Homotopy theory of coalgebras

V. A. Smirnov

Izvestiya. Mathematics , Tome 27 (1986) no. 3, pp. 575-592.

Résumé

This paper makes a study of operads and of coalgebras over operads. Certain operads $E_n$ and $E$ are defined, constituting the algebraic analogues of the "little $n$-cube" operads; it is then shown that the singular chain complex $C_*(X;R)$ of a topological space $X$ is a coalgebra over the operad $E$, and that this structure completely determines the weak homotopy type of the space. Bibliography: 26 titles.

Bibliographie

[1] Adams J. F., “On the cobar construction”, Colloque de topologie algebrique (Louvin, 1956), 1957, 81–87 | MR | Zbl

[2] Braun E., “Skreschennye tenzornye proizvedeniya”, Matematika, 6:1 (1962), 33–59 | MR

[3] Massey W. S., Peterson F. P., The mod 2 cohomology structure, Mem. Am. Math. Soc., 74, 1967 | MR | Zbl

[4] Bousfield A. K., Curtis E. B., “A spectral sequence for the homotopy of nice space”, Trans. Amer. Math. Soc., 151 (1970), 457–479 | DOI | MR | Zbl

[5] Curtis E. B., “Some relations between homotopy and homology”, Ann. of Math., 83 (1965), 386–413 | DOI | MR

[6] Bousfield A. K., Kan D. M., “The homotopy spectral sequence of a space with coefficient in a ring”, Topology, 11 (1972), 79–106 | DOI | MR

[7] Bousfield A. K., Curtis E. B., Kan D. M., Quillen D. G., Rector D. L, Schlesinger J. W., “The mod $p$ lower central series and the Adams spectral sequence”, Topology, 5 (1966), 331–342 | DOI | MR | Zbl

[8] Postnikov M. M., Issledovaniya po gomotopicheskoi teorii nepreryvnykh otobrazhenii, Tr. Matemat. instituta im. V. A. Steklova AN SSSR, 46, M., 1955 | MR | Zbl

[9] Mosher R., Tangora M., Kogomologicheskie operatsii i ikh prilozheniya v teorii gomotopii, Mir, M., 1966

[10] Quillen B. G., Homotopical algebra, Lecture Notes in Mathematics, 43, 1967 | MR | Zbl

[11] Quillen B. G., “Rational homotopy theory”, Ann. of Math., 90 (1969), 205–295 | DOI | MR | Zbl

[12] “Gomotopicheskaya teoriya differentsialnykh form”, Sbornik statei, Mir, M., 1981 | MR

[13] “Proceedings of Symposia in Pure mathematics”, Problems, colloquium in algebraic topology, 22, 1971

[14] Bordman Dzh., Fogt R., Gomotopicheski invariantnye algebraicheskie struktury na topologicheskikh prostranstvakh, Mir, M., 1977 | MR

[15] Mei Dzh., “Teoriya beskonechnokratnykh prostranstv petel”, Uspekhi matem. nauk, 36:6 (1981), 137–195 | MR

[16] May J. P., “Homology operations on infinite loop spaces”, Proc. Symp. in Pure Math., 22 (1971), 171–185 | MR | Zbl

[17] May J. P., Simplicial objects in algebraic topology, Princeton, 1967 | MR

[18] Smirnov V. A., “O kotsepnom komplekse topologicheskikh prostranstv”, Matem. sb., 115:1 (1981), 146–158 | MR | Zbl

[19] Maklein S., Gomologiya, Mir, M., 1966

[20] Spener E., Algebraicheskaya topologiya, Mir, M., 1971 | MR

[21] Bukur I., Delyanu A., Vvedenie v teoriyu kategorii i funktorov, Mir, M., 1972 | MR

[22] Dold A., Puppe D., “Homologie nichtadditiver Functoren anvendungen”, Annals of Institute Fourier, 11 (1961), 201–312 | MR | Zbl

[23] Uaitkhed Dzh., Noveishie dostizheniya v teorii gomotopii, Mir, M., 1974 | MR

[24] Stasheff J. B., “Homotopy associativity of $H$-spaces”, Trans. Amer. Math. Soc., 108:2 (1963), 275–312 | DOI | MR | Zbl

[25] Berikashvili N. A., Differentsialy spektralnoi posledovatelnosti, Tr. Tbilisskogo Matem. instituta im. A. M. Razmadze, LI, 1976

[26] Smirnov V. A., “Funktor $D$ i silnaya gomotopiya”, Matemat. zametki, 21:4 (1977), 557–564 | MR | Zbl