Compact flat manifolds with non-vanishing Stiefel–Whitney classes

✍ Scribed by Sung Mo Im; Heung Ki Kim

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 91 KB
• Volume: 96
• Category: Article
• ISSN: 0166-8641
• DOI: 10.1016/s0166-8641(98)00056-x

✦ Synopsis

We construct a class of compact flat Riemannian manifolds M of dimension 2n + 1 with the following properties:

(1) M has holonomy group (Z 2 ) n+1 , (2) M is not a (non-trivial) flat toral extension of a compact flat manifold, (3) the first Betti number of M is 0, and (4) Stiefel-Whitney classes w 2j (M) are non-zero for 0 2j n. This is in the spirit of Vasquez's second example which is in error (Vasquez, 1970).

For each finite group Φ, there is a positive integer N(Φ) such that every flat manifold with holonomy group Φ has dimension higher than N(Φ), then M must be a non-trivial flat toral extension of a compact flat manifold. Vasquez pointed out that N((Z 2 ) n ) n or n -1 depending on n being even or odd. Our result shows that N((Z 2 ) n ) > 2n -1, a much sharper result.