Browsing by Author "Haavisto, Mikael"

Characteristicclassesofferanindispensablefamilyoftopological invariantsinthetheoryofvector bundles, appearing inthe intersectionofdifferential geometry, algebraic topology, andalgebraic geometry. Asthemainfocusof thisworkaretwofundamental setsof characteristicclasses: the ChernclassesandtheCherncharacterclasses. Thetwosetsascharacteristicclassesof complex vectorbundlesareabundantlyencounteredinvariousmathematical contexts. Furthermore,both enjoyaxiomaticdefinitionswhichcharacterizethemuniquelyascohomologyclasses. Forvectorbundlesoversmoothmanifolds,Chern–Weil theoryisanentirelygeometricframeworkin concretelyreproducingcharacteristiccohomologyclasses.ThefundamentalconstructioninChern WeiltheoryforvectorbundlesistheChern–Weilhomomorphism.Thisisanalgebrahomomorphism mappingelementsofthealgebraofinvariantpolynomialsongl(n,C)todeRhamcohomologyclasses onthebasemanifoldof avectorbundle, obtainedfromevaluatingthe invariantpolynomialson localcurvatureformsassociatedtoaconnection. Inthiswork,wereviewthenecessarybackgroundinordertodefinetheaforementionedmapprop erly, andwebrieflystudy its implications. Thecontentof thework includesanintroductionto smoothfiberbundles, thebasicsof thenotionsof connectionsandcurvatureonvectorbundles, aswell as the theoryof thealgebraof invariantpolynomials ongl(n,C). After this, thework finallyculminates intheChern–Weilconstructionanditsconsequences.Morespecifically,wewill verifythatthecharacteristicclassesobtainedviatheChern–Weilmapsatisfyageneralaxiomatic definitionthatisindependentofthegeometricapproachchosenhere.Afterwards,theremainingdis cussiontreatssomebasiccharacteristicsandapplicationsoftheChernclasses,aswellastheChern characterasanaturalringhomomorphismfromtopologicalK-theorytodeRhamcohomology.