Rational homotopy of the polyhedral product functor

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.