Characteristic Classes in K-Theory General Theory

Transcription

1 Characteristic Classes in K-Theory General Theory Robert Bruner Department of Mathematics Wayne State University Topology Seminar Universitetet i Bergen 10 August 2011 Robert Bruner (Wayne State University) Char classes: general Bergen 1 / 49

2 Outline 1 Chern Classes 2 Equivariant Connective K-Theory 3 The Real Case Robert Bruner (Wayne State University) Char classes: general Bergen 2 / 49

3 Chern Classes Chern classes in general If α : G U(n) we have compatible Chern classes ci H (α) c ku v (α) i ci K (α) ci R (α) i H 2i BG defined by universal example. ku 2i v BG i K 0 BG R(G) All but the ci R example. are familiar, and those we ll derive from the universal Robert Bruner (Wayne State University) Char classes: general Bergen 4 / 49

4 Chern Classes Under restriction to the maximal torus T n U(n), we have Λ i σ i (l 1,..., l n ). We want Then we ll have c i = c i (Λ 1 ) σ i (c 1 (l 1 ),..., c 1 (l n )) c H i ci ku v i ci K ci R H 2i (BU(n); Z) ku 2i v BU(n) i K 0 BU(n) R(U(n)) H 2i (BT n ; Z) ku 2i BT n v i K 0 BT n R(T n ) Robert Bruner (Wayne State University) Char classes: general Bergen 5 / 49

5 Chern Classes Chern classes in representation theory Definition Let n = dim(v ). Then c R k (V ) = k i=0 ( ) n i ( 1) i Λ i (V ) n k That this restricts as desired follows from the requirement that n c i t n i = 0 n (t + c 1 (l i )) = 1 n (t + 1 l i ) = Observe that, since t 1 t is an involution, we also have Λ k (V ) = k i=0 1 ( ) n i ( 1) i ci R (V ) n k n ( 1) i (t + 1) n i Λ i 0 Robert Bruner (Wayne State University) Char classes: general Bergen 6 / 49

6 Chern Classes Definition The Chern (or gamma ) filtration of representation theory: JU i (G) R(G) is the ideal generated by all products c i1 (V 1 ) c ik (V k ) with i 1 + i k i. This is multiplicative: JU j (G)JU k (G) JU j+k (G) Robert Bruner (Wayne State University) Char classes: general Bergen 7 / 49

7 Chern Classes Modified Rees ring Given a ring R and a multiplicative filtration F = {R = F 0 F 1 } the Modified Rees ring MRees(R, F) = { i= N r i t i r i F i for i > 0} The usual Rees ring construction uses F i = I i for an ideal I R. If C is the Chern filtration of R(G) then MRees(RU(G)) := MRees(RU(G), C) is a very good approximation to the image of ku G in KU G. Robert Bruner (Wayne State University) Char classes: general Bergen 8 / 49

8 Equivariant Connective K-Theory The construction The crude truncation ku = KU[0, ) used in the non-equivariant case will not produce an interesting result in the equivariant case. In particular it will not have Euler classes, and would not be complex orientable. Solution: observe that any equivariant ku G should sit in a commutative square F (EG +, ku G ) ku G F (EG +, inf G 1 ku) KU G F (EG +, inf G 1 KU) and define ku G to be the pullback. Robert Bruner (Wayne State University) Char classes: general Bergen 10 / 49

9 Equivariant Connective K-Theory Greenlees [JPAA 2004] showed this has good properties: 1 ku G is a (strict) commutative ring G-spectrum. 2 If H G then res G H ku G = ku H. 3 ku G is a split ring G-spectrum. 4 ku G [v 1 ] = KU G. 5 kug is Noetherian. 6 ku G is complex orientable. 7 kug ku BG is completion 8 There is a local cohomology spectral sequence. Robert Bruner (Wayne State University) Char classes: general Bergen 11 / 49

10 Equivariant Connective K-Theory The same construction works in the real case. We define ko G to be the pullback. ko G F (EG +, inf G 1 ko) KO G F (EG +, inf G 1 KO) Robert Bruner (Wayne State University) Char classes: general Bergen 12 / 49

