#### Volume 8, issue 2 (2008)

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The decomposition of the loop space of the mod $2$ Moore space

### Jelena Grbić, Paul Selick and Jie Wu

Algebraic & Geometric Topology 8 (2008) 945–951
##### Abstract

In 1979 Cohen, Moore and Neisendorfer determined the decomposition into indecomposable pieces, up to homotopy, of the loop space on the mod $p$ Moore space for primes $p>2$ and used the results to find the best possible exponent for the homotopy groups of spheres and for Moore spaces at such primes. The corresponding problems for $p=2$ are still open. In this paper we reduce to algebra the determination of the base indecomposable factor in the decomposition of the mod $2$ Moore space. The algebraic problems involved in determining detailed information about this factor are formidable, related to deep unsolved problems in the modular representation theory of the symmetric groups. Our decomposition has not led (thus far) to a proof of the conjectured existence of an exponent for the homotopy groups of the mod $2$ Moore space or to an improvement in the known bounds for the exponent of the $2$–torsion in the homotopy groups of spheres.

##### Keywords
mod $2$ Moore spaces, homotopy decomposition, modular representation theory of the symmetric groups
Primary: 55P35
Secondary: 16W30
##### Publication
Received: 26 December 2007
Revised: 28 March 2008
Accepted: 23 April 2008
Published: 20 June 2008
##### Authors
 Jelena Grbić School of Mathematics University of Manchester M12 9PL Manchester United Kingdom http://www.maths.manchester.ac.uk/~jelena Paul Selick Department of Mathematics University of Toronto Toronto, Ontario M5S 3G3 Canada http://www.math.toronto.edu/selick Jie Wu Department of Mathematics National University of Singapore Singapore 119260 Singapore http://www.math.nus.edu.sg/~matwujie