Gottlieb Groups of Function Spaces

TL;DR

This paper provides explicit decompositions of Gottlieb groups for various function spaces, including free loop spaces, in terms of the Gottlieb groups of the target space, with applications to classification and homotopy groups.

Contribution

It introduces explicit formulas and decompositions for Gottlieb and generalized Gottlieb groups of function spaces, extending previous understanding and connecting to homotopy and classification problems.

Findings

Decompositions of Gottlieb groups of map(X,Y) in terms of Y's Gottlieb groups.

Formulas for ranks of Gottlieb groups based on Betti numbers and ranks.

Finite group structure of Gottlieb groups in most degrees for certain spaces.

Abstract

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.