Embeddings of moment-angle manifolds and sequences of Massey products

Abstract

We show that for any face $F$ of a simple polytope $P$ the canonical equivariant homeomorphisms $h_P:\,\mathcal Z_P\to\mathcal Z_{K_P}$ and $h_F:\,\mathcal Z_F\to\mathcal Z_{K_F}$ are linked in a pentagonal commutative diagram with the maps of moment-angle manifolds and moment-angle-complexes, induced by a face embedding $i_{F,P}:\,F\to P$ and a simplicial embedding $\Phi_{F,P}:\,K_F\to K_{F,P}\to K_P$, where $K_{F,P}$ is the full subcomplex of $K_P$ on the same vertex set as $\Phi_{F,P}(K_F)$. We introduce the explicit constructions of the maps $i_{F,P}$, $\Phi_{F,P}$ and show that a polytope $P$ is flag if and only if the induced embedding $\hat{i}_{F,P}:\,\mathcal Z_{F}\to\mathcal Z_P$ of moment-angle manifolds has a retraction and thus induces a split ring epimorphism in cohomology for any face $F\subset P$. As the applications of these results we obtain the sequences $\{P^n\}$ of flag simple polytopes such that there exists a nontrivial $k$-fold Massey product in $H^*(\mathcal Z_{P^n})$ with $k\to\infty$ as $n\to\infty$ and, moreover, the existence of a nontrivial $k$-fold Massey product in $H^*(\mathcal Z_{P^n})$ implies existence of a nontrivial $k$-fold Massey product in $H^*(\mathcal Z_{P^l})$ for any $l>n$.