Cyclopermutohedron: geometry and topology

Nekrasov, Ilia; Panina, Gaiane; Zhukova, Alena

doi:10.1007/s40879-016-0107-3

Cited by 4 publications

(7 citation statements)

References 8 publications

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“…With her coauthors Panina also computed the homology groups of CP n+1 in [4]. This was done by finding an optimal discrete Morse function and then computing the number of critical cells.…”

Section: The Cyclopermutohedronmentioning

confidence: 99%

“…This was done by finding an optimal discrete Morse function and then computing the number of critical cells. In the remaining of this section we recall some relevant results proved in [4] and provide different proof of vanishing of boundary maps in the Morse complex. In particular, we find a closed form formula for the degree of the attaching homeomorphisms.…”

Section: The Cyclopermutohedronmentioning

confidence: 99%

“…In this section we recall the homology computations of CP n+1 done in [4]; most results in this section are not new. However, we give a slightly different proof of vanishing of boundary homomorphisms in the Morse complex of CP n+1 .…”

Section: A Discrete Morse Function For the Cyclopermutahedronmentioning

confidence: 99%

“…Interestingly, the complex CP n+1 is neither a topological ball (possibly) nor a wedge of spheres; its homotopy type is much more complicated. In [4], using discrete Morse theory, the authors showed that H i (CP n+1 ) are torsion free for i ≥ 0 and computed their Betti numbers. We recall their result: Theorem 1.1 ([4, Theorem 2]).…”

Section: Introductionmentioning

confidence: 99%

“…The new result here is a closed form formula for the degree of the attaching homeomorphisms in terms of underlying combinatorics. In Section 4 we recall the discrete Morse function on this complex first described in [4]. We also give a different proof of the fact that this Morse function is optimal, i.e., in the Morse complex all boundary homomorphisms vanish.…”

Section: Introductionmentioning

confidence: 99%

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On the topology of bi-cyclopermutohedra

Deshpande¹,

Manikandan²,

Singh³

2019

Preprint

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Motivated by the work of Panina and her coauthors in on cyclopermutohedron we study a poset whose elements correspond to equivalence classes of partitions of the set {1, • • • , n + 1} up to cyclic permutations and orientation reversion. This poset is the face poset of a regular CW complex which we call bi-cyclopermutohedron and denote it by QP n+1 . The complex QP n+1 contains subcomplexes homeomorphic to moduli space of certain planar polygons with n + 1 sides up to isometries. In this article we find an optimal discrete Morse function on QP n+1 and use it to compute its homology with Z as well as Z 2 coefficients.2010 Mathematics Subject Classification. 51M20.

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“…With her coauthors Panina also computed the homology groups of CP n+1 in [4]. This was done by finding an optimal discrete Morse function and then computing the number of critical cells.…”

Section: The Cyclopermutohedronmentioning

confidence: 99%

Section: The Cyclopermutohedronmentioning

confidence: 99%

Section: A Discrete Morse Function For the Cyclopermutahedronmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the topology of bi-cyclopermutohedra

Deshpande¹,

Manikandan²,

Singh³

2019

Preprint

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On the topology of bi-cyclopermutohedra

Deshpande

Manikandan

Singh

2022

Indian J Pure Appl Math

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Cluster-Permutohedra and Submanifolds of Flag Varieties with Torus Actions

Ayzenberg

Buchstaber

2023

International Mathematics Research Notices

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In this paper, we describe a relation between the notion of graphicahedron, introduced by Araujo-Pardo, Del Río-Francos, López-Dudet, Oliveros, and Schulte in 2010 and toric topology of manifolds of sparse isospectral Hermitian matrices. More precisely, we recall the notion of a cluster-permutohedron, a certain finite poset defined for a simple graph $\Gamma $. This poset is build as a combination of cosets of the symmetric group, and the geometric lattice of the graphical matroid of $\Gamma $. This poset is similar to the graphicahedron of $\Gamma $, in particular, 1-skeleta of both posets are isomorphic to Cayley graphs of the symmetric group. We describe the relation between cluster-permutohedron and graphicahedron using Galois connection and the notion of a core of a finite topology. We further prove that the face poset of the natural torus action on the manifold of isospectral $\Gamma $-shaped Hermitian matrices is isomorphic to the cluster-permutohedron. Using recent results in toric topology, we show that homotopy properties of graphicahedra may serve an obstruction to equivariant formality of isospectral matrix manifolds. We introduce a generalization of a cluster-permutohedron and describe the combinatorial structure of a large family of manifolds with torus actions, including Grassmann manifolds and partial flag manifolds.

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Cyclopermutohedron: geometry and topology

Cited by 4 publications

References 8 publications

On the topology of bi-cyclopermutohedra

On the topology of bi-cyclopermutohedra

On the topology of bi-cyclopermutohedra

Cluster-Permutohedra and Submanifolds of Flag Varieties with Torus Actions

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