On the coverings of Hantzsche-Wendt manifold

DOI10.2748/TMJ.20210308arXiv2009.06691MaRDI QIDQ6349081

Author name not available (Why is that?)

Publication date: 14 September 2020

Abstract: There are only 10 Euclidean forms, that is flat closed three dimensional manifolds: six are orientable

m a t h c a l G_{1}, d o t s, m a t h c a l G_{6}

and four are non-orientable

m a t h c a l B_{1}, d o t s, m a t h c a l B_{4}

. In the present paper we investigate the manifold

m a t h c a l G_{6}

, also known as Hantzsche-Wendt manifold; this is the unique Euclidean

3

-form with finite first homology group

H_{1} (m a t h c a l G_{6}) = m a t h b b Z_{4}^{2}

. The aim of this paper is to describe all types of

n

-fold coverings over

m a t h c a l G_{6}

and calculate the numbers of non-equivalent coverings of each type. We classify subgroups in the fundamental group

p i_{1} (m a t h c a l G_{6})

up to isomorphism. Given index

n

, we calculate the numbers of subgroups and the numbers of conjugacy classes of subgroups for each isomorphism type and provide the Dirichlet generating series for the above sequences.

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