The Gottlieb group of a wedge of suspensions. (English) Zbl 1314.55009

Arkowitz, Martin; Maruyama, Ken-Ichi

J. Math. Soc. Japan 66, No. 3, 735-743 (2014).

Let \(X\) be a connected space and \(\pi_n(X)\) its \(n\)-th homotopy group. The Gottlieb groups \(G_n(X)\subseteq \pi_n(X)\) were introduced and studied by D. H. Gottlieb [Am. J. Math. 91, 729–756 (1969; Zbl 0185.27102)] and have been shown to have many topological applications.
The authors study \(G_n(\Sigma X_1\vee\cdots\vee \Sigma X_k)\), where \(\Sigma X_i\) is the suspension of the space \(X_i\) for \(i=1,\ldots, k\) with particular attention to the case \(k=2\) and \(X_i\) a sphere. Necessary and sufficient conditions for an element of \(\pi_n(\Sigma X_1\vee\cdots\vee \Sigma X_k)\) to be in \(G_n(\Sigma X_1\vee\cdots\vee \Sigma X_k)\) are presented and \(G_n(M(A,n))\) for some Moore spaces \(M(A,n)\) is computed.

Reviewer: Marek Golasiński (Olsztyn)

MSC:

55Q20

Homotopy groups of wedges, joins, and simple spaces

Keywords:

homotopy group; Gottlieb group; wedge sum; Hopf invariant; Whitehead product

Citations:

Zbl 0185.27102

References:

[1]	M. Arkowitz, The generalized Whitehead product, Pacific J. Math., 12 (1962), 7-23. · Zbl 0118.18404
[2]	W. D. Barcus and M. G. Barratt, On the homotopy classification of the extensions of a fixed map, Trans. Amer. Math. Soc., 88 (1958), 57-74. · Zbl 0095.16801
[3]	Y. Félix, S. Halperin and J.-C. Thomas, Rational Homotopy Theory, Grad. Texts in Math., 205 , Springer-Verlag, 2001.
[4]	M. Golasiński and J. Mukai, Gottlieb groups of spheres, Topology, 47 (2008), 399-430. · Zbl 1172.55004
[5]	D. H. Gottlieb, Evaluation subgroups of homotopy groups, Amer. J. Math., 91 (1969), 729-756. · Zbl 0185.27102
[6]	P. J. Hilton, On the homotopy groups of the union of spheres, J. London Math. Soc., 30 (1955), 154-171. · Zbl 0064.17301
[7]	H. Toda, Composition Methods in Homotopy Groups of Spheres, Ann. of Math. Stud., 49 , Princeton University Press, Princeton, N.J., 1962. · Zbl 0101.40703
[8]	G. W. Whitehead, Elements of Homotopy Theory, Grad. Texts in Math., 61 , Springer-Verlag, 1978. · Zbl 0406.55001