This week I have been reviewing the paper: Platonic polyhedra tune the 3-sphere: Harmonic analysis on simplices by Kramer. This Harmonic analysis can be applied to the cosmic microwave background observed in astrophysics. The Selection rules found in this analysis can detect the multiple connectivity of spherical
3-manifolds on the space part of cosmic space-time.
Harmonic analysis on topological 3-manifolds has been invoked
in cosmological models of the space part of space-time. A direct experimental access to the topology from the autocorrelation of the cosmic matter distribution is difficult. As an alternative, the data from fluctuations of the Cosmic Microwave Background radiation can be examined by harmonic analysis. It is hoped to find in this way the characteristic selection rules and tuning for a specific nontrivial
topology, distinct from the standard simply-connected one.
Viewed on its universal cover S(n−1), a spherical topological manifold of dimension n − 1 forms a prototile on its cover the (n-1)-sphere. The tiling is generated by the fixpoint free action of the group of deck transformations. This group is isomorphic to the first homotopy group π1(M) and hence is a topological invariant.
A basis for the harmonic analysis on the (n-1)-sphere is given by the spherical harmonics which transform according to irreducible representations of the orthogonal group. Multiplicity and selection rules appear in the form of reduction of group representations.
The deck transformations form a subgroup, and so the representations of the orthogonal group can be reduced to those of this subgroup. Upon reducing to the identity representation of the subgroup, the reduced subset of spherical harmonics becomes periodic on the tiling and tunes the harmonic analysis on the (n-1)-sphere to the manifold.
A particular class of spherical 3-manifolds arises from the five Platonic polyhedra. The harmonic analysis on the Poincare dodecahedral 3-manifold was analyzed along these lines. The authors construct the harmonic analysis on simplicial spherical manifolds of dimension n = 1, 2, 3.
Below is listed the five polyhedra and the known order of their homotopy group and the volume fraction frac(M) = |π1(M))|−1 of the prototile with respect to the volume of the 3-sphere. .
The tetrahedron and the dodecahedron display extremal values of the frac(M).
The harmonic analysis on these manifolds can be started from S(n−1). There its basis is the complete, orthonormal set 〈Y λ 〉 of spherical harmonics, the square integrable eigenmodes of S(n−1). To pass
to a 3-manifold M universally covered by S(n−1), the author considers the maximal subset 〈Y λ0 〉 of this basis periodic with respect to deck transformations. Due to the periodicity, it can be restricted to the prototile M and forms its eigenmodes. These periodic eigenmodes tune the sphere S(n−1) to the topology of M.
Of the Platonic 3-manifolds, the Poincare dodecahedral manifold of minimal volume fraction and its eigenmodes have found particular attention. Representation theory was applied to the harmonic analysis on Poincare’s dodecahedral 3-manifold. A comparative study of the harmonic analysis, tuned to different topological 3-manifolds, can provide clues for future applications.
The Platonic tetrahedral 3-manifold
With regard to simplicial manifolds on S(n−1), where n− 1 = 1, 2, 3, the diagram below illustrates symbolically the tilings and simplicial manifolds for n − 1 = 1, 2, 3.
The tetrahedral 3-simplex S0(3) on the sphere S³
Consider the 3-sphere S³ < E4 and an inscribed regular 4-simplex with its vertices enumerated as 1, 2, 3, 4, 5. The full point symmetry of the 4-simplex is S(5). Central projection of the 3-faces of this simplex to S³ yields a tiling with 5 tetrahedral tiles. Choose the tetrahedron obtained by dropping the vertex 5 as the simplicial manifold S0(3). Its internal point symmetry group is S(4). The homotopy group π1( S0(3)) of the Platonic tetrahedron is described by a graph algorithm. Its prime dimension 5 identifies it and the group of deck transformations as the cyclic group C5. The group/subgroup analysis can be used to characterize the harmonic analysis on S0(3).
The reduction S(5) > C5
The cyclic group C5 has the elements
They belong to the classes (4)(1) or (5) of S(5).
The computation of the multiplicity m(f, 0) of the identity representation D0(C5) is straightforward and we include it
in the last column of the table below. The representation D0 is contained once in the representations f =  ,  ,  , , twice in the representation f = , but not in the representations f =  , f = .
Harmonic analysis on S0(3)
Summary the basis construction for the harmonic analysis on S0(3) in terms of C5-periodic states on the sphere S³.
The spherical harmonics for fixed degree 2j = 0, 1, 2, . . . are the Wigner Dj(u) functions. The Wigner Dj functions are the irreducible representations of SU(2,C).
which are homogeneous polynomials with real coefficients of degree 2j in the complex matrix elements they are explicitly given by
The methods of group theory allow the construction and analysis of the harmonic analysis on topological manifolds. This is demonstrated for the simplicial manifold S0(3). The multiplicities provide the specific selection rules for the chosen simplex
topology. The symmetric group S(5) plays a key role. Its representations f =  ,  are eliminated from the harmonic analysis.
In general, the harmonic analysis on two different manifolds M, M′ covered by the sphere S(n−1) is unified by the spherical harmonics and corresponding representations. The differences between topologies appear in the form of different subgroups of deck
transformations. In the harmonic analysis these involve different group/subgroup representations and reductions in O(n,R) > deck(M), O(n,R) > deck(M′).
Intermediate subgroups as S(n + 1) in can dominate the harmonic analysis on spherical manifolds. The reduction O(n,R) > S(n + 1), n > 2 for simplicial manifolds may require generalized Casimir operators.
Selection rules for S(n + 1) > Cn+1 eliminate complete representations Df of the group S(n + 1) from the harmonic analysis on the sphere S(n−1) when restricted to the simplicial manifold.
To see the topological variety of the harmonic analysis, compare the tetrahedral Platonic 3-manifold M analyzed here with the dodecahedral Platonic 3-manifold M′. The homotopy group of Poincare’s dodecahedral 3-manifold M′ is, compare the binary icosahedral group. It ha been found that the isomorphic group deck(M′) acts exclusively as a subgroup of SU(2,C)r from the right on the sphere S³ in the coordinates,
with the consequence of a degeneracy of the dodecahedral eigenmodes. The multiplicity in the reduction from O(4,R) to the
subset of eigenmodes for the dodecahedral 3-manifold is completely resolved by a generalized Casimir operator. Multiplicity analysis in shows that the lowest dodecahedral eigenmodes are of degree (2j) = 12.
Comparison with the harmonic analysis for the simplicial 3-manifold demonstrates a dependence of the selection rules and the spectrum of eigenmodes on the topology and on the topologically invariant subgroups involved. Corresponding implications can be drawn for the use of harmonic analysis in the cosmic topology of 3-space.
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