2018/08/27 by Victor Matveevich Buchstaber, Buchstaber, Victor, Ivan Limonchenko +1
Mathematics · #13F55 #55S30 (Primary) 52B11 (Secondary) #Advanced Combinatorial Mathematics #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1808.08851
openalex publication_date 2018/08/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that for any face F of a simple polytope P the canonical equivariant homeomorphisms hP: \mathcal ZP→\mathcal ZKP and hF: \mathcal ZF→\mathcal ZKF are linked in a pentagonal commutative diagram with the maps of moment-angle manifolds and moment-angle-complexes, induced by a face embedding iF,P: F→ P and a simplicial embedding ΦF,P: KF→ KF,P→ KP, where KF,P is the full subcomplex of KP on the same vertex set as ΦF,P(KF). We introduce the explicit constructions of the maps iF,P, ΦF,P and show that a polytope P is flag if and only if the induced embedding iF,P: \mathcal ZF→\mathcal ZP of moment-angle manifolds has a retraction and thus induces a split ring epimorphism in cohomology for any face F⊂ P. As the applications of these results we obtain the sequences \Pn\ of flag simple polytopes such that there exists a nontrivial k-fold Massey product in H^*(\mathcal ZPn) with k→∞ as n→∞ and, moreover, the existence of a nontrivial k-fold Massey product in H^*(\mathcal ZPn) implies existence of a nontrivial k-fold Massey product in H^*(\mathcal ZPl) for any l>n.