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

 

Rational homotopy of the polyhedral product functor


Authors: Yves Félix and Daniel Tanré
Journal: Proc. Amer. Math. Soc. 137 (2009), 891-898
MSC (2000): Primary 13F55, 55P62, 55U10
Published electronically: September 24, 2008
MathSciNet review: 2457428
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Abstract: Let $ (X,\ast)$ be a pointed CW-complex, $ K$ be a simplicial complex on $ n$ vertices and $ X^K$ be the associated polyhedral power. In this paper, we construct a Sullivan model of $ X^K$ from $ K$ and from a model of $ X$.

Let $ \mathcal{F}(K,X)$ be the homotopy fiber of the inclusion $ X^K\to X^n$. Recent results of Grbić and Theriault, on one side, and of Denham and Suciu, on the other side, show the diversity of the possible homotopy types for $ \mathcal{F}(K,X)$. Here, we prove that the corresponding map between Sullivan models is Golod attached, generalizing a result of J. Backelin. This property is deduced from the existence of a succession of fibrations whose fibers are suspensions.

We consider also the Lusternik-Schnirelmann category of $ X^K$. In the case that $ \operatorname{cat}X^n=n\,\operatorname{cat}X$, we prove that $ \operatorname{cat}X^K =(\operatorname{cat}X)(1+\dim K)$.

Finally, we mention that this work is written in the case of a sequence of pairs, $ \underline{X}=(X_i,A_i)_{1\leq i\leq n}$, as in a recent work of Bahri, Bendersky, Cohen and Gitler.


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Additional Information

Yves Félix
Affiliation: Département de Mathématiques, Université Catholique de Louvain, 2, Chemin du Cyclotron, 1348 Louvain-La-Neuve, Belgium
Email: felix@math.ucl.ac.be

Daniel Tanré
Affiliation: Département de Mathematiques, UMR 8524, Université de Lille 1, 59655 Villeneuve d’Ascq Cedex, France
Email: Daniel.Tanre@univ-lille1.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09591-9
PII: S 0002-9939(08)09591-9
Received by editor(s): January 22, 2008
Received by editor(s) in revised form: March 21, 2008
Published electronically: September 24, 2008
Communicated by: Paul Goerss
Article copyright: © Copyright 2008 American Mathematical Society