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.
@article{bwmeta1.element.doi-10_2478_BF02476541,
author = {Masahito Toda},
title = {Representation of finite groups and the first Betti number of branched coverings of a universal Borromean orbifold},
journal = {Open Mathematics},
volume = {2},
year = {2004},
pages = {218-249},
zbl = {1078.57017},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_BF02476541}
}
Masahito Toda. Representation of finite groups and the first Betti number of branched coverings of a universal Borromean orbifold. Open Mathematics, Tome 2 (2004) pp. 218-249. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_BF02476541/
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