Gottlieb groups of spheres

Abstract

This paper takes up the systematic study of the Gottlieb groups G n+k (S n ) of spheres for k ≤ 13 by means of the classical homotopy theory methods. We fully determine the groups G n+k (S n ) for k ≤ 13 except for the 2-primary components in the cases: k = 9, n = 53; k = 11, n = 115. In particular, we show [ι n , η 2The Gottlieb groups G k (X ) of a pointed space X have been defined by Gottlieb in [9] and [10]; first G 1 (X ) and then G k (X ) for all k ≥ 1. The higher Gottlieb groups G k (X ) are related in [10] to the existence of sectioning fibrations with fiber X . For instance, if G k (X ) is trivial then there is a crosssection for every fibration over the (k + 1)-sphere S k+1 , with fiber X . This paper grew out of our attempt to develop techniques in calculating G n+k (S n ) for k ≤ 13 and any n ≥ 1. The composition methods developed by Toda [36] are the main tools used in the paper. Our calculations also deeply depend on the results of [13,16,21].

Determination of the order of the P–image by Toda brackets

“…We also recall ∆(ν 2 8n+k ) = 0 if n ≥ 0 and k = 4, 5. The following is one of the main results in [5]: 1, ≡ 2 13 − 26 (2 13 ) 2, ≡ 2 13 − 26 (2 13 ) ≥ 2 14 − 26…”

“…This paper is a sequel to [5] by Golasiński and the author in the stable case. The methods are to use those of [5].…”

“…This paper is a sequel to [5] by Golasiński and the author in the stable case. The methods are to use those of [5]. In particular, the EHP sequence, the method and result of Toda [18,Chapter 11] and the result of Nomura [15] are essentially used.…”

“…1 Results from [5] In this section, we shall collect the result of [5] that we need. We denote by SO(n) the n-th rotation group and by ∆ : π k (S n ) → π k−1 (SO(n)) the connecting homomorphism.…”

Evaluation Subgroups of Homogeneous Spaces of Compact Lie Groups

“…In [9], Mimura, Woo and the second author provided some lower bounds of Gottlieb groups of quaternionic projective spaces and Stiefel manifolds, and used them to compute some Gottlieb groups of their homogeneous spaces. Golasinski and Mukai [4] fully determine the groups G n+k (S n ) for k = 13, except for the 2-primary components in the cases: k = 9, n = 53; k = 11, n = 115 by means of the classical homotopy theory.…”