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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
DOI: https://doi.org/10.1090/S0002-9947-1990-0956029-6
MathSciNet review: 956029
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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.


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DOI: https://doi.org/10.1090/S0002-9947-1990-0956029-6
Keywords: Derived functors, homology, homotopy, Hurewicz homomorphism, $ \Pi $-algebras, spectral sequences
Article copyright: © Copyright 1990 American Mathematical Society