覆盖、对应和非交换几何

Inthisthesisweconstructanadditivecategorywhoseobjectsareembeddedgraphs(orinparticularknots)inthe3-sphereandwheremorphismsareformallinearcombinationsof3-manifolds.OurdefinitionofcorrespondencesreliesontheAlexanderbranchedcoveringtheorem[1],whichshowsthatallcompactoriented3-manifoldscanberealizedasbranchedcoveringsofthe3-sphere,withbranchedlocusanembedded(notnecessarilyconnected)graph.Thewayinwhichagiven3-manifoldisrealizedasabranchedcoverishighlynotunique.Itispreciselythislackofuniquenessthatmakesitpossibletoregard3-manifoldsascorrespondences.Infact,weshowthat,byconsideringa3-manifoldMrealizedintwodifferentwaysasacoveringofthe3-sphereasdefiningacorrespondencebetweenthebranchlociofthetwocoveringmaps,weobtainawelldefinedassociativecompositionofcorrespondencesgivenbythefiberedproduct.