The cohomology
H
xlarge955.gif" alt="lambda" align="BASELINE" BORDER="0">
xlarge969.gif" alt="ohgr" align="BASELINE" BORDER="0">* (
G/
![](/ft/mat/26_00014346/x<font color=)
xlarge915.gif" alt="Gamma" align="BASELINE" BORDER="0">,
![](/ft/mat/26_00014346/x<font color=)
xlarge8450.gif" alt="Copf" align="BASELINE" BORDER="0">) of the de Rham complex
![](/ft/mat/26_00014346/x<font color=)
xlarge923.gif" alt="Lambda" align="BASELINE" BORDER="0">*(
G/
![](/ft/mat/26_00014346/x<font color=)
xlarge915.gif" alt="Gamma" align="BASELINE" BORDER="0">)
![](/ft/mat/26_00014346/x<font color=)
xlarge8855.gif" alt="otimes" align="MIDDLE" BORDER="0">
![](/ft/mat/26_00014346/x<font color=)
xlarge8450.gif" alt="Copf" align="BASELINE" BORDER="0"> of a compact solvmanifold
G/
![](/ft/mat/26_00014346/x<font color=)
xlarge915.gif" alt="Gamma" align="BASELINE" BORDER="0"> with deformed differential
d
xlarge955.gif" alt="lambda" align="BASELINE" BORDER="0">
xlarge969.gif" alt="ohgr" align="BASELINE" BORDER="0"> =
d +
![](/ft/mat/26_00014346/x<font color=)
xlarge955.gif" alt="lambda" align="BASELINE" BORDER="0">
![](/ft/mat/26_00014346/x<font color=)
xlarge969.gif" alt="ohgr" align="BASELINE" BORDER="0">, where
![](/ft/mat/26_00014346/x<font color=)
xlarge969.gif" alt="ohgr" align="BASELINE" BORDER="0"> is a closed 1 -form, is studied. Such cohomologies naturally arise in Morse-Novikov theory. It is shown that, for any completely solvable Lie group
G containing a cocompact lattice
![](/ft/mat/26_00014346/x<font color=)
xlarge915.gif" alt="Gamma" align="BASELINE" BORDER="0">
![](/ft/mat/26_00014346/x<font color=)
xlarge8834.gif" alt="sub" align="MIDDLE" BORDER="0">
G, the cohomology
H
xlarge955.gif" alt="lambda" align="BASELINE" BORDER="0">
xlarge969.gif" alt="ohgr" align="BASELINE" BORDER="0">*(
G/
![](/ft/mat/26_00014346/x<font color=)
xlarge915.gif" alt="Gamma" align="BASELINE" BORDER="0">,
![](/ft/mat/26_00014346/x<font color=)
xlarge8450.gif" alt="Copf" align="BASELINE" BORDER="0">) is isomorphic to the cohomology
H
xlarge955.gif" alt="lambda" align="BASELINE" BORDER="0">
xlarge969.gif" alt="ohgr" align="BASELINE" BORDER="0">*(
![$$\mathfrak{g}$$](/ft/mat/26_00014346/11006_2005_Article_6_TeX2GIFIE1.gif)
) of the tangent Lie algebra
![$$\mathfrak{g}$$](/ft/mat/26_00014346/11006_2005_Article_6_TeX2GIFIE2.gif)
of the group
G with coefficients in the one-dimensional representation
xlarge961.gif" alt="rgr" align="MIDDLE" BORDER="0">
xlarge955.gif" alt="lambda" align="BASELINE" BORDER="0">
xlarge969.gif" alt="ohgr" align="BASELINE" BORDER="0"> :
![](/ft/mat/26_00014346/x<font color=)
xlarge8594.gif" alt="rarr" align="BASELINE" BORDER="0">
![$$\mathbb{K}$$](/ft/mat/26_00014346/11006_2005_Article_6_TeX2GIFIE4.gif)
defined by
xlarge961.gif" alt="rgr" align="MIDDLE" BORDER="0">
xlarge955.gif" alt="lambda" align="BASELINE" BORDER="0">
xlarge969.gif" alt="ohgr" align="BASELINE" BORDER="0">(
![](/ft/mat/26_00014346/x<font color=)
xlarge958.gif" alt="xgr" align="MIDDLE" BORDER="0">) =
![](/ft/mat/26_00014346/x<font color=)
xlarge955.gif" alt="lambda" align="BASELINE" BORDER="0">
![](/ft/mat/26_00014346/x<font color=)
xlarge969.gif" alt="ohgr" align="BASELINE" BORDER="0">(
![](/ft/mat/26_00014346/x<font color=)
xlarge958.gif" alt="xgr" align="MIDDLE" BORDER="0">). Moreover, the cohomology
H
xlarge955.gif" alt="lambda" align="BASELINE" BORDER="0">
xlarge969.gif" alt="ohgr" align="BASELINE" BORDER="0">*(
G/
![](/ft/mat/26_00014346/x<font color=)
xlarge915.gif" alt="Gamma" align="BASELINE" BORDER="0">,
![](/ft/mat/26_00014346/x<font color=)
xlarge8450.gif" alt="Copf" align="BASELINE" BORDER="0">) is nontrivial if and only if -
![](/ft/mat/26_00014346/x<font color=)
xlarge955.gif" alt="lambda" align="BASELINE" BORDER="0">[
![](/ft/mat/26_00014346/x<font color=)
xlarge969.gif" alt="ohgr" align="BASELINE" BORDER="0">] belongs to a finite subset
![$$\tilde \<font color=](/ft/mat/26_00014346/11006_2005_Article_6_TeX2GIFIE5.gif)
Omega _\mathfrak{g} $$
" align="middle" border="0"> of
H1(
G/
![](/ft/mat/26_00014346/x<font color=)
xlarge915.gif" alt="Gamma" align="BASELINE" BORDER="0">,
![](/ft/mat/26_00014346/x<font color=)
xlarge8450.gif" alt="Copf" align="BASELINE" BORDER="0">) defined in terms of the Lie algebra
![$$\mathfrak{g}$$](/ft/mat/26_00014346/11006_2005_Article_6_TeX2GIFIE6.gif)
.