11 Equivariant Connective K-Theory Calculational consequence The coefficient rings sit in pullback squares ku G KU G ko G KO G ku (BG) KU (BG) ko (BG) KO (BG) Robert Bruner (Wayne State University) Char classes: general Bergen 13 / 49

12 Equivariant Connective K-Theory The coefficients We easily compute the kug i for i near 0 in terms of dimensions and determinants, but our goal is to relate them to the Chern filtration, so we start by studying that. Lemma JU 1 (G) is the augmentation ideal JU(G), consisting of representations of virtual dimension 0. Proof. Since JU is generated by first Chern classes, c1 R (V ) = dim(v ) V, JU JU 1. Robert Bruner (Wayne State University) Char classes: general Bergen 14 / 49

13 Equivariant Connective K-Theory Proof. (Cont.) Conversely, for k > 0, k ( ) n i dim(ck R (V )) = dim ( 1) i Λ i (V ) n k i=0 k ( ) n i = ( 1) i dim Λ i (V ) n k i=0 k ( )( ) n i n = ( 1) i n k i i=0 ( ) k ( ) n k = ( 1) i n k i = 0 i=0 so JU 1 JU. Robert Bruner (Wayne State University) Char classes: general Bergen 15 / 49

14 Equivariant Connective K-Theory Lemma JU 2 consists of representations of virtual dimension 0 and virtual determinant 1. For the defining representation of U(n), det = Λ n = 1 c1 R + + ( 1) n cn R = 1 vc1 ku + + ( v) n cn ku. Since SU(n) is the fiber of det : U(n) U(1), ku BSU(n) = ku BU(n)/(c1 ku (det ) = ku [[c 1,..., c n ]]/((1 Λ n )/v) = ku [[c 1,..., c n ]]/(c 1 vc 2 + v 2 c 3 + ( v) n 1 c n ). ( Unnaturally isomorphic to ku [[c 2,..., c n ]].) Robert Bruner (Wayne State University) Char classes: general Bergen 16 / 49

15 Equivariant Connective K-Theory JU 2 is generated by products of Chern classes and by c i (V ) for i 2. Let JU 2 be the ideal of virtual dimension 0, determinant 1 representations, i.e., those whose classifing map lifts over BSU BU BU Z. For V W JU 2, let δ = det(v ) = det(w ) and n = dim(v ) = dim(w ). δ R(G), so it suffices to show V δ 1 W δ 1 JU 2. So we may assume δ = 1. Robert Bruner (Wayne State University) Char classes: general Bergen 17 / 49

16 Equivariant Connective K-Theory Since V W = (n W ) (n V ), it suffices to show c 1 (W ) = n W JU 2. det(w ) = 1 so the representation W lifts to SU(n). In ku BSU(n), c 1 = vc 2 v 2 c 3 + ± v n 1 c n and it follows that JU 2 JU 2. Robert Bruner (Wayne State University) Char classes: general Bergen 18 / 49

17 Equivariant Connective K-Theory Conversely, we must show that any Chern class c k (V ), k > 1, and any product c 1 (V )c 1 (W ), have dimension 0 and determinant 1. The first was shown already. For the second, recall that det(kv ) = (det(v )) k and det(λ i (V )) = (det(v )) (n 1 i 1) for i > 0. Thus det(c k (V )) = = This is det(v ) raised to the power k i=1 ( 1) i ( n i n k since k 1 > 0. k (det Λ i (V )) (( 1)i ( n i i=1 k (det V ) (( 1)i ( n i i=1 )( ) n 1 = i 1 n k)) n k)( n 1 i 1)) ( ) k ( ) n 1 k 1 ( 1) i = 0, k 1 i 1 i=1 Robert Bruner (Wayne State University) Char classes: general Bergen 19 / 49

18 Equivariant Connective K-Theory Finally, if m = dim(v ) and n = dim(w ). Then dim(c 1 (V )c 1 (W )) = dim((m V )(n W )) = 0 and det(m V )(n W )) = det(mn nv mw + VW ) = 1 since det(vw ) = (det(v )) n (det(w )) m. Robert Bruner (Wayne State University) Char classes: general Bergen 20 / 49

