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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Regularity in the growth of the loop space homology of a finite CW complex


Authors: Yves Félix, Steve Halperin and Jean-Claude Thomas
Journal: Proc. Amer. Math. Soc. 142 (2014), 1025-1033
MSC (2010): Primary 55P35; Secondary 17B70
Published electronically: November 15, 2013
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Abstract: To any path connected topological space $ X$, such that $ \mbox {\rm rk}\, H_i(X) <\infty $ for all $ i\geq 0$, are associated the following two sequences of integers: $ b_i=$$ \mbox {\rm rk}\, H_i(\Omega X)$ and $ r_i=$$ \mbox {\rm rk}\, \pi _{i+1}(X)$. If $ X$ is simply connected, the Milnor-Moore theorem together with the Poincaré-Birkoff-Witt theorem provides an explicit relation between these two sequences. If we assume moreover that $ H_i(X;\mathbb{Q})=0 $, for all $ i\gg 0$, it is a classical result that the sequence of Betti numbers $ (b_i)$ grows polynomially or exponentially, depending on whether the sequence $ (r_i)$ is eventually zero or not. The purpose of this note is to prove, in both cases, that the $ r^{\mbox {\scriptsize\rm th}} $ Betti number $ b_r$ is controlled by the immediately preceding ones. The proof of this result is based on a careful analysis of the Sullivan model of the free loop space $ X^{S^1}$.


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Additional Information

Yves Félix
Affiliation: Institut Mathematique, Université Catholique de Louvain, 2, Chemin du Cyclotron, 1348 Louvain-La-Neuve, Belgium

Steve Halperin
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742

Jean-Claude Thomas
Affiliation: Department of Mathematics, University of Angers, 49100 Angers, France

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11796-X
PII: S 0002-9939(2013)11796-X
Received by editor(s): February 22, 2012
Received by editor(s) in revised form: April 2, 2012
Published electronically: November 15, 2013
Communicated by: Brooke Shipley
Article copyright: © Copyright 2013 American Mathematical Society