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230 PAULO TIRAO
being those in which the holonomy representation has indecomposable summands
of dimension 3 all new.
Integral Homology. Since we have explicit realizations for all the manifolds
classified it is not difficult to compute their first integral homology
group by the formula H1(M; Z) = Γ/[Γ, Γ].
In all cases Γ = 〈ω1, ω2, Λ〉, where Λ = 〈Le1 , . . . , Len〉, ω1 = B1Lb1 and
ω2 = B2Lb2 (see (5.1) and (5.2)). Hence, [Γ, Γ]=〈[ω1, Lei ]; [ω2, Lei ]; [ω1, ω2]〉.
We have,
[ω1, Lei ] = B1ei − ei = (B1 − I)ei,
[ω2, Lei ] = B2ei − ei = (B2 − I)ei.
Since B1 and B2 are block diagonal, with blocks of rank 1, 2 and 3, we
proceed block by block.
Rank 1. Let Λ = 〈e〉 and suppose B1 = (1) and B2 = (−1). We then have
(B1 − I)e = 0; (B2 − I)e = −2e.
Rank 2. Let Λ = 〈e, f〉 and suppose B1 = J and B2 = −J. We get
immediately
(B1 − I)e = −e + f, (B1 − I)f = e − f;
(B2 − I)e = −e − f, (B2 − I)f = −e − f.
Rank 3. Let Λ = 〈e, f, g〉.
Let B1 and B2 be the first two matrices describing µ as in (3.4). Then,
In this case writing
(B1 − I)e = −2e, (B2 − I)e = −2e,
(B1 − I)f = −f + g, (B2 − I)f = e − f − g,
(B1 − I)g = f − g, (B2 − I)g = −e − f − g.
Λ
=
Λ ′
〈e, f, g〉
= 〈e, f, g〉,
〈Im (B1 − I), Im (B2 − I)〉
it turns out that |e| = 2, 4f = 0, f = g and e = f, from which we conclude
that
Λ
Λ ′ Z2 ⊕ Z4.
Now, let B1 and B2 be the first two matrices describing ν as in (3.4).
Then,
(B1 − I)e = −2e, (B2 − I)e = −2e,
(B1 − I)f = e, (B2 − I)f = −2f,
(B1 − I)g = −2g, (B2 − I)g = −e.