2,2

Rossetti and Szczepanski study the family of closed Riemannian n-manifolds with holonomy group isomorphic to (Z_2)^(n-1), which they call generalized Hantzsche-Wendt manifolds. They prove results on their structure, compute some invariants, find relations between them which they illustrate with a graph connecting the family. A flat manifold is a closed Riemannian manifold with zero sectional curvature. From Bieberbach's theorems, we know that in each dimension there are only a finite number of such manifolds (up to affine equivalence), and efforts are underway to classify them. Recently this has been completed up through dimension 6. In dimension 2, the Klein bottle belongs in this family, and in dimension 3 there are 3 of them: a classical flat manifold first described by Hantzsche and Wendt (now called "didicosm") and 2 nonorientable ones.

Juan P. Rossetti and Andrzej Szczepanski, Generalized Hantzsche-Wendt flat manifolds, Rev. Mat. Iberoamericana 21 (2005) no. 3, pp. 1053-1070.

This is adapted from the table on p. 1061; beta is first Betti number.

dim.|.beta=0.|.beta=1.|.orient.|.nonorient.|.total.|.holonomy reps|

.2.|..0.....|..1.....|..0.....|..1........|..1....|.1............|

.3.|..1.....|..2.....|..1.....|..2........|..3....|.2............|

.4.|..2.....|..10....|..0.....|..12.......|..12...|.2............|

.5.|..23....|..100...|..2.....|..121......|..123..|.3............|

.6.|..352...|..2184..|..0.....|..2536.....|..2536.|.3............|

Sequence in context: A133553 A010571 A067124 this_sequence A073987 A138392 A035087

Adjacent sequences: A124150 A124151 A124152 this_sequence A124154 A124155 A124156

hard,nonn

Jonathan Vos Post (jvospost3(AT)gmail.com), Dec 01 2006