Topology of moment-angle manifolds arising from flag nestohedra

Limonchenko, Ivan

doi:10.1007/s11401-017-1037-1

Cited by 12 publications

(7 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, there are other nontrivial triple Massey products in H * (Z P ), as illustrated in Example 4.13. Via an explicit example, it was also shown in [19,Proposition 4.1] and [20,Lemma 4.9] that there are triple Massey products of three-dimensional classes in H * (Z P ) for n dimensional permutahedra P with n > 3. Here we will generalise this and show that Z P , for the n-dimensional permutahedron P , has a non-trivial k-Massey product for k n. .…”

Section: Permutahedramentioning

confidence: 97%

“…Alternatively nestohedra are interpreted as hypergraph polytopes [13]. The first examples of Massey products in momentangle manifolds associated to nestohedra were in [19,Proposition 4.1] and [20,Lemma 4.9] and were triple Massey products constructed either by explicit calculation or using the classification of lowest degree Massey products [12,15]. We will use Theorems 3.17 and 4.12 to construct families of new non-trivial higher Massey products in moment-angle manifolds associated to certain nestohedra.…”

Section: Proposition 53 ([24]mentioning

confidence: 99%

“…For example, from each of the obstruction graphs in the classification of lowest-degree triple Massey products in moment-angle complexes [12,15], we obtain infinite families of non-trivial triple Massey products of higher dimensional classes. We give an alternative proof of known examples of non-trivial triple Massey products in moment-angle manifolds, such as those associated with Pogorelov polytopes [26] and permutahedra or stellohedra [19,20] using "stretched" obstruction graphs. Also, the two constructions, Constructions 3.5 and 4.6, can be combined to create new higher Massey products.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Postgraduate students’ beliefs about and confidence for academic writing in the field of applied linguistics

Zotzmann

Sheldrake

2021

Journal of Second Language Writing

View full text Add to dashboard Cite

As part of various obstruction theories, non-trivial Massey products have been studied in symplectic and complex geometry, commutative algebra and topology for a long time. We introduce a general approach to constructing nontrivial Massey products in the cohomology of moment-angle complexes, using homotopy theoretical and combinatorial methods. Our approach sets a unifying way of constructing higher Massey products of arbitrary cohomological classes and generalises all existing examples of non-trivial Massey products in momentangle complexes. As a result, we obtain explicit constructions of infinitely many non-formal manifolds that appear in topology, complex geometry and algebraic geometry.

show abstract

Section: Permutahedramentioning

confidence: 97%

Section: Proposition 53 ([24]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Postgraduate students’ beliefs about and confidence for academic writing in the field of applied linguistics

Zotzmann

Sheldrake

2021

Journal of Second Language Writing

View full text Add to dashboard Cite

show abstract

“…Remark. The polytope P and its generalizations were considered by Limonchenko in [26]. Limonchenko constructed a rich set of complex moment-angle manifolds with nontrivial Massey product.…”

Section: Proof Of Theorem 17mentioning

confidence: 99%

Zoo of monotone Lagrangians in $\mathbb{C}P^n$

Oganesyan¹

2021

Preprint

View full text Add to dashboard Cite

Let P ⊂ R m be a polytope of dimension m with n facets and a 1 , . . . , a n be the normal vectors to the facets of P . Assume that P is Delzant, Fano, and a 1 + . . . + a n = 0. We associate a monotone embedded Lagrangian L ⊂ CP n−1 to P . As an abstract manifold, the Lagrangian L fibers over (S 1 ) n−m−1 with fiber R P , where R P is defined by a system of quadrics in RP n−1 . The manifold R P is called the real toric space.We find an effective method for computing the Lagrangian quantum cohomology groups of the mentioned Lagrangians. Then we construct explicitly some set of wide and narrow Lagrangians.Our method yields many different monotone Lagrangians with rich topological properties, including non-trivial Massey products, complicated fundamental group and complicated singular cohomology ring.Interestingly, not only the methods of toric topology can be used to construct monotone Lagrangians, but the converse is also true: the symplectic topology of Lagrangians can be used to study the topology of R P . General formulas for the rings H * (R P , Z), H * (R P , Z 2 ) are not known. Since we have a method for constructing narrow Lagrangians, the spectral sequence of Oh can be used to study the singular cohomology ring of R P . This idea will be developed in a further paper.

show abstract

“…Recently, it was shown in [51] that there exists a nontrivial triple Massey product in H * (Z P ) for any Pogorelov polytope P (the latter class of 3dimensional flag simple polytopes contains, in particular, all fullerenes). First examples of polyhedral products with nontrivial n-fold Massey products in cohomology for any n ≥ 2 were constructed in [38,39].…”

Section: Introductionmentioning

confidence: 99%

Topology of polyhedral products over simplicial multiwedges

Limonchenko¹

2017

Preprint

Self Cite

View full text Add to dashboard Cite

We prove that certain conditions on multigraded Betti numbers of a simplicial complex K imply existence of a higher Massey product in cohomology of a moment-angle-complex Z K , which contains a unique element (a strictly defined product). Using the simplicial multiwedge construction, we find a family F of polyhedral products being smooth closed manifolds such that for any l, r ≥ 2 there exists an l-connected manifold M ∈ F with a nontrivial strictly defined r-fold Massey product in H * (M ). As an application to homological algebra, we determine a wide class of triangulated spheres K such that a nontrivial higher Massey product of any order may exist in Koszul homology of their Stanley-Reisner rings. As an application to rational homotopy theory, we establish a combinatorial criterion for a simple graph Γ to provide a (rationally) formal generalized moment-angle manifold Z J P = (D ¯2ji , S ¯2ji−1 ) ∂P * , J = (j 1 , . . . , j m ) over a graphassociahedron P = P Γ and compute all the diffeomorphism types of formal moment-angle manifolds over graph-associahedra.

show abstract

Topology of moment-angle manifolds arising from flag nestohedra

Cited by 12 publications

References 20 publications

Postgraduate students’ beliefs about and confidence for academic writing in the field of applied linguistics

Postgraduate students’ beliefs about and confidence for academic writing in the field of applied linguistics

Zoo of monotone Lagrangians in $\mathbb{C}P^n$

Topology of polyhedral products over simplicial multiwedges

Contact Info

Product

Resources

About