Rational homotopy of the polyhedral product functor

Abstract

Let (X, *) 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 XK from K and from a model of X. Let F(K, X) be the homotopy fiber of the inclusion X-K -> X-n. Recent results of Grbic and Theriault, on one side, and of Denham and Suciu, on the other side, show the diversity of the possible homotopy types for 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 vibrations whose fibers are suspensions. We consider also the Lusternik-Schnirelmann category of XK. In the case that cat X-n = n cat X, we prove that cat X-K = (cat X)(1 + dimK). Finally, we mention that this work is written in the case of a sequence of pairs, (X) under bar = (X-i, A(i)) 1 <= i <= n, as in a recent work of Bahri, Bendersky, Cohen and Gitler.