Characteristic Classes in K-Theory Connective K-theory of BG, G Compact Lie

Characteristic Classes in K-Theory Connective K-theory of BG, G Compact Lie Robert Bruner Department of Mathematics Wayne State University Topology Seminar Universitetet i Bergen 10 August 2011 Robert Bruner (Wayne State University) Char classes: k of BG Bergen 1 / 54

Outline 1 Notation 2 Tori 3 Special Unitary Groups 4 Unitary Groups 5 Symplectic Groups 6 Orthogonal Groups 7 Special Orthogonal Groups Robert Bruner (Wayne State University) Char classes: k of BG Bergen 2 / 54

Notation For tori and symplectic tori we have: RU(T n ) = Z[t 1 ±1,..., t±1 n ] RU(Sp(1) n ) = Z[s 1,..., s n ] If λ i is the i th exterior power of the defining representation, then: RU(Sp(n)) = Z[λ 1,..., λ n ] RU(SU(n)) = Z[λ 1,..., λ n 1 ] RU(U(n)) = Z[λ 1,..., λ n, λ 1 n ] RU(O(n)) = Z[λ 1,..., λ n ]/(λ 2 n 1, λ i λ n λ n i ) RU(SO(2n + 1)) = Z[λ 1,..., λ n ] with λ n+i = λ n+1 i RU(SO(2n)) = Z[λ 1,..., λ n 1, λ + n, λ n ]/R with λ n+i = λ n i, λ n = λ + n + λ n and R = ((λ + n + i λ n 2i)(λ n + i λ n 2i) ( λ n 1 2i ) 2 ) Robert Bruner (Wayne State University) Char classes: k of BG Bergen 4 / 54

Tori RU(T n ) = Z[t ±1 1,..., t±1 n ] All simple representations but the trivial one are complex. The integral cohomology ring is with y i = c 1 (t i ). Theorem with y i = c ku 1 (t i) and H BT n = Z[y 1,..., y n ] ku BT n = ku [[y 1,..., y n ]] ku T n = MRees(RU(T n )) = ku [y 1, y 1..., y n, y n ] (vy i y i = y i + y i ) with y i = c ku 1 (t 1 i ) y i /(1 vy i ). Robert Bruner (Wayne State University) Char classes: k of BG Bergen 6 / 54

Tori The calculation of ku T n uses the pullback square ku T n ku BT n KU T n KU BT n We have the relations t i = 1 vy i and t 1 i = 1 vy i, hence KUT n = KU [t 1 ±1,..., t±1 n ] = KU [y 1, y 1,..., y n, y n ]/R, where R is the ideal generated by the relations vy i y i = y i + y i. Then ku T n = ku [[y 1,..., y n ]] KU [y 1, y 1,..., y n, y n ]/R Robert Bruner (Wayne State University) Char classes: k of BG Bergen 7 / 54

Tori Real connective K-theory H BT n is free over E[Sq 2 ] = H Cη, and ko Cη = ku. Hence, the Adams spectral sequence for ko BT n collapses and is concentrated in even degrees. Hence η acts trivially and the η c R sequence is just a short exact sequence 0 ko BT n c ku BT n R ko +2 BT n 0 Complex conjugation acts by τ(v) = v τ(y i ) = τ( 1 t i v ) = 1 t 1 i v = y i Robert Bruner (Wayne State University) Char classes: k of BG Bergen 8 / 54

Tori The Bockstein differential cr is then exact and ko -linear, hence 2v 2 linear. But ku BT n is 2-torsion-free, so cr is v 2 -linear. Hence v 2 cr(x) = cr(v 2 x) = cr(vx) = (1 + τ)(vx) = vx vτ(x) or Theorem ko T n = (ku T n ) C 2 cr(x) = x τ(x) v where C 2 acts by conjugation. Proof. ko T n = im(c) = ker(r) = ker(vcr) = ker(1 τ). Robert Bruner (Wayne State University) Char classes: k of BG Bergen 9 / 54

Special Unitary Groups RU(SU(n)) = Z[λ 1,..., λ n 1 ]. We have λ i = λ n i, so the λ i are all complex unless n = 2m, when λ m is real if m is even and quaternionic if m is odd. The integral cohomology is H BSU(n) = Z[c 2,..., c n ] with c i = c i (λ 1 ). The connective complex K-theory is easy to compute. Theorem ku BSU(n) = ku [[c 2,..., c n ]] and ku SU(n) = MRees(RU(SU(n))) = ku [c 2,..., c n ]. Robert Bruner (Wayne State University) Char classes: k of BG Bergen 11 / 54

