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Representation of finite groups and the first Betti number of branched coverings of a universal Borromean orbifold

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Toda M.

Open Mathematics

2004

tom 2

nr 2

218-249

The paper studies the first homology of finite regular branched coverings of a universal Borromean orbifold called B 4,4,4ℍ3. We investigate the irreducible components of the first homology as a representation space of the finite covering transformation group G. This gives information on the first betti number of finite coverings of general 3-manifolds by the universality of B 4,4,4. The main result of the paper is a criterion in terms of the irreducible character whether a given irreducible representation of G is an irreducible component of the first homology when G admits certain symmetries. As a special case of the motivating argument the criterion is applied to principal congruence subgroups of B 4,4,4. The group theoretic computation shows that most of the, possibly nonprincipal, congruence subgroups are of positive first Betti number.

Ograniczanie wyników

1 Open Mathematics

1 Toda M.

1 2004

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Representation of finite groups and the first Betti number of branched coverings of a universal Borromean orbifold