文档介绍：AnintroductorycourseindifferentialgeometryandtheAtiyah-SingerindextheoremPaulLoyaBinghamtonUniversity,BinghamtonNY13902-6000E-mailaddress:******@-Singerindextheorem?“index”:Thestudentwillbeableto...•heindexofalinearmap.•computetheindexfor“simple”linearmaps.•,notnecessarilyfinite-dimensional,andletL:(ornullspace)ofLisdefinedas→kerL:=vVLv=0.{∈|}ThecokernelofLisbydefinitionWmodulotheimageofL:cokerL:=W/ImL;therepresentsthespace“missedbyL”(notintheimageofL).-dimensional,,thismeansthatLis“almostbijective”inthesensethatLis“almostinjective”becauseLisinjectiveexceptonthefinite-dimensionalpartkerL,andLis“almostsurjective”becauseLonlyafinite-,notethatifLisanisomorphismofthevectorspacesVandW,thenboththekernelandcokernelofLarezero,,thedimensionsofthevectorspacekerLandcokerLareintegers,andtheindexofLisdefinedasthedifference:indL:=dimkerLdimcokerLZ.−∈Inparticular,-dimensional,observethatkerLVisautomati-callyfinite-,ifWisfinite-dimensional,thencok⊂erL=W/ImLisautomaticallyfinite-,ifbothVandWarefinite-dimensional,thenkerLandcokerLarefinite-dimensionalforanygivenlinearmapL:,anylinearmapbetweenfinite-dimensionalvectorspacesisFredholm;→inTheorem??,considertheexample12L=:→-SINGERINDEXTHEOREM?Onecancheckthat21kerL=spanandImL=span.−12Thus,dimkerL=1anddimcokerL=dim(C2/ImL)=1,soindL=11==Id:−=0andcokerL=C2