Special Unitary Groups Proof. Since H BSU(n) is concentrated in even degrees, the Atiyah-Hirzebruch spectral sequence implies ku BSU(n) must be the complete ku algebra freely generated by c 2,..., c n. In KUSU(n), we have λ i = i j=0 ( ) n j ( 1) j cj R = n i i j=0 ( ) n j ( 1) j v j cj ku. n i and λ n = 1. Hence, the c j = cj ku generate, and c 1 vc 2 + + ( v) n 1 c n = 0. Thus KUSU(n) is polynomial on any n 1 of c 1,..., c n. In particular, KUSU(n) = KU [c 2,..., c n ]. Robert Bruner (Wayne State University) Char classes: k of BG Bergen 12 / 54

Special Unitary Groups Proof. (Cont.) The pullback square ku SU(n) KU [c 2,..., c n ] ku [[c 2,..., c n ]] KU [[c 2,..., c n ]] shows that ku SU(n) = ku [c 2,..., c n ]. Robert Bruner (Wayne State University) Char classes: k of BG Bergen 13 / 54

Unitary Groups RU(U(n)) = Z[λ 1,..., λ n, λ 1 n ] The integral cohomology is H BU(n) = Z[c 1,..., c n ] where c i = c i (λ 1 ). Again, the complex connective K-theory follows immediately. Theorem ku BU(n) = ku [[c 1,..., c n ]] and ku U(n) = MRees(RU(U(n))) = ku [c 1,..., c n, 1 ] where = λ n = 1 vc 1 + v 2 c 2 + ( v) n c n. Proof. The argument is nearly the same as for SU(n), except that KUU(n) polynomial, but is instead KU [c 1,..., c n, 1 ]. is not Robert Bruner (Wayne State University) Char classes: k of BG Bergen 15 / 54

Unitary Groups In cohomology, restriction along the inclusion SU(n) U(n) is the quotient which sends c 1 to 0. The proper way to think of this is that we are taking the quotient by the determinant of the defining representation. In K-theory, the map this induces is more interesting. Theorem The restriction homomorphism kuu(n) ku SU(n) is the quotient ku [c 1,..., c n, 1 ] ku [c 2,..., c n ] which sends to 1 and c 1 to vc 2 v 2 c 3 + ( v) n 1 c n. Proof. SU(n) is the kernel of the determinant U(n) U(1). The determinant sends y = (1 λ 1 )/v kuu(1) 2 to (1 λ n)/v = c 1 vc 2 + v 2 c 3, so this must go to zero in kusu(n). After dividing by this, we have an isomorphism, by the calculation of kusu(n). Robert Bruner (Wayne State University) Char classes: k of BG Bergen 16 / 54

Unitary Groups Consider the conjugate Chern classes c i (V ) = c i (V ). Proposition Restriction kuu(n) ku T (n) sends c i to σ i (y 1,..., y n ). There is a relation n ( ) k c i = ( 1) k v k i c k i k=i The conjugate = λ n = 1 vc 1 + v 2 c 2 ± v n c n satisfies = 1. Collecting terms we find Proposition In ku U(n), c 1 + c 1 = 2n ( v) k 1 c i c j k=2 i+j=k Robert Bruner (Wayne State University) Char classes: k of BG Bergen 17 / 54

Symplectic Groups RU(Sp(n)) = Z[λ 1,..., λ n ] The λ 2i are real and the λ 2i+1 are quaternionic, hence all are self conjugate. Note that λ 1 = H n = C 2n, which is 2n dimensional, but its higher exterior powers λ n+1,... λ 2n can be expressed in terms of the first n. The integral cohomology is with p i = 4i. H BSp(n) = Z[p 1,..., p n ] Restriction along Sp(1) n Sp(n) will play much the same role for Sp(n) as restriction along T n = U(1) n U(n) plays for U(n), so we start by considering Sp(1) n. Robert Bruner (Wayne State University) Char classes: k of BG Bergen 19 / 54

