Gottlieb groups of spheres and projective spaces
by
Marek Golasiński
Faculty of Maths and Computer Science, 87-100 Torun, Chopina 12/18, Poland
Coauthors: Juno Mukai

The Gottlieb groups G_k(X) of a pointed space X have been defined by Gottlieb in [G] and [G1]; first G₁(X) and then G_k(X) for all k ≥ 1. This is the subgroup of the k-th homotopy group p_k(X) containing all elements which can be represented by a map f : S^k→ X such that id_X∨f: X∨S^k→ X extends (up to homotopy) to a map F : X×S^k→ X, where S^k is the k-sphere.

The higher Gottlieb groups G_k(X) are related in [G1] to the existence of sectioning fibrations with fiber X. For instance, if G_k(X) is trivial then there is a cross-section for every fibration over the (k+1)-sphere S^k+1, with fiber X.

Basing on [GM], we take up the systematic study of the Gottlieb groups G_n+k(Sⁿ) of spheres for k ≤ 13 by means of the classical homotopy theory methods. We fully determine the groups G_n+k(Sⁿ) for k ≤ 13 except for the 2-primary components in the cases: k=9, n=53; k=11, n=115.

Next, by use the classical results of homotopy groups of spheres and Lie groups, we determine some Gottlieb groups of projective spaces or give the lower bounds of their orders.

References:

[GM] M. Golasi\'nski and J. Mukai, Gottlieb groups of spheres, Topology (to appear).

[G] D. Gottlieb, A certain subgroup of the fundamental group, Amer. J. of Math. 87 (1965), 840-856.

[G1] ---- , Evaluation subgroups of homotopy groups, Amer. J. of Math. 91 (1969), 729-756.

Date received: April 15, 2008