Invariant operator due to F. Klein quantizes H. Poincare's dodecahedral 3-manifold

Abstract

The eigenmodes of the Poincaré dodecahedral 3-manifold $M$ are constructed as eigenstates of a novel invariant operator. The topology of $M$ is characterized by the homotopy group ${\pi }_1(M)$, given by loop composition on $M$, and by the isomorphic group of deck transformations $deck(\tilde{M})$, acting on the universal cover $\tilde{M}$. (${\pi }_1(M)$, $\tilde{M}$) are known to be the binary icosahedral group ${\mathcal{H}}_3$ and the sphere $S^3$ respectively. Taking $S^3$ as the group manifold $SU(2,C)$ it is shown that $deck(\tilde{M})\sim {\mathcal{H}}_3^r$ acts on $SU(2,C)$ by right multiplication. A semidirect product group is constructed from ${\mathcal{H}}_3^r$ as normal subgroup and from a second group ${\mathcal{H}}_3^c$ which provides the icosahedral symmetries of $M$. Based on F. Klein's fundamental icosahedral ${\mathcal{H}}_3$-invariant, we construct a novel hermitian ${\mathcal{H}}_3$-invariant polynomial (generalized Casimir) operator $\mathcal{K}$. Its eigenstates with eigenvalues $\kappa$ quantize a complete orthogonal basis on Poincaré's dodecahedral 3-manifold. The eigenstates of lowest degree $\lambda =12$ are 12 partners of Klein's invariant polynomial. The analysis has applications in cosmic topology \cite{LA},\cite{LE}. If the Poincaré 3-manifold $M$ is assumed to model the space part of a cosmos, the observed temperature fluctuations of the cosmic microwave background must admit an expansion in eigenstates of $\mathcal{K}$.

31 pages, 1 figure, revised version

References in corpus (4)

• Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background
• Topology and the Cosmic Microwave Background
• Topological Lensing in Spherical Spaces
• Eigenmodes of 3-dimensional spherical spaces and their application to cosmology

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