Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A Hurewicz spectral sequence for homology

Author: David A. Blanc
Journal: Trans. Amer. Math. Soc. 318 (1990), 335-354
MSC: Primary 55T99; Secondary 55Q35
MathSciNet review: 956029
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For any connected space $ {\mathbf{X}}$ and ring $ R$, we describe a first-quadrant spectral sequence converging to $ {\tilde H_*}({\bf {X}};R)$, whose $ {E^2}$-term depends only on the homotopy groups of $ {\mathbf{X}}$ and the action of the primary homotopy operations on them. We show that (for simply connected $ {\mathbf{X}}$) the $ {E^2}$-term vanishes below a line of slope $ 1/2$; computing part of the $ {E^2}$-term just above this line, we find a certain periodicity, which shows, in particular, that this vanishing line is best possible. We also show how the differentials in this spectral sequence can be used to compute certain Toda brackets.

References [Enhancements On Off] (What's this?)

  • [1] J. F. Adams, Lectures on generalised cohomology, Category Theory, Homology Theory and their Applications, III (Battelle Institute Conference, Seattle, Wash., 1968, Vol. Three), Springer, Berlin, 1969, pp. 1–138. MR 0251716
  • [2] A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972. MR 0365573
  • [3] Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
  • [4] P. J. Hilton, On the homotopy groups of the union of spheres, J. London Math. Soc. 30 (1955), 154–172. MR 0068218
  • [5] P. J. Hilton, Calculation of the homotopy groups of 𝐴_{𝑛}²-polyhedra. I, Quart. J. Math., Oxford Ser. (2) 1 (1950), 299–309. MR 0039251
  • [6] P. J. Hilton, Calculations of the homotopy groups of 𝐴_{𝑛}²-polyhedra. II, Quart. J. Math., Oxford Ser. (2) 2 (1951), 228–240. MR 0043462
  • [7] Daniel M. Kan, Minimal free c.s.s. groups, Illinois J. Math. 2 (1958), 537–547. MR 0126273
  • [8] Daniel M. Kan, A relation between 𝐶𝑊-complexes and free c.s.s. groups, Amer. J. Math. 81 (1959), 512–528. MR 0111036
  • [9] T. Y. Lin, Homological algebra of stable homotopy ring 𝜋_{*} of spheres, Pacific J. Math. 38 (1971), 117–143. MR 0307233
  • [10] Saunders Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998. MR 1712872
  • [11] -, Homology, Springer-Verlag, Berlin and New York, 1963.
  • [12] J. Peter May, Simplicial objects in algebraic topology, Van Nostrand Mathematical Studies, No. 11, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0222892
  • [13] J. C. Moore, Homotopie des complexes monoïdaux, I, Sèm. Henri Cartan 7 (1954-55), $ \S18$.
  • [14] Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. MR 0223432
  • [15] Daniel Quillen, On the (co-) homology of commutative rings, Applications of Categorical Algebra (Proc. Sympos. Pure Math., Vol. XVII, New York, 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 65–87. MR 0257068
  • [16] D. G. Quillen, Spectral sequences of a double semi-simplicial group, Topology 5 (1966), 155–157. MR 0195097
  • [17] Graeme Segal, Classifying spaces and spectral sequences, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 105–112. MR 0232393
  • [18] Christopher R. Stover, A van Kampen spectral sequence for higher homotopy groups, Topology 29 (1990), no. 1, 9–26. MR 1046622, 10.1016/0040-9383(90)90022-C
  • [19] George W. Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, New York-Berlin, 1978. MR 516508

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 55T99, 55Q35

Retrieve articles in all journals with MSC: 55T99, 55Q35

Additional Information

Keywords: Derived functors, homology, homotopy, Hurewicz homomorphism, $ \Pi $-algebras, spectral sequences
Article copyright: © Copyright 1990 American Mathematical Society