Sur Les nombres de James des variétés de Stiefel complexes

Sigrist, François

doi:10.1215/ijm/1256053753

Cited by 5 publications

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“…. v p (O*{n,k}) = v p (O{n,k}) = 0 for a primes > dk/2, by(32) and(27).(5-3). 0»{n, 1} = 0{n, 1} = 1 and W s {n, 2} = W{n, 2} = 2/{n, 2).…”

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confidence: 81%

Some James numbers of Stiefel manifolds

Ōshima

1982

Math. Proc. Camb. Phil. Soc.

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The purpose of this note is to determine some unstable James numbers of Stiefel manifolds. We denote the real numbers by R, the complex numbers by C, and the quaternions by H. Let F be one of these fields with the usual norm, and d = dimRF. Let On, k = On, k(F) be the Stiefel manifold of all orthonormal k–frames in Fn, and q: On, k → Sdn−1 the bundle projection which associates with each frame its last vector. Then the James number O{n, k} = OF{n, k} is defined as the index of q* πdn−1(On, k) in πdn−1(Sdn−1). We already know when O{n, k} is 1 (cf. (1), (2), (3), (13), (33)), and also the value of OK{n, k} (cf. (1), (13), (15), (34)). In this note we shall consider the complex and quaternionic cases. For earlier work see (11), (17), (23), (27), (29), (31) and (32). In (27) we defined the stable James number , which was a divisor of O{n, k}. Following James we shall use the notations X{n, k}, Xs{n, k}, W{n, k} and Ws{n, k} instead of OH{n, k}, , Oc{n, k} and respectively. In (27) we noticed that O{n, k} = Os{n, k} if n ≥ 2k– 1, and determined Xs{n, k} for 1 ≤ k ≤ 4, and also Ws{n, k} for 1 ≤ k ≤ 8. On the other hand Sigrist (31) calculated W{n, k} for 1 ≤ k ≤ 4. He informed the author that W{6,4} was not 4 but 8. Since Ws{6,4} = 4 (cf. § 5 below) this yields that the unstable James number does not equal the stable one in general.

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“…. v p (O*{n,k}) = v p (O{n,k}) = 0 for a primes > dk/2, by(32) and(27).(5-3). 0»{n, 1} = 0{n, 1} = 1 and W s {n, 2} = W{n, 2} = 2/{n, 2).…”

mentioning

confidence: 81%