Cohomology of Groups
Kenneth S.BrownThisbookisbasedonacoursegivenatCornellUniversityandintended
primarilyforsecond-
yeargraduatestudents.Thepurposeofthecoursewas
tointroducestudentswhoknewalittlealgebraandtopology10asubjectin
whichthereisaveryrichinterplaybetweenthetwo.ThusItakeneithera
purelyalgebraicnorapurelytopologicalapproach.butratherIuseboth
algebraicandtopologicaltechniquesastheyseemappropriate.
ThefirstsixchapterscontainwhatIconsidertobethebasicsofthesubject.
Theremainingfourchaptersaresomewhatmorespecializedandreflectmy
ownresearchinterests.Forthemostpart,theonlyprerequisitesforreading
thenotesaretheelementsofalgebra(groupsrings,andmodules,including
tensorproductsovernon-commutativerings)andtheelementsofalgebraic
topology(fundamentalgroup,coveringspaces,simplicialandCW-
com-
plexes,andhomology).Therearehowever,afewtheoremsespeciallyin
thelaterchapters,whoseproofsuseslightlymoretopology(suchasthe
HurewicztheoremorPoincaréduality).Thereaderwhodoesnothavethe
requiredbackgroundintopologycansimplytakethesetheoremsonfaith.
Thereareanumberofexercises,someofwhichcontainresultswhichare
referredtointhetext.Afewortheexerdsesafemarkedwithanasteriskto
warnthereaderthattheyaremoredifficultthantheothersorthattheyrequire
morebackground.
Iamverygratefu1toR.Bieri,l-
P.Serre,U.Stammbach,R.Strebel,and
C.T.C.Wallforhelpfulcommentsonapreliminaryversionorthisbook.