Symplectic Groups where s i : Sp(1) n Hence c ku 1 (s i) = vc ku 2 (s i) and RU(Sp(1) n ) = Z[s 1,..., s n ] p i Sp(1) = SU(2) U(2) v 2 c ku 2 (s i ) = vc ku 1 (s i ) = c R 2 (s i ) = c R 1 (s i ) = 2 s i. Thus, we have classes z i = c ku 2 (s i) ku 4 (BSp(1) n ) which satisfy v 2 z i = 2 s i. We will see that z i comes from ko. The integral cohomology ring is H BSp(1) n = Z[z 1,..., z n ] with z i = p 1 (s i ), the first Pontrjagin class of s i. Robert Bruner (Wayne State University) Char classes: k of BG Bergen 20 / 54

Symplectic Groups Theorem There are compatible generators z i so that ku BSp(1) n = ku [[z 1,..., z n ]] ko BSp(1) n = ko [[z 1,..., z n ]] ku Sp(1) n = ku [z 1,..., z n ] = MRees(RU(Sp(1) n )) ko Sp(1) n = ko [z 1,..., z n ] In particular, z ku i ku 4 Sp(1) n and z ko i ko 4 Sp(1) n satisfy v 2 z ku i = 2 s i ku 0 Sp(1) n = RU(Sp(1) n ), αz ko i βz ko i = 2(2 s i ) ko 0 Sp(1) n = RO(Sp(1) n ) and = 2 s i ko 4 Sp(1) n = RSp(Sp(1) n ). Robert Bruner (Wayne State University) Char classes: k of BG Bergen 21 / 54

Symplectic Groups Proof. The Adams spectral sequence collapses at E, 2 = H BSp(1) n Ext, A(1) (F 2, F 2 ) = ko BSP(1) n and similarly for E(1) and ku. The equivariant cases then follow by the defining pullback squares ku Sp(1) n KU [z 1,..., z n ] ku [[z 1,..., z n ]] KU [[z 1,..., z n ]] The periodic groups are as claimed because we can change generators from the s i to the z i = (2 s i )/v 2. This is MRees(RU(Sp(1) n )): all irreducible representations are two dimensional, so JU 2n = JU 2n 1 = (JU 2 ) n. Robert Bruner (Wayne State University) Char classes: k of BG Bergen 22 / 54

Symplectic Groups Pontrjagin classes Definition The k th representation theoretic Pontrjagin class of an n-dimensional symplectic representation V : G Sp(n) is Proposition p R k (V ) = k j=0 ( ) n j ( 1) j 2 k j Λ j (V ) n k The restriction RU(Sp(n)) RU(Sp(1) n ) sends pk R to σ k (2 s 1,..., 2 s n ). The representation pk R is real if k is even, and quaternionic if k is odd. Accordingly, we shall generally consider p2i R as an element of RO(G) and p2i+1 R as an element of RSp(G). Note, however, that representations which are not irreducible can be both real and quaternionic. Robert Bruner (Wayne State University) Char classes: k of BG Bergen 23 / 54

Symplectic Groups Theorem We have ku BSp(n) = ku [[p 1,..., p n ]] ko BSp(n) = ko [[p 1,..., p n ]] ku Sp(n) = ku [p 1,..., p n ] ko Sp(n) = ko [p 1,..., p n ]. In each case, p k restricts to σ k (z 1,..., z n ). In ku, v 2k p ku k = p R k ku0 Sp(n) = RU(Sp(n)). In ko, β k p2k ko = pr 2k ko0 Sp(n) = RO(Sp(n)) and β k p2k+1 ko = pr 2k+1 ko4 Sp(n) = JSp(Sp(n)). Robert Bruner (Wayne State University) Char classes: k of BG Bergen 24 / 54

Symplectic Groups Definition Let V : G Sp(n) be a symplectic representation. For E = RU, ko, KO, ku, KU or H, we define the Pontrjagin class pi E (V ) EG 4i to be V (p i ). It is convenient to collect these into the total Pontrjagin class p E (V ) = 1 + p1 E (V ) + p2 E (V ) + + pn E (V ) and to let pi E (V ) = 0 if i > n. Corollary p E (V W ) = p E (V )p E (W ) Robert Bruner (Wayne State University) Char classes: k of BG Bergen 25 / 54

