Cellularization for exceptional spherical space forms and the flag manifold of \(\mathsf{SL}_3 (\mathbb{R})\). (English) Zbl 1502.57016
J. W. Milnor [Am. J. Math. 79, 623–630 (1957; Zbl 0078.16304)] classified the finite groups that act freely on \(S^3\). For many of these actions equivariant cell decompositions are know. The authors provide such a decomposition in the remaining cases, for the free action of the octahedral group \(\mathcal O\) and the binary icosahedral group \(\mathcal I\). Using curved joins these decompositions extend to higher dimensions. The authors’ method also provides a decomposition of the action of the tetrahedral group \(\mathcal T\). The main result is:
Theorem. Every sphere \(S^{4n-1}\), endowed with the natural free action of \(\mathcal O\) (resp., of \(\mathcal I\) or \(\mathcal T\)), admits an explicit equivariant cell decomposition. As a consequence, the associated cellular homology chain complex is explicitly given in terms of matrices with entries in the group algebras \(\mathbb Z[\mathcal O]\), \(\mathbb Z[\mathcal I]\), and \(\mathbb Z[\mathcal T]\), respectively.
Theorem. Every sphere \(S^{4n-1}\), endowed with the natural free action of \(\mathcal O\) (resp., of \(\mathcal I\) or \(\mathcal T\)), admits an explicit equivariant cell decomposition. As a consequence, the associated cellular homology chain complex is explicitly given in terms of matrices with entries in the group algebras \(\mathbb Z[\mathcal O]\), \(\mathbb Z[\mathcal I]\), and \(\mathbb Z[\mathcal T]\), respectively.
Reviewer: Karl Heinz Dovermann (Honolulu)
MSC:
| 57N60 | Cellularity in topological manifolds |
| 57R91 | Equivariant algebraic topology of manifolds |
| 57M60 | Group actions on manifolds and cell complexes in low dimensions |
| 52B11 | \(n\)-dimensional polytopes |
| 57S17 | Finite transformation groups |
| 57S25 | Groups acting on specific manifolds |
Citations:
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