The manifold of isospectral symmetric tridiagonal matrices and realization of cycles by aspherical manifolds

Gaifullin, Alexander A.

doi:10.1134/s0081543808040044

Cited by 11 publications

(21 citation statements)

References 12 publications

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“…In this section we describe an auxiliary combinatorial construction that will be used in the next section in the proof of Theorem 2.5. This construction generalizes the construction suggested by the author in [19] and developed in [20], [21], [22], and [23].…”

Section: A Combinatorial Constructionsupporting

confidence: 69%

“…This construction does not allow to obtain effective estimates for the multiplicity k. But, unlike the classical algebraic topological approach, it allows us to control the topology of the obtained manifold N n . Namely, it was shown in [20], [21] that in every dimension n there exists a single manifold M n 0 -the so-called Tomei manifold -such that for any homology class z ∈ H n (X; Z) of any topological space X, the manifold N n that realizes a multiple kz of z can be chosen to be a (non-ramified) finitefold covering of M n 0 . The Tomei manifold M n 0 first appeared in the paper [39] by Tomei as the isospectral manifold of symmetric tridiagonal real matrices.…”

Section: Introductionmentioning

confidence: 99%

“…We shall present an explicit construction that, for a singular cycle representing the given homology class, builds a manifold being a finite-fold covering of M P Γ ,λ that realizes a multiple of the given homology class. This construction is a further modification of the construction suggested by the author in [19] and developed in [20], [21], and [23]. Notice that another modification of this construction was applied to solve the problem of constructing of an oriented simplicial manifold with the prescribed set of links of vertices, see [22].…”

Section: Introductionmentioning

confidence: 99%

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Small covers of graph-associahedra and realization of cycles

Gaifullin¹

2016

Sb. Math.

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Abstract. An oriented connected closed manifold M n is called a URC-manifold if for any oriented connected closed manifold N n of the same dimension there exists a non-zero degree mapping of a finite-fold covering M n of M n onto N n . This condition is equivalent to the following: For any n-dimensional integral homology class of any topological space X, a multiple of it can be realized as the image of the fundamental class of a finite-fold covering M n of M n under a continuous mapping f : M n → X. In 2007 the author gave a constructive proof of the classical result by Thom that a multiple of any integral homology class can be realized as an image of the fundamental class of an oriented smooth manifold. This construction yields the existence of URC-manifolds of all dimensions. For an important class of manifolds, the so-called small covers of graph-associahedra corresponding to connected graphs, we prove that either they or their two-fold orientation coverings are URC-manifolds. In particular, we obtain that the two-fold covering of the small cover of the usual Stasheff associahedron is a URC-manifold. In dimensions 4 and higher, this manifold is simpler than all previously known URC-manifolds.

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Section: A Combinatorial Constructionsupporting

confidence: 69%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Small covers of graph-associahedra and realization of cycles

Gaifullin¹

2016

Sb. Math.

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“…Colored pseudomanifolds were introduced by Pezzana andFerri in 1975-1976, see [16], [3], [4], [5]. ♦ 1.5.…”

Section: Colored Pseudomanifoldsmentioning

confidence: 99%

“…Such graphs correspond to colored pseudomanifolds. In [5]- [7] there was considered an action of free product Z 2 * · · · * Z 2 of n copies of Z 2 on the set of chambers of a colored pseudomanifold. A construction relative to the present construction was considered in [1].…”

Section: A Correspondence Between Pseudomanifolds and Symmetric Groupsmentioning

confidence: 99%

Infinite symmetric group, pseudomanifolds, and combinatorial cobordism-like structures

Gaifullin

Neretin

2018

J. Topol. Anal.

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We consider a category whose morphisms are bordisms of n-dimensional pseudomanifolds equipped with a certain additional structure (coloring). On the other hand, we consider the product G of (n + 1) copies of infinite symmetric group. We show that unitary representations of G produce functors from the category of (n−1)-dimensional bordisms to the category of Hilbert spaces and bounded linear operators.

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Right-Angled Spaces and Groups

Davis

2024

Infinite Group Actions on Polyhedra

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The manifold of isospectral symmetric tridiagonal matrices and realization of cycles by aspherical manifolds

Cited by 11 publications

References 12 publications

Small covers of graph-associahedra and realization of cycles

Small covers of graph-associahedra and realization of cycles

Infinite symmetric group, pseudomanifolds, and combinatorial cobordism-like structures

Right-Angled Spaces and Groups

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