Self-equivalences of dihedral spheres

• Autores: Davide L. Ferrario
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 53, Fasc. 3, 2002, págs. 251-264
• Idioma: inglés
• Títulos paralelos:
  • Autoequivalencias de esferas diédricas
• Enlaces
  • Texto completo
• Resumen

  • Let $G$ be a finite group. The group of homotopy self-equivalences $\mathcal{E}_G(X)$ of an orthogonal $G$-sphere $X$ is related to the Burnside ring $A(G)$ of $G$ via the stabilization map $I:\mathcal{E}_G(X)\subset [X,X]_G\rightarrow \{X,X\}_G = A(G)$ from the set of $G$-homotopy classes of self-equivalences of $X$ to the ring of stable $G$-homotopy classes of self-maps of $X$ (that is, the 0-th dimensional $G$-homotopy group of $S^0$, which is isomorphic to the Burnside ring). As a consequence of the properties of $I$, $\mathcal{E}_G(X)$ is equal to an extension of a subgroup of the group of units in $A(G)$ with the kernel of $I$. The aim of the paper is to give examples of a family of equivariant (dihedral) spheres with the property that the kernel of $I$ is a non-abelian torsion-free group with many generators, and to give estimates on the structure of $\mathcal{E}_G(X)$ itself.

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