Stems and spectral sequences

Baues, Hans Joachim; Blanc, David

doi:10.2140/agt.2010.10.2061

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Stems and spectral sequences

Hans Joachim Baues and David Blanc

Algebraic & Geometric Topology 10 (2010) 2061–2078

DOI: 10.2140/agt.2010.10.2061

Abstract

We introduce the category $P s t e m [n]$ of $n$ –stems, with a functor $P [n]$ from spaces to $P s t e m [n]$ . 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 $Q_{∙}$ an $(n + 1)$ –truncated spectral sequence. Moreover, if $Q_{∙} = P [n] X_{∙}$ is the Postnikov $n$ –stem of a simplicial space $X_{∙}$ , the truncated spectral sequence for $Q_{∙}$ is the truncation of the usual homotopy spectral sequence of $X_{∙}$ . 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

References

Publication

Received: 1 April 2010

Revised: 10 August 2010

Accepted: 20 August 2010

Published: 8 October 2010

Authors

Hans Joachim Baues
Max-Planck-Institut für Mathematik Vivatsgasse 7 PO Box 7280 D-53111 Bonn Germany

David Blanc
Department of Mathematics University of Haifa 31905 Haifa Israel

Volume 10, issue 4 (2010)