Univ. Strasbourg XIII, Hermann, Paris 1964.
T.G. Goodwillie, “Cyclic homology, derivations, and the free loopspace” Topology24 (1985) No. 2, pp. 187–216.
S. Halperin &J.D. Stasheff, “Obstructions to homotopy equivalences”, Adv. in Math.32 (1979) No. 3, pp. 233–279.
Y. Haralambous, “Perturbation d’algèbres de Lie différentielles et coformalité modérée” Bull. Soc. Math. Belg. Sér. B43 (1991) No. 1, pp. 59–67.
T. Jozefiak, “Tate resolutions for commutative graded algebras over a local ring” Fund. Math.74 (1972) No. 3, pp. 209–231.
T.J. Lada & M. Markl, “Strongly homotopy Lie algebras” Comm.in Alg.23 (1995) No. 6, pp. 2147–2161.
J.-M. Lemaire & F. Sigrist, “Dénombrement de types d’homotopie rationelle” C. R. Acad. Sci. Paris 1287 (1978) No. 3, pp. 109–112.
J.-L. Loday Cyclic HomologySpringer-Verlag Grund. math. Wissens.301, Berlin-New York, 1992.
S. Mac Lane HomologySpringer-Verlag Grund. math. Wissens.114, Berlin-New York, 1963.
S. Mac Lane Categories for the Working MathematicianSpringer-Verlag Grad. Texts Math.5, Berlin-New York, 1971.
J.P. May Simplicial Objects in Algebraic TopologyU. Chicago Press, Chicago-London, 1967.
W. Meier, “Minimal models for nonnilpotent spaces” Comm. Math. Hely.55 (1980), No. 4, pp. 622–633.
T.J. Miller &J A. Neisendorfer. “Formal and coformal spaces” Ill. J. Math.22 (1978) No. 4, pp. 565–580.
J.A. Neisendorfer, “Lie algebras, coalgebras and rational homotopy theory for nilpotent spaces” Poe. J. Math.74 (1978) No. 2, pp. 429–460.
A. Oukili Sur l’homologie d’une algèbre différentielle de LiePh.D. thesis, Univ. de Nice, 1978.
G.J. Porter. “Higher order Whitehead products” Topology3 (1965) pp. 123–165.
D.G. Quillen Homotopical AlgebraSpringer-Verlag Lec. Notes Math. 20, Berlin-New York, 1963.
D.G. Quillen, “Spectral sequences of a double semi-simplicial group” Topology5 (1966), pp. 155–156.
D.G. Quillen, “Rational homotopy theory”, Ann. Math.90 (1969) No. 2, pp. 205–295.
D.G. Quillen, “On the (co-)homology of commutative rings” Applications of Categorical AlgebraProc. Symp. Pure Math. 17, AMS, Providence, RI, 1970, pp. 65–87.
V.S. Retakh, “Massey operations in Lie superalgebras and differentials of the Quillen spectral sequence” Funk. Ann. i Pril.12 (1978) No. 4, pp. 91–92 [transl. in Funct. Anal. 4 Applic.12 (1978) No. 4, pp. 319–321.
V.S. Retakh, “Lie-Massey brackets and n-homotopically multiplicative maps of differential graded Lie algebras” J. Pure Appl. Alg.89 (1993) No. 1–2, pp. 217–229.
M. Schlessinger &.J.D. Stasheff. “Deformation theory and rational homotopy type”, to appear in Pub. Math. Inst. Hautes Et. Sci..
C.R. Stover, “A Van Kampen spectral sequence for higher homotopy groups” Topology29 (1990), pp. 9–26.
D. Tanré Homotopie Rationelle: Modèles de Chen Quillen SullivanSpringer-Verlag Lec. Notes Math.1025, Berlin-New York, 1983.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer Basel AG
About this chapter
Cite this chapter
Blanc, D. (2003). Homotopy Operations and Rational Homotopy Type. In: Arone, G., Hubbuck, J., Levi, R., Weiss, M. (eds) Categorical Decomposition Techniques in Algebraic Topology. Progress in Mathematics, vol 215. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7863-0_4
Download citation
DOI: https://doi.org/10.1007/978-3-0348-7863-0_4
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9601-6
Online ISBN: 978-3-0348-7863-0
eBook Packages: Springer Book Archive