KO -theory of complex Stiefel manifolds
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1 KO -theory of complex Stiefel manifolds Daisuke KISHIMOTO, Akira KONO and Akihiro OHSHITA 1 Introduction The purpose of this paper is to determine the KO -groups of complex Stiefel manifolds V n,q which is q-frames in C n. We compute it by using the Atiyah- Hirzebruch spectral sequence of KO (V n,q ) and obtain the following. Theorem. Let P (t) be the polynomial n (1+t 2i 1 ), Q(t) be the following and a k, b k be the sum of coefficients of t 4i+k in P (t), t 8i+k+1 in Q(t) respectively. (1 + t 2(n q)+1 ) Q(t) = n 2 (1 + t 2n 1 ) (1 + t 2(n q)+1 )(1 + t 2n 1 ) n 1 n 1 i=n q+2 n 2 i=n q+2 (1 + t 4i ) (n, q) = (2k, 2l) (1 + t 4i ) (n, q) = (2k + 1, 2l) (1 + t 4i ) (n, q) = (2k, 2l + 1) (1 + t 4i ) (n, q) = (2k + 1, 2l + 1) Then the KO i -groups of V n,q is rz sz 2 for (r, s) below. i (r, s) (a 0, b 7 + b 0 ) (a 1, b 1 + b 0 ) (a 2, b 1 + b 2 ) (a 3, b 3 + b 2 ) i (r, s) (a 0, b 3 + b 4 ) (a 1, b 5 + b 4 ) (a 2, b 5 + b 6 ) (a 3, b 7 + b 6 ) 1
2 2 The Atiyah-Hirzebruch spectral sequence First we recall that the coefficient ring of KO -theory is that KO = Z[α, x, β, β 1 ]/(2α, α 3, αx, x 2 4β), where α = 1, x = 4 and β = 8. Let X be a finite CW-complex. The Atiyah-Hirzebruch spectral sequence of KO (X) is the spectral sequence with E p,q 2 = H p (X; KO q ) converging to KO (X). It is well known that the differential d 2 of the Atiyah-Hirzebruch spectral sequence of KO (X) is given by the following. (See [F]) Sq 2 π 2 q 0 (8) d,q 2 = Sq 2 q 1 (8) 0 otherwise, where π 2 is the modulo 2 reduction. In this paper we compute the Atiyah-Hirzebruch spectral sequence of KO (X) with X in two special classes of CW-complexes. Let E be the class of CW-complexes with only even cells and O be the one with only odd cells and 0-cells. The Atiyah-Hirzebruch spectral sequence of KO (X) for X in E is considered in [H-K]. It is easily seen that [HK, Proposition 1] is valid for a CW-complex in O and we have the following. Proposition 1. Let X be a finite CW-complex in either E or O and E r (X) be the r-th term of the Atiyah-Hirzebruch spectral sequence of KO (X). Then we have the following. 1. E p, 1 3 (X) = H p (H (X; Z 2 ); Sq 2 ). 2. Let d r be the first non-trivial differential for r 3. (a) r 2 (8). (b) There exists x E,0 r (X) such that αx 0 and αd r x 0. 3 The Sq 2 -cohomology of V n,q It is well known that V n,q U(n)/U(n q) and H (V n,q ; Z) = (e 2(n q)+1, e 2(n q)+3,..., e 2n 1 ), where U(k) is the k-dimensional unitary group and e i = i. Since Sq 2 e 4i 1 = e 4i+1 (4i + 1 2n 1), we have the following. Proposition 2. H (H (V n,q ; Z 2 ); Sq 2 ) is the exterior algebra generated by the elements in the table below. 2
3 (n, q) = (2k, 2l) (n, q) = (2k + 1, 2l) (n, q) = (2k, 2l + 1) (n, q) = (2k + 1, 2l + 1) e 2(n q)+1, e 2(n q)+3 e 2(n q)+5,..., e 2n 5 e 2n 3, e 2n 1 e 2(n q)+1 e 2(n q)+3,..., e 2n 3 e 2n 1 e 2(n q)+1 e 2(n q)+3,..., e 2n 5 e 2n 3, e 2n 1 e 2(n q)+1, e 2(n q)+3 e 2(n q)+5,..., e 2n 3 e 2n 1 4 Collapse problem of E r (V n,q ) Let G be the complex Grassmannian of k-planes in C q which is the homogeneous space U(q)/U(k) U(q k). Let ad k : U(k) GL(k 2, R) and can k : U(k) GL(2k, R) be the adjoint and the canonical representation. By abuse of notation, ad k mcan k denotes the real vector bundle associated to the representation ad k mcan k and the U(k)-principal bundle U(k) V G. In [M] it is shown that there exists a stable homotopy equivalence as follows. V n,q s q k=1 G ad k (n q)can k, ( ) where G ad k (n q)can k is the Thom space of the real vector bundle ad k (n q)can k on G. Then E r (V n,q ) splits into E r (G ad k (n q)can k ). Note that G ad k (n q)can k is either in E or in O. Proposition 3. Let E G be a real vector bundle with w 2 (E) = 0 and either k be even or q be odd. Then E r (G E ) collapses at the third term. Proof. By