For any connected space <i>X</i> the James construction shows that Ω<span style='font-family:Symbol'>S<i>X</i> is universal in the category of homotopy associative <i>H</i>-spaces in the sense that any map <i>f</i>: <i>X </i><i><span style='font-family:Symbol'>® Y </i>to a homotopy associative <i>H</i>-space factors through a uniquely determined H-map. Let <i>p</i> be a fixed prime number, and <i>X</i> a space localised at <i>p</i>. We study the possibility of generating a universal space <i>U(X)</i> from <i>X</i> which is universal in the category of homotopy associative, homotopy commutative <i>H</i>-spaces in the same way as the James construction of a connected space is universal in the category of homotopy associative <i>H-</i>spaces. We develop a method for constructing certain universal spaces. This method is used to show that the universal space <i>U(X)</i> exists for a certain three-cell complex X. We use this specific example to derive some consequences for the calculation of the unstable homotopy groups of spheres, namely, we obtain a formula for the <i>d</i><sub>1</sub>-differential of the <i>EHP-</i>spectral sequence valid in a certain range. Finally, we apply the developed method to the family of certain two-cell complexes and obtain their universal spaces. This result generalises the result of Cohen, Moore, Neisendorfer and Gray on the universal space of an odd dimensional <i>p-</i>primary Moore space.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:401586 |
| Date | January 2004 |
| Creators | GrbicÌ, Jelena |
| Publisher | University of Aberdeen |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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