19 Equivariant Connective K-Theory Theorem ku G KU G is a monomorphism in codegrees 5 and kug i = 0 i 0 odd RU(G) i 0 even 0 i = 1 JU(G) i = 2 0 i = 3 JU 2 (G) i = 4 0 i = 5 Use the long exact sequence induced by the cofiber sequence Σ 2 ku ku HZ Σ 3 ku Robert Bruner (Wayne State University) Char classes: general Bergen 21 / 49

20 Equivariant Connective K-Theory Proof. We immediately see that ku 1 G v ku 1 G = 0 is an isomorphism. Restriction to the trivial group shows that ku 0 G H0 BG computes the dimension of the virtual representation, so that ku 2 G = JU(G) = JU 1(G). We then have H 0 BG 0 ku 3 G so that ku 3 G = ku1 G = 0. H 1 BG = 0 also implies that (kug 4 that kug 4 = JU 2(G). v ku 1 G H1 (BG, Z) = 0 v ku 2 G ) = ker(ku2 G H2 BG) so Since every element of H 2 (BG, Z) is a first Chern class, H 2 BG ku 5 G is zero since it is for G = U(1), and hence ku 5 G = ku3 G = 0. Robert Bruner (Wayne State University) Char classes: general Bergen 22 / 49

21 Equivariant Connective K-Theory Beyond this, complications set in. Theorem There are exact sequences 0 H 3 BG Q 1 kug 6 v ku 4 G = JU 2(G) H 4 BG Q 1 kug 7 0 and 0 H 5 BG Q 1 kug 8 v ku 6 G H6 BG Q 1 ku 9 G Robert Bruner (Wayne State University) Char classes: general Bergen 23 / 49

22 The Real Case Representation Rings Restriction, induction and conjugation induce natural transformations between the real, complex, and quaternionic representation rings: RU τ RO c r RU c q RSp rc = 2 cr = 1 + τ τc = c cq = 1 + τ q c = 2 rτ = r τ c = c qτ = q Robert Bruner (Wayne State University) Char classes: general Bergen 25 / 49

23 The Real Case Representation Rings We may describe all these maps quite explicitly. For any compact Lie group, we may choose so that irreducible real representations U i, irreducible complex representations V j, and irreducible quaternionic representations W k RU = Z cu i Z V j, τv j Z cw k RO = Z U i Z rv j Z rqw k RSp = Z qcu i Z qv j Z W k Robert Bruner (Wayne State University) Char classes: general Bergen 26 / 49

24 The Real Case Atiyah-Segal Theorem The Atiyah-Segal Theorem asserts that the natural map S = Ee + EG + induces completion at the augmentation ideal KUG 0 = RU(G) KU0 G (EG +) = KU 0 (BG) KOG 0 = RO(G) KO0 G (EG +) = KO 0 (BG) KSpG 0 = RSp(G) KSp0 G (EG +) = KSp 0 (BG) They also show that [BG, U] = 0 = [BG, O] = [BG, Sp]. Write RO for RO b I and similarly for related rings and ideals. Robert Bruner (Wayne State University) Char classes: general Bergen 27 / 49

25 The Real Case Using representations of G on Clifford modules, Atiyah, Bott and Shapiro show: Theorem 1 KU G = RU(G)[v, v 1 ]. 2 KO G = RO (G)[β, β 1 ] where RO 0 = RO = Z{Ui, rv j, r cw k } RO 1 = RO/RU = F2 {U i } RO 2 = RU/RSp = F2 {cu i } Z{V j } RO 3 = 0 RO 4 = RSp = Z{qcUi, qv j, W k } RO 5 = RSp/RU = F 2 {W k } RO 6 = RU/RO = Z{V j } F 2 { cw k } RO 7 = 0 Robert Bruner (Wayne State University) Char classes: general Bergen 28 / 49

26 The Real Case Coefficients The action of the coefficients, and KO = KU = Z[v, v 1 ] Z[η, α, β, β 1 ] (2η, η 3, ηα, α 2 4β) with v KU 2, η KO 1, α KO 4, and β KO 8, coincide with natural maps in representation theory. Robert Bruner (Wayne State University) Char classes: general Bergen 29 / 49