Thom isomorphism, we have E 2 (G E ) = KO φ E H (G ; KO ) = KO φ E E 2 (G ), where φ E is the Thom class of E. Since d 2 φ E = Sq 2 π 2 φ E = w 2 (E)π 2 φ E = 0, we have E 3 (G E ) = KO φ E E 3 (G ). It is shown in [HK] that E r (G ) collapses at the third term for any k, q and H (H (G ; Z 2 ); Sq 2 ) has only elements of 8i degree if k is even or q is odd. Then we see d r φ E = 0 for r 3 by degree argument and Proposition 1,2,(a). Therefore we obtain that d r = 0 for r 3 by Proposition 1,2,(b). By the naturality of the Thom class, we have the following. Corollary 1. Let E G be a real vector bundle with w 2 (E) = 0, either k be even or q be odd and ı : G q 1,k G be the natural inclusion. Then E r (G ı E q 1,k ) collapses at the third term. 3
4 Lemma 1. E r (V n,q ) collapses at the third term. Proof. We show that the elements of E, 1 3 (V n,q ) = H (H (V n,q ; Z 2 ); Sq 2 ) in the table of Proposition 2 are permanent cycles. It is easily seen that w 2 (can k ) 0 and { 0 k is even w 2 (ad k ) = 0 k is odd. Since w 2 (ad k (n q)can k ) = w 2 (ad k )+(n q)w 2 (can k ) and H 2 (G ; Z 2 ) = Z 2, E r (G ad k (n q)can k ) collapses at r = 3 when n q is even and k is odd, or, n q is odd and k is even by Proposition 3 and Corollary 1. Note that G ad k (n q)can k is in E (resp. O) if k is even (resp. odd) and that Er 2l+1, (X) = 0 (Er 2l, (X) = 0) if X is in E (resp. O), then we see that x E, 1 3 (V n,q ) is a permanent cycle if n q is even and x is odd, or, n q is odd and x is even. Therefore e 2(n q)+1 is a permanent cycle. We also see that e 4i 1 e 4i+1 is permanent cycle for any n, q by considering the homomorphisms E 3 (V n,q ) E 3 (V n,q+1 ) and E 3 (V n+1,q+1 ) E 3 (V n,q ) induced by the natural projection V n,q+1 V n,q and the natural inclusion V n,q V n+1,q+1. Note that we have the homomorphism E 3 (S 2n 1 ) E 3 (V n,q ) induced from the projection V n,q V n,1 = S 2n 1, then we see that e 2n 1 is the permanent cycle. 5 Proof of Theorem It is easily seen that K n (X) is torsion free and concentrated in even (odd) dimension, if X is in E (resp.o). Consider the Bott sequence K n (X) KO n+2 (X) KO n+1 (X) c K n+1 (X), where c : KO i (X) K i (X) is the complexification map. Since rc = 2 we have the following, where r : K i (X) KO i (X) is the realization map. (See [H].) Proposition 4. If X is in E, we have If X is in O, we have KO 2i+1 (X) = sz 2 KO 2i (X) = rz sz2. KO 2i (X) = sz2 KO 2i 1 (X) = rz sz 2. Proof of Theorem. By Proposition 4 we have E p,q (X) = E 2n+8i, 1 = KO 2n 1 (X), for X in E p+q=2n 1 i 4
5 p+q=2n E p,q (X) = i E 2n+8i+1, 1 = KO 2n (X), for X in O. Note that the Thom space of a vector bundle on G as in the stable splitting ( ) is either in E or O and that Er 2i 1, (X) = 0 (Er 2i, (X) = 0 for i > 0) if X is in E (resp. O). Then we obtain that KO i (V n,q ) = rz sz 2 for (r,s) below, where s k = i rankh4i+k (V n,q ; Z), t k = i dim Z 2 E 8i+k+1, 1 (V n,q ). i (r, s) (s 0, t 7 + t 0 ) (s 1, t 1 + t 0 ) (s 2, t 1 + t 2 ) (s 3, t 3 + t 2 ) i (r, s) (s 0, t 3 + t 4 ) (s 1, t 5 + t 4 ) (s 2, t 5 + t 6 ) (s 3, t 7 + t 6 ) It is easily seen that the Poincaré series P t (H (V n,q ; Z)) = i rankh (V n,q ; Z)t i, (V n,q )) = i dim Z 2 E i, 1 (V n,q )t i = i dim Z 2 H i (H (V n,q ; Z 2 ); Sq 2 )t i P t (E, 1 are P (t), Q(t) respectively by Proposition 2. Then we have a i = s i, b i = t i and complete the proof. References [F] M. Fujii, KO -groups of projective spaces, Osaka J. Math., 4 (1967), [HK] S. Hara and A. Kono, KO -theory of complex Grassmannians, J. Math. Kyoto Univ., 31 (1991), [H] S.G. Hogger, On KO theory of Grassmannians, Quart. J. Math. Oxford (2), 20 (1969), [M] H. Miller, Stable splitting of Stiefel manifolds, Topology, 24 (1985), Department of Mathematics, Kyoto University Department of Mathematics, Kyoto University Osaka University of Economics 5
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