Gottlieb Groups of Function Spaces
Gregory Lupton, Samuel Bruce Smith

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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
