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Gottlieb groups of function spaces

Published online by Cambridge University Press:  10 April 2015

GREGORY LUPTON  and
SAMUEL BRUCE SMITH
GREGORY LUPTON
Affiliation:
Department of Mathematics, Cleveland State University, Cleveland, OH 44115, U.S.A. e-mail: G.Lupton@csuohio.edu
SAMUEL BRUCE SMITH
Affiliation:
Department of Mathematics, Saint Joseph's University, Philadelphia, PA 19131, U.S.A. e-mail: smith@sju.edu
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Abstract

We analyse 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 generalised Gottlieb groups of map(X, Y) directly in terms of generalised 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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 159 , Issue 1 , July 2015 , pp. 61 - 77
Copyright
Copyright © Cambridge Philosophical Society 2015 

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