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# Risk Theory MATH 523

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This 4 page Class Notes was uploaded by Mrs. Preston Lehner on Thursday October 29, 2015. The Class Notes belongs to MATH 523 at University of Michigan taught by Staff in Fall. Since its upload, it has received 7 views. For similar materials see /class/231505/math-523-university-of-michigan in Mathematics (M) at University of Michigan.

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Date Created: 10/29/15

Chow groups of classifying spaces B urt Tot aro Based on lecture notes by Sam Payne University of Michigan Ann Arbor April 20 2005 1 Classifying spaces in topology First let us make a de nition For any reasonable topological group G eigi a linear algberaic group the classifying space BC is the quotient EG where E is a contractible space with a free G actioni If G is reasonable we can nd such an E and the homotopy type of BC is wellde ned It does not depend on the choice of E The name classifying space77 comes from the classi cation of Gbundles in topology Recall that a principal Gbundle E i X is a free G space E such that X EGi In particular there is a quotient map E A X with every ber homeomorphic to Cl Then BC classi es principal Gbundles in the following sense for any space X the set of isomorphism classes of principal Gbundles on X is in natural l l correspondence with the set X BC of homotopy classes of maps from X to Cl Example 1 For any space X principal GLnCbundles over X correspond naturally and bijectively to ran n complex vector bundles on X up to isomor phismi Since we can put a metric on any complex vector bundle these also correspond to isomorphism classes of principal Unbundles over Xi Similarly principal Onbundles correspond to real rank n vector bundles 2 Transition to algebraic geometry Now let G be a linear algebraic group over a eld k usually C We can try to de ne a variety BGk but typically it will be in nite dimensional For example in topology the classifying space of the circle group B51 or BC has the homotopy type of ClP m which is not homotopy equivalent to any nite dimensional algebraic varietyi In the theory developed by Morel and Voevodsky MV there is an object B tak which is some in nite dimensional object obtained as a limit of algebraic varieties Before Morel7s and Voevodsky7s theory existed I still found a de ni tion of the Chow groups CH BGk using nite dimensional varieties Objects in Morel7s and Voevodskyls theory also have Chow groups but I will explain that in each degree these Chow groups agree in a natural way with the Chow groups of some nite dimensional varieties Let V be any krepresentation of Gk and think of G as a variety Let S C V be a closed Ginvariant subset such that G acts freely on V S Then VSG is some sort of approximation to BGki If V S were contractible this would be perfect but as long as the codimension of S is high this is good enough for low degrees De nition 1 fi lt codimS C V then CHiBGk CHiV SG It is a fact that CHiBGk is wellde ned It is independent of the choices of V and S In Morel7s and Voevodsky7s theory B tak limH V SGi To see that such a V and 5 always exist let W be any faithful krepresenta tion of Gki Look at V WEBNi Then the codimension of the bad set in WEBN increases to in nity as N increases to in nity Example 2 Consider the multiplicative group Gm if you think over the com plex numbers then Gm C What is CH BGm Let W be the standard ldimensional representation of Gmi Then WEBN CN with Gm acting by scalar multiplicationi So Gm acts freely away from 0 and CN 0Gm ClP N li Thus one could say BGm Clme In particular CH BGm ZM with z E CHli From my oldfashioned point of view one should never talk about in nite dimensional spaces but rather one should say that the Chow groups of BGm are the Chow groups of lP N for N gtgt 0 We can always do our computations on nite dimensional varieties Example 3 Consider the group GLnC with W C the standard rep resentation Then V WEBN HomCNC and GLnC acts freely on V S SueromCNCni Think of N increasing to in nity Here V SGLnC GrnCNi So one could say BGLnC Grn Cmyh In particular CH BGLQL C gt H BGLn C Z Zcli i i on where the Ci are Chern classes h This tells you that vector bundles