COMBINATORIAL EXTENSION COHOMOLOGY I GROUPS.pdfVIP

COMBINATORIALEXTENSION COHOMOLOGY IGROUPSA.M. DuPreRutgers UniversityJanuary 7,1993Abstract. A new method is proposed for calculating the measurable, continuous or dier-entiable cohomology of a group extension which involves deriving functional equations for therestrictions of cocycles to certain well-behaved subsets of its domain, and showing that thecocycle can be written as a certain sum of such restrictions. This technique is capable of de-termining how the quotients of the ltration given by the spectral sequence t together, andis applied to the case of the Heisenberg group Hn to yield extremely explicit cocycle represen-tatives, culminating in a stability theorem regarding multilinearizability of cocycles. One ofthe main tools for doing this is the derivation of a formula for trivializing the product of twoalternating multilinear functions, one of which is the nondegenerate bilinear one de ning theHeisenberg group, which has interesting connections with Hodge theory.I. IntroductionThis is the rst in a series of three projected papers. The second will treat Lie algebrasand the third associative algebras.Whenever the subject of cohomology of group extensions arises, it almost automaticallyinvokes the concept of a spectral sequence in response, and the two are uniformly conceivedof as inseparable. This was not always so. Just as Eilenberg and MacLane were enteringinto a longlasting and fruitful collaboration on a series of papers, and before even the rstof this series saw print, an algorithmically inclined Ph.D. student of MacLanes at Harvard,Roger Lyndon, was working on a dissertation which he completed on May 18, 1946 [Ln1].The subject of Lyndons thesis was the problem of computing the cohomology groupsHn(G,A), when G was given as an extension1 !N i!G js H! 1 :Lyndon was only able to completely solve this problem for G the direct product of the nitely generated abelian groups H and N operating trivially on the coecient group Z of1991 Mathematics Subje