206 D.V. Millionschikov
−k[e^1 ]0 k[e^1 ] H^1 (G 1 /Γ 1 ,R)=RFig. 11.3.The finite subsetΩG 1 /Γ 1and therefore the cohomologyHω∗(G 1 /Γ 1 ,R) is trivial if [ω]=0,±k[e^1 ].(a)Hk∗[e (^1) ](G 1 /Γ 1 ,R) is spanned by two classes:
e^2 =e−kzdx, e^1 ∧e^2 =dz∧e−kzdx.
(b)H∗−k[e (^1) ](G 1 /Γ 1 ,R) is spanned by two classes:
e^3 =ekzdy, e^1 ∧e^3 =dz∧ekzdy.
Hence we have the following Betti numbersbpω=dimHωp(G 1 /Γ 1 ,R)ofthe
solvmanifoldG 1 /Γ 1 :
1 .b^0 ±ke 1 =0,b^1 ±ke 1 =b^2 ±ke 1 =1,b^3 ±ke 1 =0.
2 .b^00 =b^10 =b^20 =b^30 =1.
(11.31)
It was proved by Mostow in [20] that any compact solvmanifoldG/Γis
a bundle with toroid as base space and nilmanifold as fibre, in particular a
solvmanifoldG/Γ is fibred over the circleπ:G/Γ→S^1. Hence the 1-form
π∗(dφ)onG/Γhas no critical points:mp(π∗(dφ)) = 0,∀p. It follows from
Pajitnov’s theorem [4] that forλsufficiently large we haveHλπp∗(dφ)(G/Γ,R)=
0 ,∀p.
Now we are going to introduce an example of solvmanifoldG/Γwith non-
completely solvable Lie groupG(see [21]). LetG 2 be a solvable Lie group of
matrices ⎛
⎜
⎜
⎝cos 2πz sin 2πz 0 x
−sin 2πzcos 2πz 0 y
001 z
0001⎞
⎟
⎟
⎠. (11.32)
A latticeΓ 2 inG 2 is generated by the following matrices:
⎛
⎜
⎜
⎝
cos^2 πnp sin^2 πnp 00
−sin^2 πnp cos^2 πnp 00
001 np
0001⎞
⎟
⎟
⎠,
⎛
⎜
⎜
⎝
100 u 1
010 v 1
001 0
000 1⎞
⎟
⎟
⎠,
⎛
⎜
⎜
⎝
100 u 2
010 v 2
001 0
000 1⎞
⎟
⎟
⎠,
wherenis an integer,p=2, 3 , 4 ,6and
∣
∣
∣
∣
u 1 v 1
u 2 v 2∣
∣
∣
∣= 0, or another type:
Γ ̃ 2 isgenerated by the following matrices:
⎛
⎜
⎜
⎝
1001
0100
0010
0001
⎞
⎟
⎟
⎠,
⎛
⎜
⎜
⎝
1000
0101
0010
0001
⎞
⎟
⎟
⎠,
⎛
⎜
⎜
⎝
100 u
010 v
001 n
0001