Modular representations of \(GL(n,p)\), splitting \(\Sigma(CP^{\infty} \times \dots \times CP^{\infty})\) and the \(\beta\)-family as framed hypersurfaces. (English) Zbl 0536.55006
The main result is that the elements \(\beta_ 1,\beta_ 2,...,\beta_{p-2}\) and \(\beta_ p\) in the \(p\)-primary component of the stable homotopy groups of spheres can be represented by framed hypersurfaces. The method relies on results of K. Knapp, which provide preimages for the \(\beta_ i\) under the double complex transfer \(Q\Sigma^ 2(CP^{\infty}\times CP^{\infty})\to Q\Sigma CP^{\infty}\to QS^ 0\) in terms of elements \(x_ 0\), \(x_ 1\) in the stable homotopy of \(CP^{\infty}\times CP^{\infty}\). The main technical tool is a decomposition of the \(p\)-local homotopy type of \(\Sigma(CP^{\infty}\times CP^{\infty})\) as a wedge of \(p(p^ 2-1)/2\) complexes. This enables \(x_ 0\) to be realised unstably in \(\Sigma(CP^{\infty}\times CP^{\infty})\), and \(x_ i\) in \(\Sigma^ p(CP^{\infty}\times CP^{\infty})\). The decomposition is achieved by analysing the action of \(GL(2,p)\) on the \({\mathbb{F}}_ p\)-cohomology of \(CP^{\infty}\times CP^{\infty}\) using modular representation theory.
MSC:
| 55Q45 | Stable homotopy of spheres |
| 55R35 | Classifying spaces of groups and \(H\)-spaces in algebraic topology |
| 57R90 | Other types of cobordism |
| 55P42 | Stable homotopy theory, spectra |
| 20C20 | Modular representations and characters |
Keywords:
stable homotopy; beta-family; framed hypersurface; double complex transfer; localisation; modular representation; primitive idempotent; stable splittings of the classifying space of the \(n\)-torus; \(p\)-primary component of the stable homotopy groups of spheresReferences:
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