K-theory， Chern-Connes Character and Algebraic Novikov Conjecture - Yihan Zhang.pdf

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1、K-theory,Chern-ConnesCharacterandAlgebraicNovikovConjectureYihanZhangJanuary16,2016Contents1K-theory12Chern-ConnesCharacter23AlgebraicK-theoryandNovikovConjecture31K-theoryRecallhistoryofK-theory.1.Grothendick,Riemann-RochTheorem(foralgebraicvariety).2.Atiyah,Hirzebruch,topologicalK-theor

2、y.3.Quillen,Milnor,Bass,algebraicK-theory.Applications:topology,operatoralgebra,algebra,numbertheory,etc.r11


r1n∪R,aunitalring.M(R)=


:r2R:M(R)=1M(R).nij1n=1nrn1


rnnr11


r1n0r11


r1n











Mn(R),!Mn+1(R);


!:Idempotentp2M1(R),rn1


rnn0rn1


rnn0





0

3、p2=p.Example1.X,acompactspace.R=C(X).Idempotentp2M1(C(X))()vectorbundleoverX.p:X!Mn(C);x7!p(x).1Idempotent(M1(R))=,abeliansemigroup.Twoidempotents[(p;qaresaidtobeequiv-)]1p0alentif9aninvertiblew2Mn(R)s.t.wpw=q.[p]+[q]=.Noticethat0q()1()()()01p001q0=:100q100pGrothendickprocessS(abelians

4、emigroup)!G(S)(abeliangroup).G(s)=f(s;t):s;t2Sg=.(s;t)(s0;t0)i9x2Ss.t.s+t0+x=s0+t+x.[(s;t)]=[(t;s)].Example2.S=N;G(s)=Z.Example3.S=N[f+1g;G(S)=f0g.Noticethat(s;t)(s0;t0);s+t0+1=s0+t+1.Denition1.K0(R)=G(Idempotent(M1(R))=).{∑},agroup.Groupring,C=g2cgg:cg2C,whereelement

5、sarenitesum.{∑}U:`2()!`2(),(U)(x)=(g1x).ThenC=cUB(`2()).ggg2gg{∑}R,aring.Groupring,R=g2rgg:rg2R.Question:whatisK0(R)?2Chern-ConnesCharacterp8p1;T:H!HiscalledaSchattenp-classoperatoriftr(TT)2<+1.T=p∑1pdiag(c1;


;cn;


),tr(TT)2=n=1jcnj<+1.∪1Denition2.Sp,theringofallSchatt

6、enp-classoperators.S=p=1Sp,theringofallSchattenclassoperators.,agroup.S,theringofSchattenclassoperators.S,thegroupring.Motivations:Connes-Moscovici'shigherindextheory(1990s,Topology).M2n,acompactmanifold.D,anellipticdierentialoperatoronM2n.Higherindex,indexD2K(S),=M.01ApproximatingK

7、0(S)usinglocallynitesimplicialhomologygroupofPF().,agroup.8F,anitesubset.Denition3.TheRips'complexPF()isasimplicialcomplexwhosesetofverticesis1andf0;


; ngspansasimplexiij2F.Example4.=Z.F=f1g.fn0;n1gspansasimplex.n1n02F=f1g.PF()formsaline.Example5.