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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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References

REFERENCES

[1] Aguadé, J. Decomposable free loop spaces. Canad. J. Math. 39 no.4 (1987), 938–955.Google Scholar
[2] Félix, Y., Halperin, S. and Thomas, J.-C. Rational homotopy theory. Graduate Texts in Math. vol. 205 (Springer–Verlag, New York, 2001).Google Scholar
[3] Felix, Y., Thomas, J.-C. and Vigué-Poirrier, M. The Hochschild cohomology of a closed manifold. Publ. Math. Inst. Hautes Études Sci. 99 (2004), 235–252.Google Scholar
[4] Fox, R. H. Homotopy groups and torus homotopy groups. Ann. of Math. (2) 49 (1948), 471–510.Google Scholar
[5] Golasiński, M., Gonçalves, D. and Wong, P. Generalisations of Fox homotopy groups, Whitehead products and Gottlieb groups. Ukraï n. Mat. Zh. 57 no.3 (2005), 320–328.Google Scholar
[6] Golasiński, M. and Mukai, J. Gottlieb groups of spheres. Topology 47 no.6 (2008), 399–430.Google Scholar
[7] Gottlieb, D. H. Evaluation subgroups of homotopy groups. Amer. J. Math. 91 (1969), 729–756.Google Scholar
[8] Gruher, K. and Salvatore, P. Generalized string topology operations. Proc. Lond. Math. Soc. (3) 96 no.1 (2008), 78–106.Google Scholar
[9] Lang, G. E., Jr. Localisations and evaluation subgroups. Proc. Amer. Math. Soc. 50 (1975), 489–494.Google Scholar
[10] McDuff, D. The symplectomorphism group of a blow up. Geom. Dedicata 132 (2008), 1–29.Google Scholar
[11] Naito, T. On the mapping space homotopy groups and the free loop space homology groups. Algebr. Geom. Topol. 11 no.4 (2011), 2369–2390.Google Scholar
[12] Spanier, E. H. Algebraic Topology (Corrected reprint). (Springer–Verlag, New York, 1981).Google Scholar
[13] Steenrod, N. E. A convenient category of topological spaces. Michigan Math. J. 14 (1967), 133–152.Google Scholar
[14] Varadarajan, K. Generalised Gottlieb groups. J. Indian Math. Soc. (N.S.) 33 (1970), 141–164.Google Scholar
[15] Woo, M. H. and Kim, J.-R. Localisations and generalised Gottlieb subgroups. J. Korean Math. Soc. 23 no.2 (1986), 151–157.Google Scholar
[16] Woo, M. H. and Yoon, Y. S. T-spaces by the Gottlieb groups and duality. J. Austral. Math. Soc. Ser. A 59 no.2 (1995), 193–203.Google Scholar