L_1hn2-Betti numbers of R-spaces and the integral foliated simplicial volume [Elektronische Ressource] / vorgelegt von Marco Schmidt
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L_1hn2-Betti numbers of R-spaces and the integral foliated simplicial volume [Elektronische Ressource] / vorgelegt von Marco Schmidt

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Marco Schmidt2L -Betti Numbers of R-Spaces and theIntegral Foliated Simplicial Volume2005Mathematik2L -Betti Numbers ofR-Spaces and theIntegral Foliated Simplicial VolumeInaugural-Dissertationzur Erlangung des Doktorgradesder Naturwissenschaften im FachbereichMathematik und Informatikder Mathematisch-Naturwissenschaftlichen Fakult¨atder Westf¨alischen Wilhelms-Universit¨at Munster¨vorgelegt vonMarco Schmidtaus Berlin– 2005 –Dekan: Prof. Dr. Klaus HinrichsErster Gutachter: Prof. Dr. Wolfgang L¨ uckZweiter Gutachter: Prof. Dr. Anand DessaiTag der mundlichen¨ Pr¨ufung: 30. Mai 2005Tag der Promotion: 13. JuliIntroductionThe origin of this thesis is the following conjecture of Gromov [26, Section 8A,(2)2p. 232] revealing a connection between the L -Betti numbers b (M)and the sim-kplicial volumeM ofaclosedorientedconnectedasphericalmanifold M.Conjecture. Let M be a closed oriented connected aspherical manifold with M = 0.Then(2)b (M)=0 forall k≥ 0.k2The first definition of L -Betti numbers for cocompact free proper G-manifoldswith G-invariant Riemannian metric (due to Atiyah [2]) is given in terms of theheat kernel. Wewill briefly recall this original definitionatthebeginningofChap-2ter 1. Today, there is an algebraic and more general definition of L -Betti numberswhich works for arbitrary G-spaces. Analogously to ordinary Bettinumbers, they2are given as the “rank” of certain homology modules.

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Published 01 January 2005
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