Operations on resolutions and the reverse Adams spectral sequence

Blanc, David A.

doi:10.1090/S0002-9947-1994-1132432-2

Operations on resolutions and the reverse Adams spectral sequence
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by David A. Blanc

Trans. Amer. Math. Soc. 342 (1994), 197-213

DOI: https://doi.org/10.1090/S0002-9947-1994-1132432-2

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Abstract:

We describe certain operations on resolutions in abelian categories, and apply them to calculate part of a reverse Adams spectral sequence, going "from homotopy to homology", for the space ${\mathbf {K}}(\mathbb {Z}/2,n)$. This calculation is then used to deduce that there is no space whose homotopy groups are the reduction $\bmod \; 2$ of ${\pi _\ast }{{\mathbf {S}}^r}$. As another application of the operations we give a short proof of T. Y. Lin’s theorem on the infinite projective dimension of all nonfree $\pi$-modules.

References

J. F. Adams, On the structure and applications of the Steenrod algebra, Comment. Math. Helv. 32 (1958), 180–214. MR 96219, DOI 10.1007/BF02564578
David A. Blanc, A Hurewicz spectral sequence for homology, Trans. Amer. Math. Soc. 318 (1990), no. 1, 335–354. MR 956029, DOI 10.1090/S0002-9947-1990-0956029-6
David Blanc, Derived functors of graded algebras, J. Pure Appl. Algebra 64 (1990), no. 3, 239–262. MR 1061301, DOI 10.1016/0022-4049(90)90060-U

Abelian

algebras and their projective dimension

Higher homotopy operations and the realizability of homotopy groups

Roger Godement, Topologie algébrique et théorie des faisceaux, Publications de l’Institut de Mathématique de l’Université de Strasbourg, XIII, Hermann, Paris, 1973 (French). Troisième édition revue et corrigée. MR 0345092
T. Y. Lin, Homological algebra of stable homotopy ring $\pi _{^{\ast } }$ of spheres, Pacific J. Math. 38 (1971), 117–143. MR 307233
T. Y. Lin, Homological dimensions of stable homotopy modules and their geometric characterizations, Trans. Amer. Math. Soc. 172 (1972), 473–490 (1973). MR 380789, DOI 10.1090/S0002-9947-1972-0380789-8
R. M. F. Moss, On the composition pairing of Adams spectral sequences, Proc. London Math. Soc. (3) 18 (1968), 179–192. MR 220294, DOI 10.1112/plms/s3-18.1.179
Hirosi Toda, Composition methods in homotopy groups of spheres, Annals of Mathematics Studies, No. 49, Princeton University Press, Princeton, N.J., 1962. MR 0143217
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