Mathematical Proceedings of The Cambridge Philosophical Society
We analyze the Gottlieb groups of function spaces. Our results lead to explicit decompositions of the Gottlieb groups of many function spaces map(X,Y)---including the (iterated) free loop space of Y---directly in terms of the Gottlieb groups of Y. More generally, we give explicit decompositions of the generalized Gottlieb groups of map(X,Y) directly in terms of generalized Gottlieb groups of Y. Particular cases of our results relate to the torus homotopy groups of Fox. We draw some consequences for the classification of T-spaces and G-spaces. For X, Y finite and Y simply connected, we give a formula for the ranks of the Gottlieb groups of map(X,Y) in terms of the Betti numbers of X and the ranks of the Gottlieb groups of Y. Under these hypotheses, the Gottlieb groups of map(X,Y) are finite groups in all but finitely many degrees.
Lupton, Gregory and Smith, Samuel Bruce, "Gottlieb Groups of Function Spaces" (2015). Mathematics Faculty Publications. 195.
The final publication is available at Springer via http://dx.doi.org/10.1017/S0305004115000201
This work was partially supported by a grant from the Simons Foundation (#209575 to Gregory Lupton).