27 The Real Case For example, η induces the natural quotients RO RO/RU and RSp RSp/RU and the evident inclusions RO/RU RU/RSp and RSp/RU RU/RO. On the level of Clifford algebras, multiplication by η is complexification. Robert Bruner (Wayne State University) Char classes: general Bergen 30 / 49

28 The Real Case Similarly multiplication by α is quaternionification. Precisely, it is qc : RO RSp in degrees 0 mod 8 r c : RSp RO in degrees 4 mod 8 multiplication by 2, F 2 {cu i } Z{V j } Z{V j } F 2 { cw k } and Z{V j } F 2 { cw k } F 2 {cu i } Z{V j } in degrees 2 mod 4, and 0 in odd degrees. Robert Bruner (Wayne State University) Char classes: general Bergen 31 / 49

29 The Real Case Bott Periodicity There is a more elementary deduction of the values of the periodic completed theory, which we will use to analyze the connective case. It uses just the spaces in Bott periodicity, together with the isomorphisms [BG, BO Z] = RO(G), [BG, BSp Z] = RSp(G), and [BG, BU Z] = RU(G), [BG, U] = 0. Robert Bruner (Wayne State University) Char classes: general Bergen 32 / 49

30 The Real Case Recall that Bott periodicity says that starting with BO Z and repeatedly taking loops gives O O/U U/Sp BSp Z Sp Sp/U U/O, and then BO Z again. Robert Bruner (Wayne State University) Char classes: general Bergen 33 / 49

31 The Real Case Theorem The values of [BG, ] on the infinite loop spaces above are as follows: 1 [BG, O] = RO(G)/ RU(G) 2 [BG, Sp] = RSp(G)/ RU(G) 3 [BG, U/Sp] = 0 4 [BG, U/O] = 0 5 [BG, O/U] = RU(G)/ RSp(G) 6 [BG, Sp/U] = RU(G)/ RO(G) Robert Bruner (Wayne State University) Char classes: general Bergen 34 / 49

32 The Real Case Proof. [BG, O] = RO(G)/ RU(G) by mapping BG into the fibration sequence Ω(U) Ω(U/O) O U. BU Z BO Z since [BG, U] = 0. Robert Bruner (Wayne State University) Char classes: general Bergen 35 / 49

33 The Real Case Proof. [BG, Sp] = RSp(G)/ RU(G) by mapping BG into the fibration sequence Ω(U) Ω(U/Sp) Sp U. BU Z BSp Z since [BG, U] = 0. Robert Bruner (Wayne State University) Char classes: general Bergen 36 / 49

34 The Real Case Proof. [BG, U/Sp] = 0 by mapping BG into the fibration sequence U U/Sp BSp Z BU Z, since c : RSp(G) RU(G) is a monomorphism. [BG, U/O] = 0 by mapping BG into the fibration sequence U U/O BO Z BU Z, since c : RO(G) RU(G) is a monomorphism. Robert Bruner (Wayne State University) Char classes: general Bergen 37 / 49

35 The Real Case Proof. [BG, O/U] = RU(G)/ RSp(G) by mapping BG into the fibration sequence Ω 2 (O/U) Ω(U) Ω(O) Ω(O/U) U O. BSp Z BU Z O/U U/Sp and using [BG, U/Sp] = 0. Robert Bruner (Wayne State University) Char classes: general Bergen 38 / 49

36 The Real Case Proof. Finally, for Part (6) we see [BG, Sp/U] = RU(G)/ RO(G) by mapping BG into the fibration sequence Ω 2 (Sp/U) Ω(U) Ω(Sp) Ω(Sp/U) U Sp. BO Z BU Z Sp/U U/O and using [BG, U/O] = 0. Robert Bruner (Wayne State University) Char classes: general Bergen 39 / 49

37 The Real Case The connective case Theorem ko G KO G is a monomorphism in codegrees 7, and kog i = 0 i = 1 JU/JO RU/RO i = 2 JSp/JU RSp/RU i = 3 JSp RSp i = 4 0 i = 5 JU 2 /JSp RU/RSp i = 6 JSpin/JU 2 RO/RU i = 7 Robert Bruner (Wayne State University) Char classes: general Bergen 40 / 49