Symplectic Groups Lemma The restriction ku Sp(1) n ku T (n) maps z i to y i y i. Write c i (V ) = c i (V ) for the Chern classes of the complex conjugate of a representation. Theorem The restriction maps kuu(2n) q kusp(n) ec kuu(n) obey c k ( ) k i v k 2i p k i c i c j. i 0 2i k i+j=k Specializing to ordinary cohomology by setting v = 0 we obtain the usual relations (up to sign): q (pn H ) = ci H c H j = ( 1) j ci H cj H, i+j=2n i+j=2n c (c 2i 1 ) = 0, and c (c 2i ) = p i. Robert Bruner (Wayne State University) Char classes: k of BG Bergen 26 / 54

Symplectic Groups For Sp(4), for example, c 1 vp 1 c 1 + c 1 c 2 p 1 +v 2 p 2 c 2 + c 1 c 1 + c 2 c 3 2vp 2 +v 3 p 3 c 3 + c 2 c 1 + c 1 c 2 + c 3 c 4 p 2 +3v 2 p 3 +v 4 p 4 c 4 + c 3 c 1 + c 2 c 2 + c 1 c 3 + c 4 c 5 3vp 3 +4v 3 p 4 c 4 c 1 + c 3 c 2 + c 2 c 3 + c 1 c 4 c 6 p 3 +6v 2 p 4 c 4 c 2 + c 3 c 3 + c 2 c 4 c 7 4vp 4 c 4 c 3 + c 3 c 4 c 8 p 4 c 4 c 4 Robert Bruner (Wayne State University) Char classes: k of BG Bergen 27 / 54

Symplectic Groups It is more difficult to get good expressions for the images of the individual p i. However, for i = 1, using the fact that v acts monomorphically on kuu(n) we have p 1 c 1 + c 1 v n = ( v) k 2 c i c j k=2 i+j=k = c 2 + c 1 c 1 + c 2 v n ( v) k 3 c i c j. k=3 i+j=k In cohomology, where v = 0 and c i = ( 1) i c i, we have p 1 c 2 + c 1 c 1 + c 2 = 2c 2 c 2 1 with our normalization of the p i. Thus, if c 1 = 0, then c 2 = p 1 /2. Robert Bruner (Wayne State University) Char classes: k of BG Bergen 28 / 54

Symplectic Groups Finally, we provide the following symplectic splitting principle. Theorem Let ξ be an Sp(n) bundle over X. Then there exists a map f : Y X such that f ξ is a sum of symplectic line bundles and f : H X H Y is a monomorphism. Proof. Let Y be the pullback Y X f BSp(1) n BSp(n) along the classifying map of the bundle ξ. The universal bundle over BSp(n) splits as a sum of line bundles over BSp(1) n, so f ξ also splits in this manner. The Serre spectral sequence gives the cohomology statement. Robert Bruner (Wayne State University) Char classes: k of BG Bergen 29 / 54

Orthogonal Groups RU(O(n)) = Z[λ 1,..., λ n ]/(λ 2 n 1, λ i λ n λ n i ). These representations are all real, so that complexification and quaternionification are isomorphisms RO(O(n)) = = RU(O(n)) RSp(O(n)). The integral cohomology is complicated. The best approach is to give the mod 2 cohomology, and if integral issues matter, the cohomology localized away from 2. We have where w i = w i (λ 1 ). HF 2BO(n) = F 2 [w 1,..., w n ] Robert Bruner (Wayne State University) Char classes: k of BG Bergen 31 / 54

Orthogonal Groups Rewrite the representation ring in terms of Chern classes, as usual: let c i = ci K (λ 1 ) KO(n) 2i, so that λ i = i j=0 ( ) n j ( 1) j v j c j. n i Rather than replace λ n by the top Chern class, c n, we use the first Chern class of the determinant representation, c = c K 1 (λ n) K 2 O(n). This satisfies vc = 1 λ n, which is much more convenient than v n c n = 1 λ 1 + + ( 1) n λ n. Proposition KU O(2n+1) = KU [c 1,..., c n, c]/(vc 2 2c) and KU O(2n) = KU [c 1,..., c n, c]/(vc 2 2c, c n ( ) 2n i ( v) i c i ) n i=0 Robert Bruner (Wayne State University) Char classes: k of BG Bergen 32 / 54

