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226 PAULO TIRAO
(c)1 B1 B2 B3
B1 0 0 0
B2 1 0 1
B3 -1 0 -1
The last two coordinates of the cocycle f ◦ c are zero, while the first two
have values
f ◦ c B1 B2 B3
B1 ( 0 0 ) ( 0 0 ) ( 0 0 )
B2 ( 1 1 ) ( 0 0 ) ( 1 1 )
−1
B3
−1 ( 0 0 )
−1
−1
Now, it is clear that [(f ◦ c)1] ∈ H 2 (Z2 ⊕ Z2; 〈em1+m2+1〉) does not vanish
and that [(f ◦ c) ↾∆s] ∈ H 2 (Z2 ⊕ Z2; ∆s) do not vanish either (see Case iii
in §3), therefore [f ◦ c] = (1, 1) ∈ H 2 (Z2 ⊕ Z2; ∆).
(4.1) The Hantzsche-Wendt module. Consider the Z2 ⊕ Z2-module Λ, of
rank 3, given by the representation χ1 ⊕ χ2 ⊕ χ3. Notice that it is a faithful
module and clearly Λ Z2⊕Z2 = 0.
The Hantzsche-Wendt manifold (see Introduction) is built on this module
(see §5). Thus we will call it the Hantzsche-Wendt module.
By Proposition 4.3 a Z2 ⊕Z2-module Λ admits a special cohomology class
if and only if Λ contains a submodule equivalent to the Hantzsche-Went
module.
We can state the main theorem which is now an immediate consequence
of Proposition 4.3, Lemma 4.4 and (4.1).
Theorem 4.5. The affine equivalence classes of compact Riemannian flat
manifolds with holonomy group Z2 ⊕ Z2 and first Betti number zero are in
a bijective correspondence with the Z[Z2 ⊕ Z2]-modules Λ, such that:
(1) As abelian group Λ is free and of finite rank;
(2) Λ Z2⊕Z2 = 0;
(3) Λ contains a submodule equivalent to the Hantzsche-Wendt module.
For completeness we should treat step (i) at the begining of §4. Actually,
after Theorem 4.5, it would suffice to determine the semi-equivalence classes
of Z2 ⊕ Z2-modules given by
(4.2)
ρ = m1χ1 ⊕ m2χ2 ⊕ m3χ3 ⊕ k1ρ1 ⊕ k2ρ2 ⊕ k3ρ3 ⊕ sµ ⊕ tν
= (m1, m2, m3, k1, k2, k3, s, t)
with m1, m2, m3 ≥ 1, which are already faithful.