38 The Real Case The argument goes as in the complex case using the following. Theorem The first nine spaces in the spectrum ko and the Moore-Postnikov factorization of ko KO are 0 BO Z BO Z 1 U/O U/O 2 Sp/U Sp/U 3 Sp Sp 4 BSp BSp Z 5 SU/Sp U/Sp 6 Spin/SU SO/U O/U 7 String Spin SO O 8 BString BSpin BSO BO BO Z Robert Bruner (Wayne State University) Char classes: general Bergen 41 / 49

39 The Real Case This is proved at the same time as we identify maps from BG into various of these homogeneous spaces. Denote the connective covers of O by SO = O 1, Spin = O 3, and String = O 7. Then Theorem For the 0, 2, and 6-connected covers of O and associated homogeneous spaces, we have: 1 Ω(SO/U) = U/Sp and [BG, SO/U] = ĴU(G)/ĴSp(G) 2 Ω(Spin/SU) = SU/Sp and [BG, Spin/SU] = ĴU2(G)/ĴSp(G) 3 ΩSO = O/U and [BG, SO] = ĴO(G)/ĴU(G) 4 ΩSpin = SO/U and [BG, Spin] = ĴSO(G)/ĴU(G) 5 ΩString = Spin/SU and [BG, String] = ĴSpin(G)/ĴU 2(G) Robert Bruner (Wayne State University) Char classes: general Bergen 42 / 49

40 The Real Case The most interesting of these is String. The relevant diagram in this case is Σ 2 HZ Σ 2 HZ/2 β Σ 3 HZ w 2 BU BSO Spin BSU BSpin String The latter fibre sequence shows that Ω(String) = Spin/SU and that [BG, String] is ĴSpin(G)/ĴU 2(G), since [BG, B 2 SU] = [BG, U 5 ] = 0. Robert Bruner (Wayne State University) Char classes: general Bergen 43 / 49

41 The Real Case To get the fiber sequence in the middle row of the previous diagram, consider: ΣHZ/2 ΣHZ/2 w 1 Ωw 2 BU BO SO BU BSO Spin The latter fibre sequence shows that [BG, Spin] = ĴSO(G)/ĴU(G) since, again, [BG, B 2 U] = [BG, SU] = 0. Robert Bruner (Wayne State University) Char classes: general Bergen 44 / 49

42 The Real Case The Bockstein spectral sequence The cofiber sequence Σko η ko c ku results in a Bockstein spectral sequence with first differential cr R Σ 2 ko ku (X )[η] = ko (X ) closely related to 1 + τ in the periodic theory, and to Sq 2 in cohomology. Better, since η 3 = 0, it collapses: E 4 = E at E it is concentrated on lines 0, 1 and 2. Robert Bruner (Wayne State University) Char classes: general Bergen 45 / 49

43 The Real Case The η-c-r sequence, its differential, and related operations Σko η ko c Σ 2 ku r v R ku Σ 2 ko d 1 Σ 2 ku c HF 2 Sq 2 Σ 2 HF 2 Robert Bruner (Wayne State University) Char classes: general Bergen 46 / 49

44 The Real Case Adams Spectral Sequence Since H ku = A//E(1) and H ko = A//A(1), we have Adams spectral sequences Ext s,t E(1) (F 2, H BG) = ku t+s BG and Ext s,t A(1) (F 2, H BG) = ko t+s BG Accounting device, drastically simplifies multiplicative information. Typical uses: show that we have enough elements to generate, show that the Bott map acts monomorphically in a range sufficiently far beyond the edge of periodicity that relations can be accurately detected in periodic K-theory verify that the implications of these relations suffice. Robert Bruner (Wayne State University) Char classes: general Bergen 47 / 49

45 The Real Case End of Part One Robert Bruner (Wayne State University) Char classes: general Bergen 48 / 49

46 The Real Case Robert Bruner (Wayne State University) Char classes: general Bergen 49 / 49