Orthogonal Groups To compute ku BO(n), we need to determine the E(1)-module structure of HF 2BO(n). We start with its stable type. Let ɛ be 0 or 1. First, the submodule F 2 [w 2 2, w 2 4,..., w 2 2n] H BO(2n + ɛ) is a trivial E(1)-submodule. Second, the reduced homology of BO(1) is the ideal (w 1 ) in F 2 [w 1 ], and as an E(1)-submodule, (w 1 ) F 2 [w 2 2, w 2 4,..., w 2 2n 2] H BO(2n ɛ) is a direct sums of suspensions of (w 1 ). The sum of these two submodules exhausts the interesting part of H BO(n), in the sense that the complementary summand is E(1)-free. Robert Bruner (Wayne State University) Char classes: k of BG Bergen 33 / 54

Orthogonal Groups Theorem The inclusions and F 2 [w 2 2, w 2 4,..., w 2 2n] (w 1 ) F 2 [w 2 2, w 2 4,..., w 2 2n 2] H BO(2n) F 2 [w 2 2, w 2 4,..., w 2 2n] (w 1 ) F 2 [w 2 2, w 2 4,..., w 2 2n] H BO(2n + 1) induce isomorphisms in Q 0 and Q 1 homology. Corollary As an E[Q 0, Q 1 ]-module, H BO(n) is the sum of trivial modules, suspensions of H BO(1), and free modules. Proof. The corollary follows by the result of Adams and Margolis, that Q 0 and Q 1 homology detects the stable isomorphism type of the module. Robert Bruner (Wayne State University) Char classes: k of BG Bergen 34 / 54

Orthogonal Groups In principle, this describes H BO(n) as an E(1)-module but finding a good parametrization of the complementary E(1)-free submodule is non-trivial. The A(1)-module structure is not as simple, as Sq 2 does not annihilate all squares. The w 2 2i detect Pontrjagin classes p i of the defining representation and w 2 1 in the (w 1 ) summand detects the first Chern class of the determinant representation. Corollary The Adams spectral sequence converging to ku BO(n) collapses at E 2, and the natural homomorphism is a monomorphism. ku BO(n) H BO(n) KU BO(n) Robert Bruner (Wayne State University) Char classes: k of BG Bergen 35 / 54

Orthogonal Groups Comments on the proof The H(, Q 0 ) isomorphism is straightforward, but the H(, Q 1 ) isomorphism requires a careful choice of generators. Once the correct generators are identified, it turns out that the general case is just a regraded version of H BO(4) tensored with an E(1)-trivial subalgebra. See the book with Greenlees for details. Robert Bruner (Wayne State University) Char classes: k of BG Bergen 36 / 54

Orthogonal Groups O(1) Recall that ku O(1) = ku [c]/(vc 2 2c) by the pullback square ku O(1) KU O(1) = KU [c]/(vc 2 2c) ku BO(1) = ku [[c]]/(vc 2 2c) KU BO(1) = KU [[c]]/(vc 2 2c) The Bockstein spectral sequence then gives Theorem There are unique elements p 0 ko 0 O(1) and p 1 ko 4 O(1) with complexifications c(p 0 ) = vc and c(p 1 ) = c 2. The ring ko O(1) = ko [p 0, p 1 ] (ηp 1, αp 1 4p 0, βp 1 αp 0, p 0 p 1 2p 1, p 2 0 2p 0) Robert Bruner (Wayne State University) Char classes: k of BG Bergen 37 / 54

Orthogonal Groups In terms of representation theory, this can be written as follows. Corollary O(1)-equivariant connective real K-theory has coefficient ring koo(1) i = To justify the notation p i : Theorem The restriction ku Sp(1) ku O(1) is: RSp i = 8k 4 0 RO/2 i = 8k 2 0 RO/2 i = 8k 1 0 RO i = 8k 0 JSp k = JSp k i = 4k > 0 0 otherwise z = p 1 (λ 1 ) p 1, αz 4p 0, and βz αp 0, Robert Bruner (Wayne State University) Char classes: k of BG Bergen 38 / 54

Orthogonal Groups Proof. That z maps to p 1 is evident by comparison with ku. The rest follows by the relations in ko O(1). Thus, p 1 really is the first Pontrjagin class of the quaternionic representation induced up from the defining representation of O(1), while p 0 is a genuinely real class. We call it p 0 because of the relations which tie it to p 1. Robert Bruner (Wayne State University) Char classes: k of BG Bergen 39 / 54

