Stems and spectral sequences

Abstract

We introduce the category $\mathcal{P}stem\left[n\right]$ of $n$–stems, with a functor $\mathcal{P}\left[n\right]$ from spaces to $\mathcal{P}stem\left[n\right]$. 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 ${\mathcal{Q}}_{\bullet }$ an $\left(n+1\right)$–truncated spectral sequence. Moreover, if ${\mathcal{Q}}_{\bullet }=\mathcal{P}\left[n\right]X_{\bullet }$ is the Postnikov $n$–stem of a simplicial space $X_{\bullet }$, the truncated spectral sequence for ${\mathcal{Q}}_{\bullet }$ 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.