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Cohomology with local coefficients of solvmanifolds and Morse-Novikov theory | Dmitri V. Millionschikov
; | Date: |
7 Mar 2002 | Subject: | Differential Geometry; Rings and Algebras MSC-class: 58A12, 17B30, 17B56, 57T15 | math.DG math.RA | Abstract: | We study the cohomology $H^*_{lambda omega}(G/Gamma, {mathbb C})$ of the deRham complex $Lambda^*(G/Gamma)otimes{mathbb C}$ of a compact solvmanifold $G/Gamma$ with a deformed differential $d_{lambda omega}=d + lambdaomega$, where $omega$ is a closed 1-form. This cohomology naturally arises in the Morse-Novikov theory. We show that for a solvable Lie group $G$ with a completely solvable Lie algebra $mathfrak{g}$ and a cocompact lattice $Gamma subset G$ the cohomology $H^*_{lambda omega}(G/Gamma, {mathbb C})$ coincides with the cohomology $H^*_{lambda omega}(mathfrak{g})$ of the Lie algebra $mathfrak{g}$ associated with the one-dimensional representation $
ho_{lambda omega}: mathfrak{g} o {mathbb K},
ho_{lambda omega}(xi) = lambda omega(xi)$. Moreover $H^*_{lambda omega}(G/Gamma, {mathbb C})$ is non-trivial if and only if $-lambda [omega]$ belongs to the finite subset ${0cup ilde Omega_{mathfrak{g}}$ in $H^1(G/Gamma, {mathbb C})$ well defined in terms of $mathfrak{g}$. | Source: | arXiv, math.DG/0203067 | Services: | Forum | Review | PDF | Favorites |
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