Orthogonal Groups O(2) Corollary Proof. The calculation KU O(2) = KU [c, c 1 ]/(vc 2 2c, c(2 vc 1 )) = KU [c, c 2 ]/(vc 2 2c, cc 2 ) v 2 c 2 = 1 λ 1 + λ 2 = 1 (2 vc 1 ) + (1 vc) = v(c 1 c) shows that c 1 = c + vc 2. Then the relation 0 = c(2 vc 1 ) becomes v 2 cc 2 = 0 since c(2 vc) = 0. Robert Bruner (Wayne State University) Char classes: k of BG Bergen 40 / 54

Orthogonal Groups The connective K-theory is similar but somewhat larger. Theorem ku O(2) = ku [c, c 2 ]/(vc 2 2c, 2cc 2, vcc 2 ) Proof. Decomposing H BO(2) as an E[Q 0, Q 1 ]-module shows that c and c 2 are algebra generators for ku BO(2). The monomorphism into H BO(2) KU BO(2) then shows the relations are complete. The pullback square then gives us ku O(2). Robert Bruner (Wayne State University) Char classes: k of BG Bergen 41 / 54

Orthogonal Groups The Bockstein spectral sequence then gives Theorem KOO(2) = KO [p 0, r 0 ]/(p0 2 2p 0, p 0 r 0 ) and koo(2) = ko [p 0, p 1, p 2, r 0, r 1, s]/i where I is the ideal generated by the relations ηp 1 = 0 αp 1 = 4p 0 βp 1 = αp 0 ηr 1 = 0 αr 1 = 4r 0 βr 1 = αr 0 ηs = 0 αs = 0 βs = η 2 r 0 p 0 p 2 = 0 p 1 p 2 = s 2 p 0 p 1 r 0 r 1 s p 0 2p 0 2p 1 0 0 0 p 1 2p 1 p1 2 0 0 r 0 0 0 βp 2 αp 2 p 1 s η 2 p 2 r 1 0 0 αp 2 4p 2 0 Robert Bruner (Wayne State University) Char classes: k of BG Bergen 42 / 54

Orthogonal Groups p 0 and r 0 are 1 det and the Euler class of the defining representation, respectively. p 1 and r 1 are their images in JSp = ko 4. This explains the similarity of the action of ko on them. The class p 2 refines the square of the Euler class in the sense that r 2 1 = 4p 2, r 0 r 1 = αp 2 and r 2 0 = βp 2. The class s is a square root of the product p 1 p 2 = s 2. The relation βs = η 2 r 0 is hidden in the Bockstein spectral sequence. Representation theory (i.e., the map into KOO(2) ) and the Adams spectral sequence each work to recover it. Robert Bruner (Wayne State University) Char classes: k of BG Bergen 43 / 54

Orthogonal Groups O(3) Corollary KU O(3) = KU [c, c 1 ]/(vc 2 2c) Proposition KO O(3) = KO [p 0, q 0 ]/(p 2 0 2p 0) where p 0 and q 0 have complexifications vc and vc 1 respectively. The restriction KO O(3) KO O(2) sends p 0 to p 0 and q 0 to p 0 + r 0. The Chern classes no longer suffice to generate kuo(n) for n > 2. Let Q 0 : H ΣHZ and Q 1 : HZ Σ 3 ku be the boundary maps in the cofiber sequences for 2 : HZ HZ and v : Σ 2 ku ku. They are lifts of the Milnor primitives Q 0 and Q 1. Robert Bruner (Wayne State University) Char classes: k of BG Bergen 44 / 54

Orthogonal Groups Definition Let q 2 = Q 1 Q 0 (w 2 ) ku 6 BO(3) and q 3 = Q 1 Q 0 (w 3 ) ku 7 BO(3). Proposition The classes q 2 and q 3 are nonzero classes annihilated by (2, v). The class q 3 is independent of c, c 2, and c 3, while q 2 = cc 2 3c 3. These are the only nonzero 2 or v-torsion classes in ku 6 BO(3) and ku 7 BO(3). Theorem ku BO(3) = ku [c, c 2, c 3, q 3 ]/R, where R is an ideal containing (vc 2 2c, 2(cc 2 3c 3 ), v(cc 2 3c 3 ), 2q 3, vq 3, vcc 3 2c 3 ). Robert Bruner (Wayne State University) Char classes: k of BG Bergen 45 / 54

