A Hurewicz spectral sequence for homology

Blanc, David A.

doi:10.1090/S0002-9947-1990-0956029-6

A Hurewicz spectral sequence for homology
HTML articles powered by AMS MathViewer

by David A. Blanc PDF

Trans. Amer. Math. Soc. 318 (1990), 335-354 Request permission

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

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
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
Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
P. J. Hilton, On the homotopy groups of the union of spheres, J. London Math. Soc. 30 (1955), 154–172. MR 68218, DOI 10.1112/jlms/s1-30.2.154
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 39251, DOI 10.1093/qmath/1.1.299
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 43462, DOI 10.1093/qmath/2.1.228
Daniel M. Kan, Minimal free c.s.s. groups, Illinois J. Math. 2 (1958), 537–547. MR 126273
Daniel M. Kan, A relation between $\textrm {CW}$-complexes and free c.s.s. groups, Amer. J. Math. 81 (1959), 512–528. MR 111036, DOI 10.2307/2372755
T. Y. Lin, Homological algebra of stable homotopy ring $\pi _{^{\ast } }$ of spheres, Pacific J. Math. 38 (1971), 117–143. MR 307233
Saunders Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998. MR 1712872

Homology

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

Homotopie des complexes monoïdaux

7

Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. MR 0223432
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
D. G. Quillen, Spectral sequences of a double semi-simplicial group, Topology 5 (1966), 155–157. MR 195097, DOI 10.1016/0040-9383(66)90016-4
Graeme Segal, Classifying spaces and spectral sequences, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 105–112. MR 232393
Christopher R. Stover, A van Kampen spectral sequence for higher homotopy groups, Topology 29 (1990), no. 1, 9–26. MR 1046622, DOI 10.1016/0040-9383(90)90022-C
George W. Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, New York-Berlin, 1978. MR 516508