in algebraic geometry have invariants in the Chow ring called Chern classes 7 this has been known for a long time Aside Note that CH BGk is the ring of characteristic classes with values in CHquot for G bundles over smooth kvarieties Is there any way of making sense of the statement XBG principal Gbundles over X in algberaic geometry Example 4 Consider the group Zp7 and let W be the standard ldimensional representation with Zp acting by roots of unity Then WGBN 0G CN 0Zp7 which is a Gm bundle over ClP N l corresponding to the line bundle C10 So CHCN 0Zp E CH ClP N lp z 0i Letting N increase to in nity7 we have CH BZp g Zzpz 0 gt H BZp7 Z where z E CHll Example 5 For p odd7 we have CHBZP P g Zpi117 v x 7171 where each 11 is in degree 1 Then there is a natural map CIWBZ10n10H HBZP 7ZP gZPlty17m7yngtlI17m7 where the yj and 11 have degrees 1 and 2 respectively7 and the natural map takes CH1 into H21quot 7 The Chow cohomology ring of a classifying space is not going to map surjec tively onto the cohomology ring because there is odd dimensional cohomology7 even for groups Still7 it looks like CH BG is the simple part77 of HquotBGZ7 at least in these examples This de nition of Chow groups of classifying spaces was generalized by Edidin and Graham to give a de nition of equivariant Chow groups for arbitrary Gvarieties EC De nition 2 IfX is a Gvarz39ety andi lt codimS C V then CHgX CHi X X g 5 Kresch generalized this de nition further to give a de nition of CH for any stack 3 Kre Roughly speaking7 a stack is something that locally looks like a quotient of a variety by a group If y is the quotient stack XG7 then you get the Edidin Graham de nition of CH It is a fact that7 for any algebraic group G the natural map with coef cients CHBG 8 Q L HBGQ is an isomorphismi One half of this has a simple explanation surjectivity follows from the fact that7 for any algebraic group G HB G is generated by Chern classes of complex representations of A representation G A GLn7 C induces a pullback map CHBG lt7 CHBGLnltC mg an So H BGQ is generated by classes that obviously come from algebraic ge ometry In some examples such as when G is abelian CH BG is generated by Chern classes of complex representations even though HBGZ might not be This ts with the idea that Chow groups are in some sense the nice part of cohomology Example 6 Consider the orthogonal group We have CH BOQL Zcl cnlQcodd 0 where the Ci are the pullbacks of the generators of CH BGLOL C under the natural map BOn A BGLn C Although H BOnZ2 is bigger than CH BOQL ZQ there is an isomorphism after inverting 2 HBOnZ12 g CH BOn mm However there are some nite groups G eg G 55 such that CH BG is not generated by Chern classes For 55 the symmetric group on 6 elements CH BG is at least generated by transfers of Chern classes These transfers are de ned as follows for an inclusion H C G of nite groups there is a natural nite map f BH A BC which is a ber bundle with ber the coset space The push forward f HiBHZ A HiBGZ is the transfer In It is often the case that the cohomology of groups is one step worse than one had reason to expect Schuster Yagita SY and I found that there exists a nite group G such that CH BG is not generated by transfers of Chern classes The example is an extension of 2groups lHZZHGHZ25gtl given by the standard alternating form on Z25 For connected Lie groups we can hope to compute CH BG This has been done for G2 and F4 Yag PGL3 Vez and PGLp for p prime Vis Although the Chow groups for these groups are not generated by Chern classes they are still generated by elements you can write down in a nice way and they are nicer than the cohomology groups There is more to say about this but I will stop for time Question from audience Why should we want to compute the Chow groups of these classifying spaces Response We hope to understand what replaces the Hodge conjecture if you want to work with integral cycles Also Chow groups of classifying spaces are a source of explicit counterexamples for instance there was a conjecture that the torsion in CHiX should map injectively into the Deligne cohomology group HgiXZi This conjecture is true for i 012 and dimX For i 2 it was proved by Merjurjev and Suslin Col and for i dimX it was

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