Regularity in the growth of the loop space homology of a finite CW complex
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- by Yves Félix, Steve Halperin and Jean-Claude Thomas PDF
- Proc. Amer. Math. Soc. 142 (2014), 1025-1033 Request permission
Abstract:
To any path connected topological space $X$, such that $\operatorname {rk} H_i(X) <\infty$ for all $i\geq 0$, are associated the following two sequences of integers: $b_i= \operatorname {rk} H_i(\Omega X)$ and $r_i= \operatorname {rk} \pi _{i+1}(X)$. 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 $r^{\mathrm {th}}$ 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 $X^{S^1}$.References
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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
- 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
- © Copyright 2013 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 142 (2014), 1025-1033
- MSC (2010): Primary 55P35; Secondary 17B70
- DOI: https://doi.org/10.1090/S0002-9939-2013-11796-X
- MathSciNet review: 3148536