Orthogonal Groups O(n) for larger n The free summands in H BO(n) begin to get more complicated at n = 4. Let us write w S for the product i S w i. Proposition Maximal E(1)-free summands of H BO(n) are: n = 4 n = 5 n = 6 F 2 [w 2 1, w 2 2, w 2 3, w 2 4 ] w 2, w 3, w 4, w 234 F 2 [w 2 1, w 2 2, w 2 4 ] w 24 F 2 [w 2 1, w 2 2, w 2 3, w 2 4, w 2 5 ] w 2, w 3, w 4, w 5, w 234, w 235, w 245, w 345 F 2 [w 2 1, w 2 2, w 2 4, w 2 5 ] w 24, w 34 F 2 [w 2 1, w 2 2, w 2 3, w 2 4, w 2 5, w 2 6 ] w 2, w 3, w 4, w 5, w 6, w 26, w 234, w 235, w F 2 [w 2 1, w 2 2, w 2 3, w 2 4, w 2 5, w 2 6 ] w 236, w 246, w 256, w 346, w 456, w 23456 F 2 [w 2 1, w 2 2, w 2 4, w 2 5, w 2 6 ] w 24, w 34, w 2456 F 2 [w 2 1, w 2 2, w 2 4, w 2 6 ] w 46 Robert Bruner (Wayne State University) Char classes: k of BG Bergen 46 / 54

Orthogonal Groups Remark Each w S generating a free E(1) will give rise to a (2, v)-annihilated class Q 1 Q 0 (w S ) ku BO(n). Robert Bruner (Wayne State University) Char classes: k of BG Bergen 47 / 54

Special Orthogonal Groups with λ n+i = λ n+1 i and RU(SO(2n + 1)) = Z[λ 1,..., λ n ] RU(SO(2n)) = Z[λ 1,..., λ n 1, λ + n, λ n ]/R with λ n+i = λ n i and λ n = λ + n + λ n. The ideal R is generated by one relation (λ + n + i λ n 2i )(λ n + i λ n 2i ) = ( λ n 1 2i ) 2 All the λ i are real. RU(SO(2n)) is free over RU(SO(2n + 1)) on {1, λ + n }. Robert Bruner (Wayne State University) Char classes: k of BG Bergen 49 / 54

Special Orthogonal Groups H BSO(n) = F 2 [w 2,..., w n ] where w i = w i (λ 1 ). We have already examined SO(2) = T (1) and found (writing c 1 rather than y 1 here) and ku SO(2) = ku [c 1, c 1 ]/(vc 1 c 1 = c 1 + c 1 ) ku BSO(2) = ku [[c 1 ]]. The maps induced in ku i by the fibre sequence SO(2) O(2) det O(1) are Proposition det (c) = c, while i (c) = 0, i (c 2 ) = c 1 c 1 and i (c 1 ) = i (c + vc 2 ) = c 1 + c 1. Robert Bruner (Wayne State University) Char classes: k of BG Bergen 50 / 54

Special Orthogonal Groups SO(3) RU(SO(3)) = Z[λ 1 ] with λ 2 = λ 1 and λ 3 = 1. Proposition KU SO(3) = KU [c 2 ] Proof. The Chern classes of the defining representation of SO(3) satisfy c 1 = vc 2, c 3 = 0 and v 2 c 2 = vc 1 = 3 λ 1. Theorem ku SO(3) = ku [c 2, c 3 ]/(2c 3, vc 3 ). The first Chern class, c 1 = vc 2. The restriction ku O(3) ku SO(3) sends c and q 3 to 0, and sends each c i to c i. Robert Bruner (Wayne State University) Char classes: k of BG Bergen 51 / 54

Special Orthogonal Groups Proof. The Adams spectral sequence again collapses and gives us a monomorphism into the sum of mod 2 cohomology and periodic K-theory. This makes it easy to show ku BSO(3) = ku [[c 2, c 3 ]]/(2c 3, vc 3 ). The pullback square then gives ku SO(3) = ku [c 2, c 3 ]/(2c 3, vc 3 ). In general, we expect c 1 = vc 2 v 2 c 3, since this is true in SU(n), but here v 2 c 3 = 0. The restriction from O(3) is computed by using the monomorphism to periodic K-theory plus mod 2 cohomology. Note that q 2 = cc 2 3c 3 restricts to c 3. Robert Bruner (Wayne State University) Char classes: k of BG Bergen 52 / 54

Special Orthogonal Groups Thank you Robert Bruner (Wayne State University) Char classes: k of BG Bergen 53 / 54

Special Orthogonal Groups Robert Bruner (Wayne State University) Char classes: k of BG Bergen 54 / 54