Volume 10, issue 4 (2010)

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Stems and spectral sequences

Hans Joachim Baues and David Blanc

Algebraic & Geometric Topology 10 (2010) 2061–2078
Abstract

We introduce the category $\mathsc{P}stem\left[n\right]$ of $n$–stems, with a functor $\mathsc{P}\left[n\right]$ from spaces to $\mathsc{P}stem\left[n\right]$. This can be thought of as the $n$–th order homotopy groups of a space. We show how to associate to each simplicial $n$–stem ${\mathsc{Q}}_{\bullet }$ an $\left(n+1\right)$–truncated spectral sequence. Moreover, if ${\mathsc{Q}}_{\bullet }=\mathsc{P}\left[n\right]{X}_{\bullet }$ is the Postnikov $n$–stem of a simplicial space ${X}_{\bullet }$, the truncated spectral sequence for ${\mathsc{Q}}_{\bullet }$ is the truncation of the usual homotopy spectral sequence of ${X}_{\bullet }$. Similar results are also proven for cosimplicial $n$–stems. They are helpful for computations, since $n$–stems in low degrees have good algebraic models.

Keywords
$n$–stem, Postnikov system, spectral sequence, mapping algebra, spiral long exact sequence
Mathematical Subject Classification 2000
Primary: 55T05
Secondary: 18G40, 18G55, 55S45, 55T15, 18G30, 18G10