Abstract
Finite type nilpotent spaces are weakly equivalent if and only if their singular cochains are quasi-isomorphic as E∞ algebras. The cochain functor from the homotopy category of finite type nilpotent spaces to the homotopy category of E∞ algebras is faithful but not full.
Similar content being viewed by others
References
A. K. Bousfield, The localization of spaces with respect to homology, Topology, 14 (1975), 133–150.
E. Dror, A generalization of the Whitehead theorem, Symposium on Algebraic Topology (Battelle Seattle Res. Center, Seattle, WA 1971), Lect. Notes Math., vol. 249, Springer, Berlin, 1971, pp. 13–22.
W. G. Dwyer and D. M. Kan, Simplicial localizations of categories, J. Pure Appl. Algebra, 17 (1980), 267–284.
W. G. Dwyer and D. M. Kan, Calculating simplicial localizations, J. Pure Appl. Algebra, 18 (1980), 17–35.
W. G. Dwyer and D. M. Kan, Function complexes in homotopical algebra, Topology, 19 (1980), 427–440.
W. G. Dwyer and J. Spaliński, Homotopy theories and model categories, Handbook of algebraic topology, North-Holland, Amsterdam, 1995, pp. 73–126.
V. Hinich, Virtual operad algebras and realization of homotopy types, J. Pure Appl. Algebra, 159 (2001), 173–185.
M. A. Mandell, Equivalence of simplicial localizations of closed model categories, J. Pure Appl. Algebra, 142 (1999), 131–152.
M. A. Mandell, E∞algebras andp-adic homotopy theory, Topology, 40 (2001), 43–94.
M. A. Mandell, Equivariantp-adic homotopy theory, Topology Appl., 122 (2002), 637–651.
J. P. May, Simplicial objects in algebraic topology, D. Van Nostrand Co., Inc., Princeton, NJ – Toronto, ON – London, 1967.
D. G. Quillen, Homotopical algebra, Lect. Notes Math., vol. 43, Springer, Berlin, 1967.
D. G. Quillen, Rational homotopy theory, Ann. Math., 90 (1969), 205–295.
J.-P. Serre, Local fields, Springer, New York, 1979. Translated from the French by M. J. Greenberg.
V. A. Smirnov, Homotopy theory of coalgebras, Math. USSR–Izv., 27 (1986), 575–592.
J. R. Smith, Operads and algebraic homotopy, preprint math.AT/0004003.
D. Sullivan, The genetics of homotopy theory and the Adams conjecture, Ann. Math., 100 (1974), 1–79.
D. Sullivan, Infinitesimal computations in topology, Publ. Math., Inst. Hautes Étud. Sci., 47 (1978), 269–331.
Author information
Authors and Affiliations
About this article
Cite this article
Mandell, M. Cochains and homotopy type. Publ.math.IHES 103, 213–246 (2006). https://doi.org/10.1007/s10240-006-0037-6
Received:
Issue Date:
DOI: https://doi.org/10.1007/s10240-006-0037-6